  Macroscopic descriptions of ferroelectrics have an obvious appeal in terms of
efficiency and physical intuition. Their predictive power, however, has often
been thwarted by the lack of a systematicp rocedure to extract the relevant
materials parameters from the microscopics. Here we address this limitation by
establishing an unambiguous two-way mapping between spatially inhomogeneous
fields and discrete lattice modes. This yields a natural treatment of gradient
couplings in the macroscopic regime via a long-wavelength expansion of the
crystal Hamiltonian. Our analysis reveals an inherent arbitrariness in both the
flexoelectric and polarization gradient coefficients, which we ascribe to a
translational freedom in the definition of the polar distortion pattern.
Remarkably, such arbitrariness cancels out in all physically measurable
properties (relaxed atomic structure and energetics) derived from the model,
pointing to a generalized translational covariance in the continuum description
of inhomogeneous ferroelectric structures. We demonstrate our claims with
extensive numerical tests on 180-degree domain walls in common ferroelectric
perovskites, finding excellent agreement between the continuum model and direct
first-principles calculations.
