Sensitivity of the NMR density matrix to pulse sequence parameters: A simplified analytic approach

We present a formalism for the analysis of sensitivity of nuclear magnetic resonance pulse sequences to variations of pulse sequence parameters, such as radiofrequency pulses, gradient pulses or evolution delays. The formalism enables the calculation of compact, analytic expressions for the derivatives of the density matrix and the observed signal with respect to the parameters varied. The analysis is based on two constructs computed in the course of modified density-matrix simulations: the error interrogation operators and error commutators. The approach presented is consequently named the Error Commutator Formalism (ECF). It is used to evaluate the sensitivity of the density matrix to parameter variation based on the simulations carried out for the ideal parameters, obviating the need for finite-difference calculations of signal errors. The ECF analysis therefore carries a computational cost comparable to a single density-matrix or product-operator simulation. Its application is illustrated using a number of examples from basic NMR spectroscopy. We show that the strength of the ECF is its ability to provide analytic insights into the propagation of errors through pulse sequences and the behaviour of signal errors under phase cycling. Furthermore, the approach is algorithmic and easily amenable to implementation in the form of a programming code. It is envisaged that it could be incorporated into standard NMR product-operator simulation packages

Proton decoupling and recoupling under double-nutation irradiation in solid-state NMR.

The effect of (1)H decoupling in magic-angle spinning solid-state NMR is studied under radiofrequency irradiation causing simultaneous nutations around a pair of orthogonal axes. Double-nutation with an arbitrary pair of nutation frequencies is implemented through modulation of the amplitude, phase, and frequency of the transmitting pulses. Similarity and difference of double-nutation decoupling and two-pulse phase-modulation decoupling schemes [A. E. Bennett, C. M. Rienstra, M. Auger, K. V. Lakshmi, and R. G. Griffin, J. Chem. Phys. 103, 6951-6958 (1995) and I. Scholz, P. Hodgkinson, B. H. Meier, and M. Ernst, J. Chem. Phys. 130, 114510 (2009)] are discussed. The structure of recoupling bands caused by interference of the (1)H spin nutation with sample spinning is studied by both experiments and numerical simulations

Terminal and bridging fluorine ligands in TiF₄ as studied by ¹⁹F NMR in solids

To examine bonding nature of fluorine ligands in a metal coordinated system, ¹⁹F high-resolution solid-state NMR has been applied to TiF₄, which bears both bridging and terminal fluorines. Observed 12 isotropic signals are assigned to 12 crystallographically different fluorines (6 terminal and 6 bridging fluorines) in TiF₄ by referring to the calculated isotropic shifts using density functional theory (DFT). The isotropic chemical shift (δiso) for terminal F (FT) appears at high frequency (420–480 ppm from δ(CCl3F) = 0 ppm) with large shielding anisotropy Δσ ∼ 850 ppm. Whereas the δiso and Δσ values for bridging F (FB) are moderate; δiso ∼ 0–25 ppm and Δσ ∼ 250 ppm. The origin of the observed high-frequency shift for FT is ascribed to the second-order paramagnetic shift with increased covalency, shorter Ti–F bonds, and smaller energy difference between the occupied and vacant orbitals. Examination of the orientation of the shielding tensor relative to the molecular structure shows that the most deshielded component of the shielding tensor is oriented along the Ti–F bond. The characteristic orientation is consistent with a Ti–F σ bond formed by dYZ of Ti and pz of F. Further, we show that the selectively observed spinning sideband patterns and the theoretical patterns with the calculated Δσ and η (shielding asymmetry) values are not consistent with each other for FB, indicating deficiency of the present DFT calculation in evaluating Δσ