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We present the optical spectra of Ge$_{1-x}$Si$_x$ alloy nanocrystalscalculated with time-dependent density-functional theory in the adiabaticlocal-density ap proximation (TDLDA). The spectra change smoothly as a functionof the compositio n $x$. On the Ge side of the composition range, the lowestexcitations at the ab sorption edge are almost pure Kohn-Shamindependent-particle HOMO-LUMO transitio ns, while for higher Si contentsstrong mixing of transitions is found. Within T DLDA the first peak is slightlyhigher in energy than in earlier independent-par ticle calculations. However,the absorption onset and in particular its composit ion dependence is similarto independent-particle results. Moreover, classical depolarization effects areresponsible for a very strong suppression of the abs orption intensity. We showthat they can be taken into account in a simpler way using Maxwell-Garnettclassical effective-medium theory. Emission spectra are in vestigated bycalculating the absorption of excited nanocrystals at their relaxe d geometry.The structural contribution to the Stokes shift is about 0.5 eV. Th edecomposition of the emission spectra in terms of independent-particle transitions is similar to what is found for absorption. For the emission, very weaktra nsitions are found in Ge-rich clusters well below the strong absorptiononset.

Alloying effects on the optical properties of Ge$_{1-x}$Si$_x$ nanocrystals from TDDFT and comparison with effective-medium theory