  Ab initio band theory including correlations due to intra-atomic repulsion is
applied to study charge disproportionation and charge- and spin-ordering in
insulating Na$_{0.5}$CoO$_2$. Various ordering patterns (zigzag and two
striped) for four-Co supercells are analyzed before focusing on the observed
"out-of-phase stripe" pattern of antiferromagnetic Co$^{4+}$ spins along
charge-ordered stripes. This pattern relieves frustration and shows distinct
analogies with the cuprate layers: a bipartite lattice of antialigned spins,
with axes at 90 degree angles. Substantial distinctions with cuprates are also
discussed, including the tiny gap of a new variant of "charge transfer" type
within the Co 3d system.
