Spin-orbit excitons and electronic configuration of the $5d^4$ insulator Sr$_3$Ir$_2$O$_7$F$_2$

Here we report on the low-energy excitations within the paramagnetic spin-orbit insulator Sr$_3$Ir$_2$O$_7$F$_2$ studied via resonant inelastic X-ray scattering, \textit{ab initio} quantum chemical calculations, and model-Hamiltonian simulations. This material is a unique $d^{4}$ Ir$^{5+}$ analog of Sr$_3$Ir$_2$O$_7$ that forms when F ions are intercalated within the SrO layers spacing the square lattice IrO$_{6}$ bilayers of Sr$_3$Ir$_2$O$_7$. Due to the large distortions about the Ir$^{5+}$ ions, our computations demonstrate that a large single-ion anisotropy yields an $S$=1 ($L{\approx}$1, $J{\approx}$0) ground state wave function. Weakly coupled, excitonic modes out of the $S_z$=0 ground state are observed and are well-described by a phenomenological spin-orbit exciton model previously developed for $3d$ and $4d$ transition metal ions. The implications of our results regarding the interpretation of previous studies of hole-doped iridates close to $d^{4}$ fillings are discussed.