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A perturbation theory of the static response of insulating crystals tohomogeneous electric fields, that combines the modern theory of polarization(MTP) with the variation-perturbation framework is developed, at unrestrictedorder of perturbation. First, we address conceptual issues related to thedefinition of such a perturbative approach. In particular, in our definition ofan electric-field-dependent energy functional for periodic systems, theposition operator appearing in the perturbation term is replaced by aBerry-phase expression, along the lines of the MTP. Moreover, due to theunbound nature of the perturbation, a regularization of the Berry-phaseexpression for the polarization is needed in order to define anumerically-stable variational procedure. Regularization is achieved by meansof discretization, which can be performed either before or after theperturbation expansion. We compare the two possibilities and apply them to amodel tight-binding Hamiltonian. Lowest-order as well as generic formulas arepresented for the derivatives of the total energy, the normalization condition,the eigenequation, and the Lagrange parameters.Comment: 52 pages + 4 figures; accepted for publication in Physical Review

Berry-phase treatment of the homogeneous electric field perturbation in insulators