  Studying the local moment and 5$f$-electron occupations sheds insight into
the electronic behavior in actinide materials. X-ray absorption spectroscopy
(XAS) has been a powerful tool to reveal the valence electronic structure when
assisted with theoretical calculations. However, the analysis currently taken
in the community on the branching ratio of the XAS spectra generally does not
account for the hybridization effects between local $f$-orbitals and conduction
states. In this paper, we discuss an approach which employs the DFT+Gutzwiller
rotationally-invariant slave boson (DFT+GRISB) method to obtain a local
Hamiltonian for the single-impurity Anderson model (SIAM), and calculates the
XAS spectra by the exact diagonalization (ED) method. A customized numerical
routine was implemented for the ED XAS part of the calculation. By applying
this technique to the recently discovered 5$f$-electron topological Kondo
insulator PuB$_4$, we determined the signature of 5$f$-electronic correlation
effects in the theoretical X-ray spectra. We found that the Pu 5$f$-6$d$
hybridization effect provides an extra channel to mix the $j=5/2$ and $7/2$
orbitals in the 5$f$ valence. As a consequence, the resulting electron
occupation number and spin-orbit coupling strength deviate from the
intermediate coupling regime.
