  We study electronic and magnetic properties of the quasi-one-dimensional
spin-1/2 magnet Ba3Cu3Sc4O12 with a distinct orthogonal connectivity of CuO4
plaquettes. An effective low-energy model taking into account spin-orbit
coupling was constructed by means of first-principles calculations. On this
basis a complete microscopic magnetic model of Ba3Cu3Sc4O12, including
symmetric and antisymmetric anisotropic exchange interactions, is derived. The
anisotropic exchanges are obtained from a distinct first-principles numerical
scheme combining, on one hand, the local density approximation taking into
account spin-orbit coupling, and, on the other hand, projection procedure along
with the microscopic theory by Toru Moriya. The resulting tensors of the
symmetric anisotropy favor collinear magnetic order along the structural chains
with the leading ferromagnetic coupling J1 = -9.88 meV. The interchain
interactions J8 = 0.21 meV and J5 = 0.093 meV are antiferromagnetic. Quantum
Monte Carlo simulations demonstrated that the proposed model reproduces the
experimental Neel temperature, magnetization and magnetic susceptibility data.
The modeling of neutron diffraction data reveals an important role of the
covalent Cu-O bonding in Ba3Cu3Sc4O12.
