Determination of the Macroscopic Electric Susceptibilities Χ (n) from the Microscopic (hyper)polarizabilities α, β and γ

In this work, we will show that it is possible to evaluate the bulk dielectric constant ϵ ⁽⁰⁾ and the electric susceptibilities Χ ⁽ⁿ⁾ of a crystalline system in two different ways, namely from the variation of the cell energy as a function of the applied electric field, or from the macroscopic response field in the solid. The two ways are obviously equivalent from a formal point of view; the numerical equivalence was till now undocumented. The use a saw-tooth like finite potential (so that a positive and a negative field is applied at the same time), required for maintaining the periodicity of the lattice, introduces some complication in the formulation of the working equations.