  We report results for the superconducting transition temperature and
anisotropic energy gap for pure Niobium based on Eliashberg's equations and
electron and phonon band structures computed from density functional theory.
The electronic band structure is used to construct the Fermi surface and
calculate the Fermi velocity at each point on the Fermi surface.The phonon
bands are in excellent agreement with inelastic neutron scattering data. The
corresponding phonon density of states and electron-phonon coupling define the
electron-phonon spectral function, $\alpha^2F({\bf p},{\bf p}';\omega)$, and
the corresponding electron-phonon pairing interaction, which is the basis for
computing the superconducting properties. The electron-phonon spectral function
is good agreement with existing tunneling spectroscopy data except for the
spectral weight of the longitudinal phonon peak at
$\hbar\omega_{\text{LO}}=23\,\mbox{meV}$. We obtain an electron-phonon coupling
constant of $\lambda=1.057$, renormalized Coulomb interaction,
$\mu^{\star}=0.218$ and transition temperature $T_c=9.33\,\mbox{K}$. The
corresponding strong-coupling gap at $T=0$ is modestly enhanced,
$\Delta_0=1.55\,\mbox{meV}$, compared to the weak-coupling BCS value
$\Delta_0^{\text{wc}}=1.78\,k_{\mbox{b}}\,T_c= 1.43\,\mbox{meV}$. The
superconducting gap function exhibits substantial anisotropy on the Fermi
surface. We analyze the distribution of gap anisotropy and compute the
suppression of the superconducting transition temperature using a
self-consistent T-matrix theory for quasiparticle-impurity scattering to
describe Niobium doped with non-magnetic impurities. We compare these results
with experimental results on Niobium SRF cavities doped with Nitrogen
impurities.
