  Electronic, magnetic, and dynamical properties of the new superconducting
$\beta$-pyrochlore KOs$_2$O$_6$ and related RbOs$_2$O$_6$ and CsOs$_2$O$_6$
compounds are calculated and compared with experiment and contrasted with
structurally related spinel pyrochlores. The calculated susceptibility Stoner
enhancement (110%) and thermal mass enhancement $\lambda$ = 2.5-3 reflect
moderate but perhaps important Coulomb correlations. The K$^{+}$ ion optic mode
is found to be unstable, allowing large excursions of 0.5-0.6 \AA from its
ideal site of the K ion along $<111>$ directions. This dynamical mode is much
less anharmonic in the isostructural Rb and Cs compounds (with larger cations),
perhaps accounting for their progressively lower values of T$_c$. Electron
scattering from this very anharmonic mode may be the cause of the anomalous
concave-downward resistivity that is seen only in KOs$_2$O$_6$.
