Heteronuclear and homonuclear radio-frequency-driven recoupling

Abstract The radio-frequency-driven recoupling (RFDR) pulse sequence is used in magic-angle spinning (MAS) NMR to recouple homonuclear dipolar interactions. Here we show simultaneous recoupling of both the heteronuclear and homonuclear dipolar interactions by applying RFDR pulses on two channels. We demonstrate the method, called HETeronuclear RFDR (HET-RFDR), on microcrystalline SH3 samples at 10 and 55.555 kHz MAS. Numerical simulations of both HET-RFDR and standard RFDR sequences allow for better understanding of the influence of offsets and paths of magnetization transfers for both HET-RFDR and RFDR experiments, as well as the crucial role of XY phase cycling.

The Supplement consists of four sections.In the first section, "1D HET-RFDR Experiments" we show additional 1D 1 H-13 C HET-RFDR spectra.The second "HET-RFDR Simulations" section provides additional HET-RFDR simulations, which were performed under conditions that closely match the experiments.The third section, "Operator Paths" shows the possible paths of RFDR and HET-RFDR transfers via heteronuclear and homonuclear operators during the first two rotor periods.The last part, "RFDR Phase Cyling", shows the formal proof of zero signal transfer for a homonuclear I2 spin system with zero offset difference and when all π-pulses have the same phase.

1D HET-RFDR Experiments
Figure S1 shows a 1D HET-RFDR pulse sequence.The sequence consists of two π/2-pulses on the 1 H channel (with two step phase cycling to eliminate the signal from directly excited spins of carbons) followed by a series of HET-RFDR pulses and finally a π/2-pulse and detection on the 13 C channel.The evolution of the magnetization from proton to carbon spins through the HET-RFDR pulse sequence (Figure S1) can be described with cartesian operators as follows: where,   (  ), is an amplitude of the transferred signal.π-pulses on the both channels follow the XY8 scheme (Gullion et al., 1990).During acquisition, SWf-TPPM (Thakur et al., 2006) at 55.555 kHz or SPINAL64 (Fung et al., 2000) at 10 kHz decoupling is applied on the proton channel to narrow the detected resonances.
Solid state NMR spectroscopy: The CP and HET-RFDR spectra of 13 C, 15 N SH3 were acquired at 14.1 T (600 MHz) using a Bruker AVIIIHD spectrometer using a MASDVT600W2 BL1.3 HXY probe.The experiments were performed at 55.555 kHz MAS with the temperature of the cooling gas set to 235 K.
For 1D 1 H 13 C spectra during the HET-RFDR periods, the widths of pulses on proton and carbon channels were 3.4 us and 5 us, respectively.13.89 kHz SWf-TPPM (Thakur et al., 2006) with 36 us pulses was used during the acquisition.Table S1 summarizes the applied experimental parameters.
Figure S4a shows the HET-RFDR polarization trasfers between a directly bonded spin pair (I1-C2, solid lines) and the remote pair (I1-C3, lines with circles).The lines with diamonds represent signals that are not transferred, but remain on the spin I1.We consider three cases: rigid C2H2 chain (black lines), dynamic C2H2 chain (red lines) and rigid N2C2 chain (blue lines).For the rigid (black solid line) and dynamic (red solid line) C2H2 chains when the heteronuclear dipolar coupling constants are larger than the homonuclear dipolar constants, the polarization transfer from H1 to C2 oscillates about ~35% efficiency.However, for the spin system with the weak heteronuclear dipolar coupling constants (blue lines), the HET fp-RFDR polarization transfer between directly bonded spins is lower (blue solid line) and achieves only ~20% transfer efficiency.
The signal that remains on the starting spin (lines with diamonds) are ~40% for first two cases (black and solid lines with diamonds) and ~70% for weak dipolar coupling constants (blue line with diamonds).
The HET-RFDR transfer between remote spins, e.g.H1 and C3 are about ~10% of the initial polarization for all these three cases (black, red and blue lines with circles).The transfer of magnetization mostly occurs via relayed transfer (I1-C2-C3) and not directly from I1 to C3, which more clearly can be seen in the simulations on Figure S4b  The lines with circles represent the transfer between N1 and H3 spins for different distances (dipolar coupling constants) between H2 and H3 spins: black lines -2 Å (15 kHz), red lines -3 Å (4.4 kHz), blue lines -4 Å (1.9 kHz) and green lines -6 Å (0.5 kHz).
The distance as well as the dipolar coupling constant between N1 and H3 were kept unchanged at 3 Å and 0.45 kHz.XY8 phase cycling was used.and Ω  ≈ 5 , Figure 3f in the main text).In all cases the initial operator was Iz and the measured operator was Sz1.For the two spin system (SI, black line), the HET-RFDR polarization transfer is negligible.However, for three spins (IS2, red line) and four spins (IS3, blue line) the transferred polarization reaches a negative value of -0.05.This suggests the involvement of three or more spins in the transfer, but without a more detailed investigation, it is not immediately obvious via which operators the signal is transferred.We consider the amplitudes of the operators that are generated as a result of the evolution of the other operators through pulses or dealys: (  ) →  1 → (  ) →  2 .We first consider the heteronuclear case of an IS spin system during HET-RFDR.In the same way we tabulate the homonuclear I2 spin system during the first two rotor periods of RFDR block in Table S3.
Table S3 Transfer paths during RFDR.The single crystal amplitudes (Euler angles: 184°; 141°; 349°) of the operators atfour time points:  the end of the first pulse;  1the end of the first delay;  the end of the second pulse;  2the end of the second delay.The first column shows the initial operators.The first, second, third and fourth subsections represent the amplitudes with the initial operators Iz1, Iz2, 2Ix1Iy2, 2Iy1Ix2, respectively.The simulated parameters were as in in Figure S6b  →  2 has 0.012854 amplitude.The total amplitude of this group is -0.01262.
The total amplitude of all four groups at the time point 2TR is 0.061, which is the same as for the heteronuclear IS spin system.

RFDR Phase Cycling
In this section we show that under the specific conditions of two spins and no chemical shift offsets, there is zero RFDR transfer between operators Iz1 and Iz2 at tmix=nTR (n=1,2,3,…) when XX phase cycling is used.The measured operator at this time is described with the Eq.: We take into account the dipolar interaction as well as the rf-field during the π-pulse.Then the unitary operator, (  ) is written as follow: (  ) =  2  1 Eq.( S3) 0 }, Eq. ( S3a) Eq. ( S3b) where  ̂ is a Dyson operator and  ,12 () is a periodic dipolar time dependent function (Olejniczak et al., 1984) between spins I1 and I2.Firstly, we can simplify Eq.S3 omitting the scalar product,  ̅ 1  ̅ 2 , since it commutes with other parts of the Hamiltonian: and the dipolar function is periodic -∫  ,12 () ̅ 1  ̅ 2   0 = 0. Eq.S3a-b can be written as follow:

Figure
Figure S2 1D HC [ 13 C, 15 N] labeled SH3 spectra at 55.555 kHz (a) HET-RFDR spectra with different mixing times: 0.576 ms, 1.728 ms, 2.888 ms, 4.032 ms, 5.184 ms, 6.336 ms.(b) Comparaison of 1D HC CP spectrum (red, 1.5 ms of CP mixing) and HET-RFDR spectrum (blue, 6.336 ms of HET-RFDR mixing).The cyan spectrum shows a HC RFDR spectrum, for which πpulses were applied only on the 13 C channel.The carbon refference frequency was set up on 40 ppm.The MAS rate was 55.555 kHz.The experimental parameters are shown in TableS1.

Figure
Figure S3 1D proton-carbon HET-RFDR spectra of [ 13 C, 15 N] labeled SH3 with a 6.336 ms transfer time as a function of the flip angle of the pulses on the carbon channel between 157.34° and 184.64° (17 spectra).The width of π-pulses on the proton channel was 3.4 us.The width of the applied pulses on the carbon channel was constant and equal to 5 us.55.555 kHz MAS was used.The rf-field values in kHz on the carbon channel from left to right were:87.41, 88.2, 89.01, 89.83, 90.66, 91.51, 92.38,

Figure S4
Figure S4 Simulated HET-RFDR signals.The simulated HET-RFDR polarization transfers for S2I2 (a) and SI2 (b) spin systems are shown as a function of mixing time.For all simulations MAS was 55.555 kHz and hard π-pulses with 5.4 us width (92.59 kHz rf-field) were applied simuntaneously every rotor period.The offset and CSA values (the offset and CSA values are defined in the same way as in (Bak et al., 2000)) of spins [I1;C2;C3;I4] are [1;2;5.5;6](kHz) and [4;1;2;3] (kHz), respectively.The initial and the final operators were in the direction .(a) The solid lines represent the HET-RFDR polarization transfers between I1 and C2 spins; the lines with circles represent the HET-RFDR polarization transfers between I1 and C3 spins and the lines with diamonds represent the decay of starting signals.The carbon-carbon distance as well as the dipolar coupling constant between C2

Figure
Figure S5 demonstrates the simulated HET-RFDR transferred signals for three different spin systems (two, three and four spin systems) with similar offset values as in the experiment (Ω  ≈ 0

Figure S6
Figure S6The operator evolution through HET-RFDR and RFDR over two rotor periods.The simulated amplitudes of the operators of a single crystal (Euler angles: 184°; 141°; 349°) for HET-RFDR (a) and RFDR (b).For the heteronuclear IS spin system, ( , = 15 kHz, the initial operator is Iz) and for the homonuclear I2 spin system, ( , = 10 kHz, the initial operator is Iz1).The MAS frequency was 10 kHz and the rf-field was 83 kHz.Black lines -Iz and Iz1; Green lines -Sz and Iz2; Blue lines -2IxSy and 2Ix1Iy2; Red lines -2IySx and 2Iy1Ix2.

Table S1
Summary of the experimental parameters used in the CP (the start and the end values are shown) and HET-RFDR H 13 C

Table S2
Transfer paths during HET-RFDR.The single crystal amplitudes (Euler angles: 184°; 141°; 349°) of the operators at four time points:  the end of the first pulse;  1the end of the first delay;  the end of the second pulse;  2in the end of the second delay.The first column shows the initial operators.The first, second, third and fourth subsections represent the amplitudes with the initial operators Iz, Sz, 2IxSy, 2IySx, respectively.The used simulated parameters were as in FigureS6a and in and Figure 5b in the main text.Unlike the IS spin system, all 64 paths have nonzero amplitudes via which the signal is transferred from homonuclear operator Iz1 to operator Iz2 during the first two rotor periods of RFDR.These 64 paths can be divided into four groups.The first group contains eight paths with combinations of Iz1, Iz2 operators only.For example, the ,12 ()3 1  2 +   ( 1 +  2 )]  1 =  ̂ {∫ [ ,12 ()3 1  2 +   ( 1 +  2 )]