Polarization-dependent density-functional theory and quasiparticle theory: Optical response beyond local-density approximations

The polarization (P) dependence of the exchange-correlation energy (${\mathit{E}}_{\mathit{xc}}$) of semiconductors results in an effective field (${\mathrm{\partial }}^2$ ${\mathit{E}}_{\mathit{xc}}$/∂${\mathit{P}}^2$)P≡${\gamma }_1$P in the Kohn-Sham equations [Gonze et al., Phys. Rev. Lett 74, 4035 (1995)]. This effective field is absent in local-density approximations such as LDA and GGA. We show that in the long-wavelength limit ${\gamma }_1$≃${\mathrm{\chi }}_{\mathit{LDA}}^{\mathrm{-}1}$-${\mathrm{\chi }}_{\mathit{expt}}^{\mathrm{-}1}$ where χ is the linear susceptibility. We find that ${\gamma }_1$ scales roughly linearly with average bond length suggesting a simple, weakly material-dependent function ${\mathit{E}}_{\mathit{xc}}$[P]. For medium-gap group IV and III-V semiconductors ${\gamma }_1$ is remarkably constant: ${\gamma }_1$=-0.25±0.05. Using the average LDA band gap mismatch Δ and the average quasiparticle gap ${\mathit{E}}_{\mathit{g}}$ a simplified quasiparticle approach yields ${\mathrm{\chi }}_{\mathit{LDA}}^{\mathrm{-}1}$-${\mathrm{\chi }}_{\mathit{QP}}^{\mathrm{-}1}$≃-Δ/(${\mathit{E}}_{\mathit{g}}$χ)=-0.27 *0.10 in good agreement with the value of ${\gamma }_1$. However, for materials containing first-row elements (B,C,N,O) ${\gamma }_1$ varies by a factor of 2 while Δ/(${\mathit{E}}_{\mathit{g}}$χ) is roughly constant. That is, the simple quasiparticle estimate fails to reproduce the polarization dependence of ${\mathit{E}}_{\mathit{xc}}$[P]. For nonlinear response functions, an analysis of ${\mathit{E}}_{\mathit{xc}}$[P] leads to Miller-like expressions ${\mathrm{\chi }}_{\mathit{expt}}^{\left(\mathit{n}\right)}$≃[${\mathrm{\chi }}_{\mathit{expt}}$/${\mathrm{\chi }}_{\mathit{LDA}}$ ${]}^{\mathit{n}+1}$ ${\mathrm{\chi }}_{\mathit{LDA}}^{\left(\mathit{n}\right)}$, n= 2, 3, where the formula for ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$ is valid only when ${\mathrm{\chi }}^{\mathrm{}\left(2\right)\mathrm{}}$=0. For ${\mathrm{\chi }}^{\mathrm{}\left(2\right)\mathrm{}}$, this estimate works well for all the materials including those containing first-row elements. © 1996 The American Physical Society.