Bulk quadrupole contribution to second harmonic generation from a microscopic response function

Even‐order nonlinear methods have proven useful tools for the analysis of centrosymmetric materials where the surface dipole contribution is often dominant. Quadrupole‐order contributions from the surface discontinuity and the bulk can be on the same order of magnitude however, yet have been largely ignored in modern numerical ab initio calculations. We find that the macroscopic bulk quadrupole term P(2Q)=χ(2Q): E∇E can be related to the first order derivatives of the microscopic response function χ(2)(k,k′,k″) at the Γ‐point and outline how a quantum‐mechanical expression can be derived for a periodic system.