  The electronic structure of quasi-one-dimensional superconductor
K$_2$Cr$_3$As$_3$ is studied through systematic first-principles calculations.
The ground state of K$_2$Cr$_3$As$_3$ is paramagnetic but very close to a
ferromagnetic instability. Close to the Fermi level, the Cr-3d$_{z^2}$,
d$_{xy}$, and d$_{x^2-y^2}$ orbitals dominate the electronic states, and three
bands cross $E_F$ to form one 3D Fermi surface sheet and two quasi-1D sheets.
The electron DOS at $E_F$ is less than 1/3 of the experimental value,
indicating an intermediate electron renormalization factor around $E_F$.
Despite of the relatively small atomic numbers, the antisymmetric spin-orbit
coupling splitting is sizable ($\approx$ 60 meV) on the 3D Fermi surface sheet
as well as on one of the quasi-1D sheets. Finally, the imaginary part of bare
electron susceptibility shows large peaks at $\Gamma$, suggesting the existence
of large ferromagnetic spin fluctuation in the compound.
