One-electron formalism for second-harmonic generation in crystalline semiconductors

The one-electron, self-consistent-field formalism for harmonic generation is presented. The solution is expanded in a Fourier series, assuming that the time-dependent perturbation is weak. The equations of harmonic response form a set of inhomogeneous Schrödinger equations which, if solved order by order, are coupled only through self-consistent-field effects. Special attention is paid to the case of second-harmonic generation in crystalline semiconductors with a longitudinal field present. The f-sum rule for crystals is exploited to eliminate certain terms which apparently diverge as ${\mathrm{\omega }}^{\mathrm{-}2}$, in perfect analogy to the linear-response theory. Additionally, terms which are in the form of numerical first-, second-, and third-order finite differences of the frequency are combined into numerically superior forms.