Relation between crystal structure and optical properties in the correlated blue pigment YIn$_{1-x}$Mn$_x$O$_3$

A material's properties and functionalities are determined by its chemical constituents and the atomic arrangement in which they crystallize. For the recently discovered pigment YIn$_{1-x}$Mn$_x$O$_3$, for instance, it had been surmised that its bright blue color owes to an unusual, trigonal bipyramidal, oxygen coordination of the manganese impurities. Here, we demonstrate that, indeed, a direct correspondence between details of the local Mn environment and the pigment's blue color holds: Combining realistic many-body calculations (dynamical mean-field theory to treat the quasi-atomic Mn-multiplets at low doping x=0.08) with an effective medium description (Kubelka-Munk model to describe scattering in a milled pigment sample), we find that only a Mn-coordination polyhedra consisting of two distorted oxygen pyramids results in a diffuse reflectance commensurate with the experimental blue color. We motivate that the distortion of the bipyramid helps circumventing atomic selection rules, allowing for dipolar $d$-$d$ transitions and creating the desired two-peak absorption profile.