Angle-resolved Resonant Inelastic X-ray Scattering in NaV$_2$O$_5$

Angle-dependent resonant inelastic x-ray scattering spectrum at the V-$L_3$ edge is analyzed to determine the origin of the V-$dd$ peak in NaV2O5 [1]. Experiment shows that as the incident photon polarization is rotated from the $b$ to $c$ axis, the V-$dd$ peak grows relative to the V-d/O-p peak and its energy loss becomes larger [2]. Our first-principles calculations demonstrate that such growth must involve both the unoccupied dxy and dxz/dyz bands. Neither the dxz/dyz nor dxy excitation alone can reproduce the ratio change. For the $bc$ scan, the light first samples the dxy orbital and then the dxz/dyz orbital. Slightly detuning the incident energy away from the resonant edge reveals that the dxy band is slightly lower in energy and much narrower than the dxz/dyz band. The results suggest that our previous analysis of the correlation splitting of the dxy band is valid [3]. [1] G. P. Zhang, T. A. Callcott, G. T. Woods, {\it et al} Phys. Rev. Lett. {\bf 88}, 077401 (2002); [2] G. P. Zhang {\it et al.}, Phys. Rev. B {\bf 65}, 165107 (2002); [3] G. P. Zhang and T. A. Callcott, Phys. Rev. B {\bf 73}, 125102 (2006).