Hidden Relaxation Term in Approximate Treatments of Responses to Electric and Magnetic Fields

Recently a generalization of the \textit{modern theory of orbital magnetization}'' to include non-local Hamiltonians (e.g. hybrid functionals of the generalized Kohn-Sham theory) was provided for magnetic response properties. Results indicated inequivalence between sampling of direct and reciprocal spaces for those calculations far from the complete basis set limit. We show that this can be explained by a hidden relaxation’’ contribution to the reciprocal-space derivatives. The missing relaxation term is shown to (generally) affect the results of calculations of not only magnetic, but also electric response properties, within the context of the ``$\mathit{\text{modern}}\mathit{\text{theory}}\mathit{\text{of}}\mathit{\text{polarization}}$‘’. Necessary conditions are provided to permit avoiding the calculation of the hidden relaxation term.