  We study the electronic structure of the Re(0001) surface by means of
ab-initio techniques based on the Fully Relativistic (FR) Density Functional
Theory (DFT) and the Projector Augmented-Wave (PAW) method. We identify the
main surface states and resonances and study in detail their energy dispersion
along the main symmetry lines of the SBZ. Moreover, we discuss the effect of
spin-orbit coupling on the energy splittings and the spin-polarization of the
main surface states and resonances. Whenever possible, we compare the results
with previously studied heavy metals surfaces. We find empty resonances,
located below a gap similar to the L-gap of the (111) fcc surfaces, that have a
downward dispersion and cross the Fermi level, similarly to the recently
studied Os(0001) surface. Their spin polarization at the Fermi level is similar
to that predicted by the Rashba model, but the usual level crossing at
$\bar{\Gamma}$ is not found with our slab thickness. Moreover, for selected
states, we follow the spin polarization along the high symmetry lines,
discussing its behavior with respect to ${\bf k}_{\parallel}$, the wave-vector
parallel to the surface.
