  Electronic structure of V$_{15}$ magnetic molecules (K$_6$ [V$_{15}$ As$_6$
O$_{42}$ (H$_2$O)] \cdot 8H$_2$O)$ has been studied using LSDA+U band structure
calculations, and measurements of X-ray photoelectron (valence band, core
levels) and X-ray fluorescence spectra (vanadium K$\beta_5$ and L$_{2,3}$, and
oxygen K$\alpha$). Experiments confirm that vanadium ions are tetravalent in
V$_{15}$, and their local atomic structure is close to that of CaV$_3$O$_7$.
Comparison of experimental data with the results of electronic structure
calculations show that the LSDA+U method provides a description of the
electronic structure of V$_{15}$ which agrees well with experiments.
