As is known, when calculating localized states in a crystal
it is roughly considered that the point charge is “immersed” on medium with
static dielectric constant ε.
However, a simple estimate, for example, shows that an impurity atom
with charge modulo equal to the electron charge creates at distances of the order of
lattice constant field with a strength of about 10 ^ 7 V / m.
This is a very large field and, in my opinion, one needs to use a nonlinear
polarization theory for calculating the energy spectrum of impurities, excitons, etc.
That is, to take into account the quadratic in the field term in the decomposition of the polarization vector by degrees of electric field strength.