  We reanalyze the condition for metallic ferromagnetism in the framework of
the tight-binding approximation and investigate the consequences of the
inter-site Coulomb interactions using the Hartree-Fock approximation. We first
consider a non-degenerate $s$ band and we show that the inter-site interactions
modify the occurrence of ferromagnetism, and we derive a generalized Stoner
criterion. We analyze the main effects due to the renormalization of the
hopping integrals by the inter-site Coulomb interactions. These effects are
strongly dependent on the relative values of the inter-site electron-electron
interactions and on the shape of the density of states as illustrated by a
study of cubic crystals from which we establish general trends. We then
investigate a realistic $spd$ tight-binding model, including intra (Coulomb and
exchange) and inter-site charge-charge Coulomb integrals. This model is used to
study the electronic structure (band structure, densities of states, magnetic
moment) of bulk ferromagnetic 3d transition metals Fe(bcc), Co(hcp and fcc) and
Ni(fcc). An excellent agreement with local spin density functional calculations
is obtained for the three metals, in particular concerning the relative widths
of the majority and minority spin bands. Thus our tight-binding Hartree-Fock
model provides a consistent interpretation of this effect.
