  The electronic structure of recently discovered $\beta$-Fe$_4$Se$_5$ with
$\sqrt{5} \times \sqrt{5}$ ordered Fe vacancies is calculated using
first-principles density functional theory. We find that the ground state is an
antiferromagnetic (AFM) insulator in agreement with the experimental
observation. In K$_2$Fe$_4$Se$_5$, it is known that the ground state is
$\sqrt{5} \times \sqrt{5}$-blocked-checkerboard AFM ordered. But for this
material, we find that the ground state is $\sqrt{5} \times
\sqrt{5}$-pair-checkerboard AFM ordered, in which the intrablock four Fe spins
exhibit the collinear AFM order and the interblock spins on any two
nearest-neighboring sites are antiparallel aligned. This state is about 130
meV/Fe lower in energy than the $\sqrt{5} \times \sqrt{5}$-blocked-checkerboard
AFM one. Electron doping can lower the energy of the $\sqrt{5} \times
\sqrt{5}$-blocked-checkerboard AFM state and introduce a transition between
these two states, suggesting that there is strong AFM fluctuation in FeSe-based
materials upon doping. This provides a unified picture to understand the AFM
orders in $\beta$-Fe$_4$Se$_5$ and in alkali-metal intercalated FeSe materials.
