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The recently developed self-consistent vertex corrected GW method is used tocalculate the 5f electronic structure in delocalized $\alpha$-U and localized$\delta$-Pu, each of which is confirmed by the historical experimentalapproaches of direct and inverse photoemission. Tender X-Ray EmissionSpectroscopy (XES), in a novel application to 5f electronic structure, is usedto experimentally prove the existence of 5f delocalization in $\alpha$-U.

The relativistic 5f electronic structure of delocalized $\alpha$-U and  localized $\delta$-Pu from the self consistent vertex corrected GW approach  and X-ray Emission Spectroscopy