  We have investigated the electronic structure of the transition metal oxide
Ca3Ru2O7 within density functional theory using the generalized gradient
approximation while considering strong Coulomb correlations in the framework of
the fully relativistic spin-polarized Dirac linear muffin-tin orbital
band-structure method. Ca3Ru2O7 can be classified as a Mott insulator since it
was expected to be metallic from band structure calculations. We have
investigated the resonant inelastic x-ray scattering spectra at the Ru and Ca
K, L3, and M3 edges as well as at the O K edge. The experimentally measured
resonance inelastic x-ray spectrum of Ca3Ru2O7 at the Ru L3 edge possesses a
sharp feature below 2 electron-volts corresponding to transitions within the Ru
t_2g levels. The excitation located from 2 to 4 electron-volts is due to t2g ->
e_g transitions. The third wide structure situated at 4.5-11 electron-volts
appears due to transitions between the Ru 4dO states derived from the tails of
oxygen 2p states and t_2g and e_g states. The measured resonance inelastic
x-ray spectra at the Ru L3 and M3 edges are very similar. However, the
corresponding measured resonance inelastic x-ray spectra at the Ca site quite
differ from each other due to the significant difference in the widths of
core-levels.
