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We present a fully relativistic first principles based theoretical approachfor the calculation of the spectral properties of magnetic impurities on thesurface of a superconducting substrate, providing a material specific frameworkfor the investigation of the Yu--Shiba--Rusinov (YSR) states. By using asuitable orbital decomposition of the local densities of states we discuss ingreat details the formation of the YSR states for an Mn adatom and for twokinds of Mn dimers placed on the Nb(110) surface and compare our results torecent experimental findings. In case of the adatom we find that the spin-orbitcoupling slightly shifts some of the YSR peaks and also the localspin-polarization on the Nb atoms have marginal effects to the their positions.Moreover, by scaling the exchange field on the Mn site we could explain thelack of the $d_{x^2-y^2}$-like YSR state in the spectrum. While our results fora close packed ferromagnetic dimer are in satisfactory agreement with theexperimentally observed splitting of the YSR states, in case of anantiferromagnetic dimer we find that the spin-orbit coupling is notsufficiently large to explain the splitting of the YSR states seen in theexperiment. Changing the relative orientation of the magnetic moments in thisdimer induces splitting of the YSR states and also shifts their energy, leadingeven to the formation of a zero bias peak in case of the deepest YSR state.

Relativistic first principles theory of Yu--Shiba--Rusinov states applied to an Mn adatom and Mn dimers on Nb(110)