  Recent experiments have shown the signatures of Majorana bound states at the
ends of magnetic chains deposited on a superconducting substrate. Here, we
employ first principles calculations to directly investigate the topological
properties of $3d$ transition metal nanochains (i.e., Mn, Cr, Fe and Co). In
contrast to the previous studies [Nadj-Perge et al. Science 346, 602 (2014) and
Ruby et al. Nano Lett. 17, 4473 (2017)], we found the exact tight binding
models in the Wannier orbital basis for the isolated chains as well as for the
surface--deposited wires. Based on these models, we calculate topological
invariant of $\mathbb{Z}_2$ phase for all systems. Our results for the isolated
chains demonstrate the existence of the topological phase only in the Mn and Co
systems. We considered also a non-collinear magnetic order as a source of the
non--trivial topological phase and found that this type of magnetic order is
not a stable ground state in the Fe and Co isolated chains. Further studies
showed that a coupling between the chain and substrate leads to strong
modification of the band structure. Moreover, the analysis of the topological
invariant indicates a possibility of emergence of the topological phase in all
studied nanochains deposited on the Pb surface. Therefore, our results
demonstrate an important role of the coupling between deposited atoms and a
substrate for topological properties of nanosystems.
