Theory

For a description of the method please see also reference [107].

A magnetic field $\mathbf{B}$ is applied to the system, which reacts by induced electronic ring currents. These currents produce an additional magnetic field by themselves, which is not homogeneous in space. Therefore, the actual magnetic field at the ionic positions is different for all atoms in the cell. This field determines the resonance frequency of the nuclear spin, and this resonance can be measured with a very high accuracy.

The perturbation Hamiltonian is given by

$$H^{(1)} = \frac 12 \frac em \mathbf{p}\times \mathbf{r}\cdot \mathbf{B}.$$ (261)

The difficult part of this Hamiltonian lies in the position operator which is ill defined in a periodic system. To get around this, the wavefunctions are localized and for each localized orbital, Eq. (261) is applied individually assuming the orbital being isolated in space. Around each orbital, a virtual cell is placed such that the wavefunction vanishes at the borders of that virtual cell.

The perturbation and therefore also the response are purely imaginary, so that there is no first order response density. This simplifies the equations and speeds up convergence.