Variations of polarisation in cerystals

Abstract: It is nowadays a quite diffuse idea that variations of electronic polarisation, as introduced by Resta[1], in condensed matter theory are related to a "Berry phase"[2], as shown by Vanderbilt. The derivation of the latter geometric phase is correct and the periodic gauge[2] is used for its derivation, aside not allowing for its calculations in finite electromagnetic fields. Its derivation has not been demonstrated in the general case of an external homogeneous electric field interacting with the electronic field. In the present paper we give a brief derivation of the algorithm for the calculations of polarisation differences in a general manner than in[2] defining it even in the presence of an external electric field interacting with the electronic field and writing equations in a way directly implementable in modern first principles codes[1,2]