^{1}and Hellmut Eckert

^{1,2,a)}

### Abstract

A new solid state NMR technique is described for measuring homonuclear dipole-dipole interactions in multi-spin-1/2 systems under magic-angle spinning conditions. Re-coupling is accomplished in the form of an effective double quantum (DQ) Hamiltonian created by a symmetry-based POST-C7 sequence consisting of two excitation blocks, attenuating the signal (intensity S′). For comparison, a reference signal S_{0} with the dipolar re-coupling absent is generated by shifting the phase of the second block by 90° relative to the first block. As in rotational echo double resonance, the homonuclear dipole-dipole coupling constant can then be extracted from a plot of the normalized difference signal (S_{0} − S′)/S_{0} versus dipolar mixing time. The method is given the acronym DQ-DRENAR (“Double-Quantum-based Dipolar Re-coupling effects Nuclear Alignment Reduction”). The method is analyzed mathematically, and on the basis of detailed simulations, with respect to the order and the geometry of the spin system, the dipolar truncation phenomenon, and the influence of the chemical shift anisotropy on experimental curves. Within the range of (S_{0} − S′)/S_{0} ≤0.3–0.5 such DRENAR curves can be approximated by simple parabolae, yielding effective squared dipole-dipole coupling constants summed over all the pairwise interactions present. The method has been successfully validated for ^{31}P–^{31}P distance determinations of numerous crystalline model compounds representing a wide range of dipolar coupling strengths.

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the programme SFB858. We thank the following collaborators for providing samples that had been previously studied through collaboration: Professor Dr. Gerhard Erker (WWU Münster), PD Dr. Mimoza Gjikaj (TU Clausthal), and Professor Dr. Rüdiger Kniep (Max-Planck Institut für Chemische Physik fester Stoffe (Dresden).

I. INTRODUCTION

II. THEORY

A. The DRENAR (Dipolar Re-coupling Effects Nuclear Alignment Reduction) pulse sequence

B. Response of two-spin (I = 1/2) systems

C. Three-spin-1/2 systems and extension to multi-spin systems

1. Principles

2. Dipolar truncation effects

3. Effect of dipolar geometry

4. Effects of distributions of dipolar coupling constants

D. Effects of the chemical shift anisotropy

III. EXPERIMENTAL

IV. VALIDATION ON CRYSTALLINE MODEL COMPOUNDS

V. CONCLUSIONS

### Key Topics

- Sodium
- 13.0
- Coherence
- 10.0
- Chemical shifts
- 9.0
- Data analysis
- 9.0
- Nuclear magnetic resonance
- 9.0

##### B01F3/00

## Figures

DQ-DRENAR pulse sequences. (a) Standard sequence, C′ means all the phases of pulses in (d) are 90° shifted. (b) DQ-DRENAR sequence with cross-polarization preparation and TPPM decoupling on the I channel used during acquisition, (c) Wideband DQ-DRENAR sequence; means all the phases of pulses in (d) are 180° shifted. (d) The pulse compositions of block C.

DQ-DRENAR pulse sequences. (a) Standard sequence, C′ means all the phases of pulses in (d) are 90° shifted. (b) DQ-DRENAR sequence with cross-polarization preparation and TPPM decoupling on the I channel used during acquisition, (c) Wideband DQ-DRENAR sequence; means all the phases of pulses in (d) are 180° shifted. (d) The pulse compositions of block C.

(a) DQ-DRENAR curves of two-spin systems with different dipolar coupling constants. (b) Systematic error introduced by the approximate formula (9) as a function of data range analyzed. The simulations assume spinning rate to be 15 kHz.

(a) DQ-DRENAR curves of two-spin systems with different dipolar coupling constants. (b) Systematic error introduced by the approximate formula (9) as a function of data range analyzed. The simulations assume spinning rate to be 15 kHz.

SIMPSON simulation results of I_{j}(I_{k})_{n} systems. (a) DQ-DRENAR curves of spin systems with different numbers of k spins n; (b) the difference between the value estimated by parabola fitting and the theoretical one for different system. The dipolar coupling constant b_{jk} and the spinning rate are set to 400 Hz and 15 kHz, respectively. Tetrahedron geometry is taken. The dipolar interaction between I_{k} spins are taken into account.

SIMPSON simulation results of I_{j}(I_{k})_{n} systems. (a) DQ-DRENAR curves of spin systems with different numbers of k spins n; (b) the difference between the value estimated by parabola fitting and the theoretical one for different system. The dipolar coupling constant b_{jk} and the spinning rate are set to 400 Hz and 15 kHz, respectively. Tetrahedron geometry is taken. The dipolar interaction between I_{k} spins are taken into account.

(a) Simulated DQ-DRENAR curves of the 3-spin system shown with different ratios of b_{ik}/b_{jk}. Inset shows initial data ranges. (b) Systematic error introduced by dipolar truncation effects as a function of data range. Simulations assume b_{jk} = b_{ji} = 400 Hz, ν_{r} = 15 kHz.

(a) Simulated DQ-DRENAR curves of the 3-spin system shown with different ratios of b_{ik}/b_{jk}. Inset shows initial data ranges. (b) Systematic error introduced by dipolar truncation effects as a function of data range. Simulations assume b_{jk} = b_{ji} = 400 Hz, ν_{r} = 15 kHz.

(a) Simulated DQ-DRENAR curves for the linear 3-spin system shown with different ratios of b_{ik}/b_{jk}. Inset shows initial data ranges. (b) Systematic error introduced by dipolar truncation effects as a function of data range. Simulations assume b_{jk} = 400 Hz, ν_{r} = 15 kHz.

(a) Simulated DQ-DRENAR curves for the linear 3-spin system shown with different ratios of b_{ik}/b_{jk}. Inset shows initial data ranges. (b) Systematic error introduced by dipolar truncation effects as a function of data range. Simulations assume b_{jk} = 400 Hz, ν_{r} = 15 kHz.

Simulated DQ-DRENAR curves of the linear 3-spin system with the observe-spin I_{j} at the center, under variation of b_{ji}/b_{jk}. Simulations assume b_{jk} = 400 Hz, ν_{r} = 15 kHz.

Simulated DQ-DRENAR curves of the linear 3-spin system with the observe-spin I_{j} at the center, under variation of b_{ji}/b_{jk}. Simulations assume b_{jk} = 400 Hz, ν_{r} = 15 kHz.

(a) Dependence of the DQ-DRENAR curves on the angle ß subtended by the two dipolar vectors in a 3-spin system; Simulations assume b_{jk} = b_{ji} = 400 Hz, ν_{r} = 15 kHz and include dipolar truncation effects. (b) Dependence of the systematic error of the parabolic fitting procedure on the values as function of data range.

(a) Dependence of the DQ-DRENAR curves on the angle ß subtended by the two dipolar vectors in a 3-spin system; Simulations assume b_{jk} = b_{ji} = 400 Hz, ν_{r} = 15 kHz and include dipolar truncation effects. (b) Dependence of the systematic error of the parabolic fitting procedure on the values as function of data range.

(a) Simulated DQ-DRENAR curves of 4-spin systems I_{j}(I_{k})_{n}: Solid squares represent a pseudo-tetrahedral geometry; the angles between dipolar vectors are 109.47°. Solid circles reflect the configuration of an equilateral triangle with the observe-spin j at the center. Simulations assume b_{jk} = 400 Hz, and a spinning rate of 15 kHz. The dipolar effects between different spins I_{k} are taken into account. (b) Dependence of the systematic error of the parabolic fitting procedure on the values as a function of data range.

(a) Simulated DQ-DRENAR curves of 4-spin systems I_{j}(I_{k})_{n}: Solid squares represent a pseudo-tetrahedral geometry; the angles between dipolar vectors are 109.47°. Solid circles reflect the configuration of an equilateral triangle with the observe-spin j at the center. Simulations assume b_{jk} = 400 Hz, and a spinning rate of 15 kHz. The dipolar effects between different spins I_{k} are taken into account. (b) Dependence of the systematic error of the parabolic fitting procedure on the values as a function of data range.

Effect of distance distributions on DQ-DRENAR data, simulated for a two-spin system with r = 3.667 Å. (a) Gaussian distance distributions considered (*σ* = 0.1, 0.2, and 0.3 Å, respectively). (b) Resultant dipolar coupling constant distributions, (c) calculated DQ-DRENAR curves (the inset showing the initial data range) and (d) dependence of the systematic error of the parabolic fitting procedure on the values as a function of data range, under different sigma values.

Effect of distance distributions on DQ-DRENAR data, simulated for a two-spin system with r = 3.667 Å. (a) Gaussian distance distributions considered (*σ* = 0.1, 0.2, and 0.3 Å, respectively). (b) Resultant dipolar coupling constant distributions, (c) calculated DQ-DRENAR curves (the inset showing the initial data range) and (d) dependence of the systematic error of the parabolic fitting procedure on the values as a function of data range, under different sigma values.

CSA effect on the apparent dipolar coupling constant in two-spin systems. (a) Decay curves in the presence of CSA of different magnitudes; *b* _{ jk } = 400 Hz; (b) ratio *b* _{ jk } ^{ app }/*b* _{ jk } predicted for different CSA values and dipolar coupling constants. Coincident dipolar and magnetic shielding tensors are assumed, but orientation effects were found unimportant. Simulations assume a spinning rate *v* _{ r } = 15 kHz.

CSA effect on the apparent dipolar coupling constant in two-spin systems. (a) Decay curves in the presence of CSA of different magnitudes; *b* _{ jk } = 400 Hz; (b) ratio *b* _{ jk } ^{ app }/*b* _{ jk } predicted for different CSA values and dipolar coupling constants. Coincident dipolar and magnetic shielding tensors are assumed, but orientation effects were found unimportant. Simulations assume a spinning rate *v* _{ r } = 15 kHz.

CSA effect on the apparent dipolar coupling constant in two-spin systems. (a) DQ-DRENAR sequence of Fig. 1(a) . (b) Sequence shown in Fig. 1(c) ; *b* _{ jk } = 400 Hz; *b* _{ jk } ^{ app }/*b* _{ jk } predicted for different spinning rate. Coincident dipolar and magnetic shielding tensors are assumed.

CSA effect on the apparent dipolar coupling constant in two-spin systems. (a) DQ-DRENAR sequence of Fig. 1(a) . (b) Sequence shown in Fig. 1(c) ; *b* _{ jk } = 400 Hz; *b* _{ jk } ^{ app }/*b* _{ jk } predicted for different spinning rate. Coincident dipolar and magnetic shielding tensors are assumed.

^{31}P DQ-DRENAR curves, simplified spin-cluster simulations (dotted curves) and predicted parabolae (solid curves) based on Eq. (24) using from the crystal structures. First row: Na_{4}P_{2}O_{6}·10H_{2}O (left) and Rb_{2}[(H_{2}P_{2}O_{6})(H_{4}P_{2}O_{6})] (middle, up and down triangles represent data for the signals at 13.8 and 12.8 ppm, respectively); CdPS_{3} (right). Second row: Ga(PO_{3})_{3} (left), KPO_{3} (middle, solid squares and empty circles represent data for signals at −18.4 and −20.2 ppm, respectively) and K_{2}MoP_{2}O_{9} (right, solid and empty circles represent data for signals −7.7 and −10.9 ppm, respectively); third row: (Ph_{2}P)_{2}C=C(C_{6}F_{5})(B(C_{6}F_{5})_{2} (left, solid star and triangles represent the data obtained for the resonances at 24.4 and −8.7 ppm, respectively. Spin-pair simulations including CSA are also shown as dotted curves). Ag_{7}P_{3}S_{11} (middle, squares, solid circles, and triangle symbols represent data for PS_{4} ^{3−} groups (103.2 ppm) and the two crystallographically inequivalent P atoms of the P_{2}S_{7} ^{4−} group (101.4 and 92.0 ppm), respectively, for the latter, spin-pair simulations including CSA are also shown as dotted curves), and BPO_{4} (right); fourth row: Ag_{3}PO_{4} (left), Ca_{5}(PO_{4})_{3}(OH) (middle) and H_{3}Mo_{12}O_{40}P·13H_{2}O (right, inset amplifies the initial data region).

^{31}P DQ-DRENAR curves, simplified spin-cluster simulations (dotted curves) and predicted parabolae (solid curves) based on Eq. (24) using from the crystal structures. First row: Na_{4}P_{2}O_{6}·10H_{2}O (left) and Rb_{2}[(H_{2}P_{2}O_{6})(H_{4}P_{2}O_{6})] (middle, up and down triangles represent data for the signals at 13.8 and 12.8 ppm, respectively); CdPS_{3} (right). Second row: Ga(PO_{3})_{3} (left), KPO_{3} (middle, solid squares and empty circles represent data for signals at −18.4 and −20.2 ppm, respectively) and K_{2}MoP_{2}O_{9} (right, solid and empty circles represent data for signals −7.7 and −10.9 ppm, respectively); third row: (Ph_{2}P)_{2}C=C(C_{6}F_{5})(B(C_{6}F_{5})_{2} (left, solid star and triangles represent the data obtained for the resonances at 24.4 and −8.7 ppm, respectively. Spin-pair simulations including CSA are also shown as dotted curves). Ag_{7}P_{3}S_{11} (middle, squares, solid circles, and triangle symbols represent data for PS_{4} ^{3−} groups (103.2 ppm) and the two crystallographically inequivalent P atoms of the P_{2}S_{7} ^{4−} group (101.4 and 92.0 ppm), respectively, for the latter, spin-pair simulations including CSA are also shown as dotted curves), and BPO_{4} (right); fourth row: Ag_{3}PO_{4} (left), Ca_{5}(PO_{4})_{3}(OH) (middle) and H_{3}Mo_{12}O_{40}P·13H_{2}O (right, inset amplifies the initial data region).

Performance comparison of the pulse sequences in Figs. 1(a) and 1(c) for two crystalline model compounds with moderately large CSAs: ^{31}P DQ-DRENAR curves, and predicted parabola (solid curves) based on Eq. (24) using from the crystal structures. First row: Na_{5}B_{2}P_{3}O_{13}, left and right parts represent the data obtained from sequence 1(a) and 1(c); squares, solid circles, and triangle symbols represent data for signals centered at −0.2, −1.6 and −7.6 ppm, respectively. Second row: Na_{2}PO_{3}F, left and right represent the data obtained from sequence 1(a) and 1(c); squares and solid circles represent data for signals centered at 8.5 and 3.5 ppm, respectively.

Performance comparison of the pulse sequences in Figs. 1(a) and 1(c) for two crystalline model compounds with moderately large CSAs: ^{31}P DQ-DRENAR curves, and predicted parabola (solid curves) based on Eq. (24) using from the crystal structures. First row: Na_{5}B_{2}P_{3}O_{13}, left and right parts represent the data obtained from sequence 1(a) and 1(c); squares, solid circles, and triangle symbols represent data for signals centered at −0.2, −1.6 and −7.6 ppm, respectively. Second row: Na_{2}PO_{3}F, left and right represent the data obtained from sequence 1(a) and 1(c); squares and solid circles represent data for signals centered at 8.5 and 3.5 ppm, respectively.

## Tables

^{31}P isotropic chemical shifts (±0.5 ppm), CSA values Δ*σ* (± 5 ppm), and experimental values of for the model compounds studied. Numbers in parentheses are theoretical values calculated from the crystal structures over a range of three times the closest P–P distance. CSAs are defined according to .

^{31}P isotropic chemical shifts (±0.5 ppm), CSA values Δ*σ* (± 5 ppm), and experimental values of for the model compounds studied. Numbers in parentheses are theoretical values calculated from the crystal structures over a range of three times the closest P–P distance. CSAs are defined according to .

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