  X-Ray Thomson Scattering (XRTS) is an important experimental technique used
to measure the temperature, ionization state, structure, and density of warm
dense matter (WDM). The fundamental property probed in these experiments is the
electronic dynamic structure factor (DSF). In most models, this is decomposed
into three terms [Chihara, J. Phys. F: Metal Phys. {\bf 17}, 295 (1987)]
representing the response of tightly bound, loosely bound, and free electrons.
Accompanying this decomposition is the classification of electrons as either
bound or free, which is useful for gapped and cold systems but becomes
increasingly questionable as temperatures and pressures increase into the WDM
regime. In this work we provide unambiguous first principles calculations of
the dynamic structure factor of warm dense beryllium, independent of the
Chihara form, by treating bound and free states under a single formalism. The
computational approach is real-time finite-temperature time-dependent density
functional theory (TDDFT) being applied here for the first time to WDM. We
compare results from TDDFT to Chihara-based calculations for experimentally
relevant conditions in shock-compressed beryllium.
