Theory of K-edge resonant inelastic x-ray scattering and its application for La$_{0.5}$Sr$_{1.5}$MnO$_{4}$

We present a formula based on tight-binding approach for the calculation of K-edge resonant inelastic x-ray scattering spectrum for transition metal oxides, by extending the previous result [K. H. Ahn, A. J. Fedro, and M. van Veenendaal, Phys. Rev. B 79, 045103 (2009).] to include explicit momentum dependence and a basis with multiple core hole sites. We apply this formula to layered charge, orbital, and spin ordered manganites, La$_{0.5}$Sr$_{1.5}$MnO$_{4}$. The K-edge RIXS spectrum is found not periodic with respect to the actual reciprocal lattice, but approximately periodic with respect to the reciprocal lattice for the hypothetical unit cell with one core hole site. With experimental strcuture and reasonable tight-binding parameters, we obtain good agreement with experimental data, in particular, with regards to the large variation of the intensity with momentum. We find that the screening in La$_{0.5}$Sr$_{1.5}$MnO$_{4}$ is highly localized around the core hole site and demonstrate the potential of K-edge RIXS as a probe for the screening dynamics in materials.