<i>Ab initio</i> derivation of low-energy Hamiltonians for systems with strong spin-orbit interaction: Application to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="normal">Ca</mml:mi><mml:mn>5</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">Ir</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>12</mml:mn></mml:msub></mml:math>

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