The Lorentz sphere visualised

From the inception of nuclear magnetic resonance as a spectroscopic technique, the local origin of chemical shifts has been a topic of discussion. A useful concept employed to describe it has been that of the "Lorentz sphere," the approximately spherical volume surrounding a given nucleus in which the electronic currents contribute significantly to the chemical shift, whereas the outside can be considered as an uniformly magnetised "bulk." In this paper, we use the output of the plane wave density functional theory code CASTEP to get a quantitative estimate of the Lorentz sphere in periodic systems. We outline a mathematical description of a radial buildup function for the magnetic shielding starting from the electronic currents and the simple assumption of periodicity. We provide an approximate upper bound for the Lorentz sphere's size in any crystal, then compute buildup functions for a number of sites in two molecular crystals, showing how various chemical features such as hydrogen bonds influence to convergence to the final shielding value.