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We studied the d-d excitations in Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ by Cu L$_3$-edge Resonant Inelastic X-ray Scattering with azimuthal rotation of the polarizations in the sample space, and analyzed the experimental response with a single cluster model. Utilizing group theory classification to incorporate two-hole Coulomb interactions via a Green's function approach, the calculations suggest that the d-d excitations can be described by a two-hole ground state composed of a Hund’s triplet mixed with a Zhang-Rice singlet state in all the studied doping regions. Our results highlight the existence of the spin-triplet two-hole state for doped cuprates, and that Hund’s triplet physics could play a role in the optimally doped and over-doped regions.
