  Transient X-ray and extreme ultraviolet (XUV) spectroscopies have become
invaluable tools for studying photoexcited dynamics due to their sensitivity to
carrier occupations and local chemical or structural changes. One of the most
studied mate-rials using transient XUV spectroscopy is $\alpha$-Fe$_2$O$_3$
because of its rich photoexcited dynamics, including small polaron formation.
The interpretation of carrier and polaron effects in $\alpha$-Fe$_2$O$_3$ is
currently done using a semi-empirical method that is not transferrable to most
materials. Here, an ab initio, Bethe-Salpeter equation (BSE) approach is
developed that can incorporate photoexcited state effects for arbitrary
materials systems. The accuracy of this approach is proven by calculating the
XUV absorption spectra for the ground, photoexcited, and polaron states of
$\alpha$-Fe$_2$O$_3$. Furthermore, the theoretical approach allows for the
projection of the core-valence excitons and different components of the X-ray
transition Hamiltonian onto the band structure, providing new insights into old
measurements. From this information, a physical intuition about the origins and
nature of the transient XUV spectra can be built. A route to extracting
electron and hole energies is even shown possible for highly angular momentum
split XUV peaks. This method is easily generalized to K, L, M, and N edges to
provide a general approach for analyzing transient X-ray absorption or
reflection data.
