Application of the linear-analytic tetrahedra method of zone integration to nonlinear response functions

We extend the linear-analytic tetrahedra method for integrating linear-response functions over the Brillouin zone to nonlinear-response functions. We explicitly consider the case of χ→ ${}_{}{}^{\mathrm{}\left(2\right)\mathrm{}}{}_{}{}^{}$ for zinc-blende crystals, and evaluate the contribution to the imaginary part of χ→ ${}_{}{}^{\mathrm{}\left(2\right)\mathrm{}}{}_{}{}^{}$. We find that, as in the linear case, the results are independent of the tetrahedral cell geometry, which renders the method useful and simple. Extensions to other nonlinear-response functions are also considered.