DFT investigation of the effect of spin-orbit coupling on the NMR shifts in paramagnetic solids

Nuclear magnetic resonance (NMR) spectroscopy is a powerful tool for studying the structural and electronic properties of paramagnetic solids. However, the interpretation of paramagnetic NMR spectra is often challenging as a result of the interactions of unpaired electrons with the nuclear spins of interest. In this work, we extend the formalism of the paramagnetic NMR shielding in the presence of spin-orbit coupling towards solid systems with multiple paramagnetic centers. We demonstrate how the single-ion electron paramagnetic resonance $g$ tensor is defined and calculated in periodic paramagnetic solids. We then calculate the hyperfine tensor and the $g$ tensor with density functional theory to show the validity of the presented model and we further demonstrate how these interactions can be combined to give the overall paramagnetic shielding tensor, ${\mathbf{\sigma }}^s$. The method is applied to a series of olivine-type $\text{Li}\mathit{\text{TM}}{\text{PO}}_4$ materials (with $\mathit{\text{TM}}=\text{Mn}$, Fe, Co, and Ni) and the corresponding ${}^7\mathrm{Li}$ and ${}^{31}\mathrm{P}$ NMR spectra are simulated. We analyze the effects of spin-orbit coupling and of the electron-nuclear magnetic interactions on the calculated NMR parameters. A detailed comparison is presented between contact and dipolar interactions across the $\text{Li}\mathit{\text{TM}}{\text{PO}}_4$ series, in which the magnitudes and signs of the nonrelativistic and relativistic components of the overall isotropic shift and shift anisotropy are computed and rationalized.

Figure 1

(a) Structure of the repeating unit of the olivine-type phase of $\text{Li}\mathit{\text{TM}}{\text{PO}}_4$ $(\mathit{\text{TM}}=\text{Mn}$, Fe, Co, and Ni) consisting of a distorted hexagonal-close-packed oxygen (red) framework. Phosphorus (pink) occupies an eighth of the tetrahedral sites, while the two octahedral sites are occupied by lithium (green) and the TM (blue). (b) The four octahedral TM sites are labeled I, II, III, and IV and occupy different spatial positions: their environments (in magenta, orange, cyan, and blue, respectively) are related to one another as according to the orthorhombic symmetry of the $Pnma$ space group.

Figure 2

(a) Repeating unit of the olivine-type $\text{Li}\mathit{\text{TM}}{\text{PO}}_4$ delimited by the dashed box. The solid lines represent the pairwise TM-O-Li bonds and denote the pathways of delocalization of unpaired-electron spin density from each TM site to the nuclear position of the OC, here the lithium site labeled $\text{A}$, as in Ref. [12]. (b) Periodic expansion of the $\text{Li}\mathit{\text{TM}}{\text{PO}}_4$ repeating unit; the arrows highlight the TM-OC pairs interacting via magnetic dipolar coupling, and specifically underline the interactions between the lithium site labeled $\text{A}$ and one of the four inequivalent TM sites, here labeled as $\text{I}$, throughout the periodically repeating units.

Figure 3

Calculated isotropic value of the $g$-tensor shift, $\mathrm{\Delta }{g}_{\mathrm{iso}}$, as a function of the $U_{\mathrm{eff}}$ correction applied on ${\mathrm{Mn}}^{2+}$ (in red prisms), ${\mathrm{Fe}}^{2+}$ (in green squares), ${\mathrm{Co}}^{2+}$ (in blue circles), and ${\mathrm{Ni}}^{2+}$ (in magenta triangles). For each case the isotropic value obtained with the $U_{\mathrm{eff}}$ value used in prior DFT+$U$ studies is indicated with a cross.

Figure 4

Experimental (in solid red line) and fitted (in dashed blue line) ${}^7\mathrm{Li}$ and ${}^{31}\mathrm{P}$ spectra of ${\mathrm{LiMnPO}}_4$ [(a) and (c), respectively] and ${\mathrm{LiFePO}}_4$ [(b) and (d), respectively]. Isotropic peaks are marked with an asterisk. The experimental spectra are adapted with permission from Ref. [36]. $\copyright 2012$, American Chemical Society.