A perturbation theory of the static response of insulating crystals to
homogeneous electric fields, that combines the modern theory of polarization
(MTP) with the variation-perturbation framework is developed, at unrestricted
order of perturbation. First, we address conceptual issues related to the
definition of such a perturbative approach. In particular, in our definition of
an electric-field-dependent energy functional for periodic systems, the
position operator appearing in the perturbation term is replaced by a
Berry-phase expression, along the lines of the MTP. Moreover, due to the
unbound nature of the perturbation, a regularization of the Berry-phase
expression for the polarization is needed in order to define a
numerically-stable variational procedure. Regularization is achieved by means
of discretization, which can be performed either before or after the
perturbation expansion. We compare the two possibilities and apply them to a
model tight-binding Hamiltonian. Lowest-order as well as generic formulas are
presented 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
