  We present an ab initio derivation method for effective low-energy
Hamiltonians of material with strong spin-orbit interactions. The effective
Hamiltonian is described in terms of the Wannier function in the spinor form,
and effective interactions are derived with the constrained random phase
approximation (cRPA) method. Based on this formalism and the developed code, we
derive an effective Hamiltonian of a strong spin-orbit interaction material
Ca5Ir3O12. This system consists of three edge-shared IrO6 octahedral chains
arranged along the c axis, and the three Ir atoms in the ab plane compose a
triangular lattice. For such a complicated structure, we need to set up the
Wannier spinor function under the local coordinate system. We found that a
density-functional band structure near the Fermi level is formed by local dxy
and dyz orbitals. Then, we constructed the ab initio dxy/dyz model. The
estimated nearest neighbor transfer t is close to 0.2 eV, and the cRPA onsite U
and neighboring V electronic interactions are found to be 2.4-2.5 eV and 1 eV,
respectively. The resulting characteristic correlation strength defined by
(U-V)/t is above 7, and thus this material is classified as a strongly
correlated electron system. The onsite transfer integral involved in the
spin-orbit interaction is 0.2 eV, which is comparable to the onsite exchange
integrals near 0.2 eV, indicating that the spin-orbit-interaction physics would
compete with the Hund physics. Based on these calculated results, we discuss
possible rich ground-state low-energy electronic structures of spin, charge and
orbitals with competing Hund, spin-orbit and strong correlation physics.
