  Spin Hall effect in metallic Pt is studied with first-principles relativistic
band calculations. It is found that intrinsic spin Hall conductivity (SHC) is
as large as $\sim 2000 (\hbar/e)(\Omega {\rm cm})^{-1}$ at low temperature, and
decreases down to $\sim 200 (\hbar/e)(\Omega {\rm cm})^{-1}$ at room
temperature. It is due to the resonant contribution from the spin-orbit
splitting of the doubly degenerated $d$-bands at high-symmetry $L$ and $X$
points near the Fermi level. By modeling these near degeneracies by effective
Hamiltonian, we show that SHC has a peak near the Fermi energy and that the
vertex correction due to impurity scattering vanishes. We therefore argue that
the large spin Hall effect observed experimentally in platinum is of intrinsic
nature.
