  We present a computational study of antiferromagnetic transition in RuO$_2$.
The rutile structure with the magnetic sublattices coupled by $\pi/2$-rotation
leads to a spin-polarized band structure in the antiferromagnetic state, which
gives rise to a $d$-wave modulation of the Fermi surface in the spin-triplet
channel. We argue a finite spin conductivity that changes sign in the $ab$
plane is expected RuO$_2$ because of this band structure. We analyze the origin
of the antiferromagnetic instability and link it to presence of a nodal line
close to the Fermi level.
