Microscopic scattering approach to in-gap states: Cr adatoms on superconducting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mtext>−</mml:mtext> <mml:msub> <mml:mi>Bi</mml:mi> <mml:mn>2</mml:mn> </mml:m

We develop a microscopic scattering formalism to describe Yu-Shiba-Rusinov (YSR) states due to a single Cr adatom on the Bi-terminated surface of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>β</a:mi> <a:mtext>−</a:mtext> <a:msub> <a:mi>Bi</a:mi> <a:mn>2</a:mn> </a:msub> <a:mi>Pd</a:mi> </a:mrow> </a:math> , by combining Wannier functions with a real-space Green's function approach in the Bogoliubov–de Gennes formalism. Our framework reproduces key scanning tunneling spectroscopy features, including a single particle-hole asymmetric YSR peak and isotropic <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>d</b:mi> <b:mi>I</b:mi> <b:mo>/</b:mo> <b:mi>d</b:mi> <b:mi>V</b:mi> </b:mrow> </b:math> maps around the impurity. Decomposing the YSR states reveals contributions from four nearly degenerate <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msub> <c:mi>C</c:mi> <c:mrow> <c:mn>4</c:mn> <c:mi>v</c:mi> </c:mrow> </c:msub> </c:math> representations, with energy broadening masking their individual signatures. Spin-orbit coupling induces partial spin polarization, while the spatial asymmetry between particle and hole components arises from Cr <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>d</d:mi> <d:mtext>−</d:mtext> <d:mi>Bi</d:mi> </d:mrow> <d:mo> </d:mo> <d:mi>p</d:mi> </d:math> hybridization. These results highlight the importance of realistic band structures and microscopic modeling for interpreting STM data and provide a foundation for studying impurity chains hosting topological excitations.