Quadratic-response theory is shown to provide a conceptually simple but accurate approximation for the self-consistent one-electron potential of semiconductor nanostructures. Numerical examples are presented for GaAs/AlAs and InGaAs/InP (001) superlattices using the local-density approximation to density-functional theory and norm-conserving pseudopotentials without spin-orbit coupling. When the reference crystal is chosen to be the virtual-crystal average of the two bulk constituents, the absolute error in the quadratic-response potential for Gamma(15) valence electrons is about 2 meV for GaAs/AlAs and 5 meV for InGaAs/InP. Low-order multipole expansions of the electron density and potential response are shown to be accurate throughout a small neighborhood of each reciprocal lattice vector, thus providing a further simplification that is confirmed to be valid for slowly varying envelope functions. Although the linear response is about an order of magnitude larger than the quadratic response, the quadratic terms are important both quantitatively (if an accuracy of better than a few tens of meV is desired) and qualitatively (due to their different symmetry and long-range dipole effects).
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