  In the Kohn-Sham orbital basis imaginary-time path integral for electrons in
a semiconductor nanoparticle has a mild Fermion sign problem and is amenable to
evaluation by the standard stochastic methods. This is evidenced by the
simulations of silicon hydrogen-passivated nanocrystals, such as
$Si_{35}H_{36},~Si_{87}H_{76},~Si_{147}H_{100}$ and $Si_{293}H_{172},$ which
contain $176$ to $1344$ valence electrons and range in size $1.0 - 2.4~nm$,
utilizing the output of density functional theory simulations. We find that
approximating Fermion action with just the leading order polarization term
results in a positive-definite integrand in the functional integral, and that
it is a good approximation of the full action. We compute imaginary-time
electron propagators in these nanocrystals and extract the energies of
low-lying electron and hole levels. Our quasiparticle gap predictions agree
with the results of high-precision calculations using $G_0W_0$ technique. This
formalism can be extended to calculations of more complex excited states, such
as excitons and trions.
