  We present the optical spectra of Ge$_{1-x}$Si$_x$ alloy nanocrystals
calculated with time-dependent density-functional theory in the adiabatic
local-density ap proximation (TDLDA). The spectra change smoothly as a function
of the compositio n $x$. On the Ge side of the composition range, the lowest
excitations at the ab sorption edge are almost pure Kohn-Sham
independent-particle HOMO-LUMO transitio ns, while for higher Si contents
strong mixing of transitions is found. Within T DLDA the first peak is slightly
higher in energy than in earlier independent-par ticle calculations. However,
the absorption onset and in particular its composit ion dependence is similar
to independent-particle results. Moreover, classical depolarization effects are
responsible for a very strong suppression of the abs orption intensity. We show
that they can be taken into account in a simpler way using Maxwell-Garnett
classical effective-medium theory. Emission spectra are in vestigated by
calculating the absorption of excited nanocrystals at their relaxe d geometry.
The structural contribution to the Stokes shift is about 0.5 eV. Th e
decomposition of the emission spectra in terms of independent-particle transit
ions is similar to what is found for absorption. For the emission, very weak
tra nsitions are found in Ge-rich clusters well below the strong absorption
onset.
