Gauge Bloch

Within the Bloch gauge for the vector potential, the perturbed magnetic moments become, according to Eqn. (29),... 

Lazzeretti has also employed the Bloch gauge, which expands the vector potential in terms of multipoles, to develop a systematic theory of the molecular response to electromagnetic fields. 

The particular merit of multipolar gauge is that is allows one to express the scalar and vector potentials directly in terms of the fields E and B, thus facilitating the identification of electric and magnetic multipoles for generally time-dependent fields. We will follow the three-vector derivation given by Bloch [68]. We will furthermore in this section make extensive use of the Einstein summation convention for coordinate indices. Consider a Taylor expansion of the scalar potential... 

The latter case involves a potential that is spatially invariant and, accordingly, attractive from a computational point of view. This was the original idea of Genkin and Mednis [14] that recently was taken up by Kirtman et al. [28,29]. Since the Coulomb gauge is based on a time-dependent external (vector) potential, the solutions become time dependent, but as argued by Genkin and Mednis they can be written as (time-dependent) Bloch waves characterized by... 

This substitution is, formally, easy to perform in the many-body and the Kohn-Sham Hamiltonians of SDFT, but the presence of the vector potential complicates the task of solving these equations. Maintaining gauge invariance is not trivial in approximate calculations. Moreover, in extended systems the vector potential breaks translational invariance, so Bloch s theorem cannot be used anymore.Still, couplings of external vector potentials become relevant in many situations, and ways to deal with them have been developed, e.g. for the calculation of nuclear magnetic shielding tensors and spin-spin coupling constants. 

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