NMR spectroscopy of paramagnetic solids provides detailed information about the local configuration and the chemical environment of the NMR observed center, as well as about the structural, magnetic and electronic properties of the coordianted paramagnetic centres. In the case of complex paramagnetic solids such as cathode materials for (rechargeable) batteries, NMR represents an invaluable tool to provide insight into the structural and electronic properties of the systems, which are at the base of the electrochemical performance of these materials. However, the paramagnetism makes the interpretation of the NMR data very challenging. This is primarily due to the interactions of the unpaired electrons with the NMR observed nucleus, and the interpretation of the NMR spectra often requires the aid of reliable theoretical and computational methods.

Often the dominant interaction contributing to the measured isotropic shifts is the hyperfine interaction between the unpaired electrons and the observed nucleus, which results from the transfer of unpaired electrons from the paramagnetic centre(s) to the NMR observed site. In systems such as the ones studied here, in which the paramagnetic ions are a major constituent of the lattice, the multitide of different local environments results in a complex distribution of resonances. As in the case of the LiV$6$O$\mathrm{<mrow\; data-mjx-texclass="ORD">13</mrow>}$ cathode material, a methodical investigation of the configurational stability from first principles gives insight into the preferred site configurations. The combination of experimental $7$Li NMR spectra and hyperfine shift DFT calculations of the so-found stable Li environments allows to unravel the complex lithiation mechanism of this material. In the other case of the LiTiMnO$4$ cathode materials, the $7$Li hyperfine shifts calculated from first principles for a variety of Li environments are combined in a lattice model which allows to assign the isotropic regions of the experimental $7$Li NMR spectra, helping to resolve the complex cation ordering as a function of Mn/Ti content in the series.

For paramagnetic centres with an unquenched orbital component of the electron magnetic moment(s), the spin-orbit coupling effects also contribute to the paramagnetic NMR shift and shift anisotropy. A first principles model is derived, which describes how spin-orbit coupling and the single-ion $g$-tensor are defined and calculated in periodic paramagnetic solids, and how they can be coupled with the hyperfine interaction to model their effects on the NMR spectrum. The method is applied to a series of olivine-type LiTMPO$4$ cathode materials (with TM = Mn, Fe, Co, and Ni) and the respective $7$Li and $\mathrm{<mrow\; data-mjx-texclass="ORD">31</mrow>}$P NMR spectra are simulated and compared with the experiments.

The other paramagnetic effect considered in this thesis involves the bulk magnetic susceptibility (BMS), which is particularly important for paramagnetic single crystals and solids of complex shape. The BMS effect results from the discontinuity of the bulk susceptibility at the surface of the crystal, inducing a demagnetizing field throughout the sample which changes the measured NMR shift and shift anisotropy. A method to analytically calculate the demagnetising field and the BMS shift in crystals of different shapes is derived, and it is applied to a series of LiFePO$4$ single crystals for which the $7$Li NMR spectra are also measured experimentally. The study confirms that, particularly for $7$Li NMR, the macroscopic shape-dependent BMS shift can indeed be a significant contribution to the measured resonances, determining the large variation in shift measured for the crystals of different shapes.