  Motivated by recent experiments, here we study the indirect interactions
between magnetic impurities deposited on top of a clean Pb(110) surface induced
by the underlying conduction electrons. Our approach makes use of \textit{ab
initio} calculations to characterize the clean Pb(110) surface and avoids
self-consistency, a feature that greatly reduces the computational cost. In
combination with 2$^{\text{nd}}$ order perturbation theory in the microscopic
\textit{s-d} exchange parameter $J_{K}$ between a magnetic adatom and the
conduction electrons, we are able to systematically derive the
Ruderman-Kittel-Kasuya-Yosida (RKKY), the Dzyaloshinskii-Moriya (DM) and the
anisotropic tensor interactions emerging at the Pb(110) surface between
magnetic impurities. The only adjustable parameter is $J_{K}$, which is fitted
to reproduce the experiments. Our results show important anisotropy effects
arising both from the rectangular geometry of the (110) unit cell, and from the
strong Rashba spin-orbit interaction due to the broken inversion symmetry at
the Pb(110) surface. In addition to Pb(110), the characterization of the
indirect spin interactions described here could be extended to other realistic
metallic surfaces for weakly-coupled impurities, and would enable to fabricate
atomic-size nanostructures with engineered interactions and on-demand magnetic
properties, anticipating useful applications in nanotechnology.
