Wannier and Bloch orbital computation of the nonlinear susceptibility

We present a method to compute high-order derivatives of the total energy of a periodic solid with respect to a uniform electric field. We apply the 2n + 1 theorem to a recently introduced total energy functional which uses a Wannier representation for the electronic orbitals and we find an expression for the static nonlinear susceptibility which is much simpler than the one obtained by standard perturbative expansions. We show that the zero-field expression of the nonlinear susceptibility can be rewritten in a Bloch representation. We test numerically the validity of our approach with a 1D model Hamiltonian.