  $\alpha$-RuCl$_3$ has been proposed recently as an excellent playground for
exploring Kitaev physics on a two-dimensional (2D) honeycomb lattice. However,
structural clarification of the compound has not been completed, which is
crucial in understanding the physics of this system. Here, using {\it
ab-initio} electronic structure calculations, we study a full three dimensional
(3D) structure of $\alpha$-RuCl$_3$ including the effects of spin-orbit
coupling (SOC) and electronic correlations. Three major results are as follows;
i) SOC suppresses dimerization of Ru atoms, which exists in other Ru compounds
such as isostructural Li$_2$RuO$_3$, and making the honeycomb closer to an
ideal one. ii) The nearest-neighbor Kitaev exchange interaction between the
$j_{\rm eff}$=1/2 pseudospin depends strongly on the Ru-Ru distance and the Cl
position, originating from the nature of the edge-sharing geometry. iii) The
optimized 3D structure without electronic correlations has $P{\bar 3}1m$ space
group symmetry independent of SOC, but including electronic correlation changes
the optimized 3D structure to either $C2/m$ or $Cmc2_1$ within 0.1 meV per
formula unit (f.u.) energy difference. The reported $P3_112$ structure is also
close in energy. The interlayer spin exchange coupling is a few percent of
in-plane spin exchange terms, confirming $\alpha$-RuCl$_3$ is close to a 2D
system. We further suggest how to increase the Kitaev term via tensile strain,
which sheds new light in realizing Kitaev spin liquid phase in this system.
