  K$_x$CoO$_2$ shares many similarities with Na$_x$CoO$_2$, as well as some
important differences (no hydration-induced superconductivity has been
reported). At $T_{c2}$=20 K, K$_{0.5}$CoO$_2$ becomes an insulator with a tiny
optical gap as happens in Na$_{0.5}$CoO$_2$ at 52 K. This similarity, with a
known common structure, enables direct comparisons to be made. Using the
K-zigzag structure recently reported and the local density approximation, we
compare and contrast these cobaltates at x=0.5. Although the electronic
structures are quite similar as expected, substantial differences are observed
near the Fermi level. These differences are found to be attributable mostly to
the chemical, rather than structural difference: although Na is normally
considered to be fully ion, K has somewhat more highly ionic character than
does Na in these cobaltates.
