  Resonant inelastic x-ray scattering (RIXS) experiment was performed at the Mn
$L_3$ edge. A 10 $\%$ Mn-doped Sr$_{3}$Ru$_{2}$O$_{7}$ compound, where the
Mn$^{3+}$ ions are in the 3$d^4$ state, were probed for $dd$ excitations. The
dilute doping concentration allows one to treat the dopant Mn$^{3+}$ ions as
effectively free in the host ruthenium compound. The local nature of $dd$ RIXS
spectroscopy permits one to use a single-site model to simulate the
experimental spectra. The simulated spectra reproduces the in-plane [100]
experimental RIXS spectrum. We also predict the intensity for the in-plane
[110] direction and the out-of-plane spin orientation configuration [001].
Based on our single-ion model we were able to fit the experimental data to
obtain the crystal field parameters, the 10Dq value, and the intra-orbital
spin-flip energy 2$\mathcal{J}$(or $3J_{H}$, where $J_{H}$ is the Hund's
energy) of the Mn$^{3+}$ ion. Utilizing our computed RIXS quantum transition
amplitudes between the various $d$ orbitals of the Mn$^{3+}$ ion, the
expression for the Kramers-Heisenberg cross section, and a self-consistent
fitting procedure we also identify the energy boundaries of the non-spin-flip
and spin-flip $dd$ excitations present in the experimental data. From our
fitting procedure we obtain $2\mathcal{J} (3J_{H})=2.06$ eV, a value which is
in excellent agreement with that computed from the free ion Racah parameters.
We also identified the charge transfer boundary. In addition to predicting the
microscopic parameters, we find a quantum spin-flip transition in the non-cross
($\sigma_{in}-\sigma_{out}$, $\pi_{in}-\pi_{out}$) x-ray polarization channels
of the $dd$ RIXS spectra. A similar transition, was previously predicted to
occur in the $\pi-\pi$ channel of the magnon spectrum in the non-collinear
non-coplanar Kagome compound composed of Cu$^{2+}$ 3d$^{9}$ ion.
