Local representation of the electronic dielectric response function

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We present a local representation of the electronic dielectric response function, based on a spatial partition of the dielectric response into contributions from each Wannier function using a generalized density functional perturbation theory. This procedure is fully ab initio, and therefore allows us to rigorously define local metrics, such as "bond polarizability," on Wannier centers. We show that the locality of the response function is determined by the locality of three quantities: Wannier functions of the occupied manifold, the density matrix, and the Hamiltonian matrix. In systems with a gap, the bare dielectric response is exponentially localized, which supports the physical picture of the dielectric response function as a collection of interacting local response that can be captured by a tight-binding model.