  We report a systematic first-principles study on the recent discovered
superconducting Ba$_{1-x}$K$_x$Fe$_2$As$_2$ systems ($x$ = 0.00, 0.25, 0.50,
0.75, and 1.00). Previous theoretical studies strongly overestimated the
magnetic moment on Fe of the parent compound BaFe$_2$As$_2$. Using a negative
on-site energy $U$, we obtain a magnetic moment 0.83 $\mu_B$ per Fe, which
agrees well with the experimental value (0.87 $\mu_B$). K doping tends to
increase the density of states at fermi level. The magnetic instability is
enhanced with light doping, and is then weaken by increasing the doping level.
The energetics for the different K doping sites are also discussed.
