  The band structures of ordered and thermally disordered
Li$_2$Mo$_{12}$O$_{34}$ are calculated by use of ab-initio DFT-LMTO method with
focus on the behavior of the Mo 3d-core levels. It is shown that thermal
disorder and zero-point motion lead to substantial core level broadening, and
the broadening at room temperature is predicted to be sufficiently larger than
at zero degrees to allow for a detection by XPS measurements. However, real
purple bronze has 10 percent of Li vacancies and static disorder will attenuate
the $T$-dependent broadening. It is argued that core level spectroscopies could
be a useful tool for measuring of thermal disorders in many materials,
especially for those with minor static disorder. Studies of core levels in
magnetic materials will be helpful for an understanding of $T$-dependent spin
moments.
