  We present results of studies of charge disproportionation (CD) and spin
differentiation in Na$_x$CoO$_2$ using the correlated band theory approach
(local-density approximation+Hubbard U:LDA+U). The simultaneous CD and gap
opening can be followed through a first order charge disproportionation
transition. By comparison with experiments, we propose a value of the Coulomb
repulsion strength U($x$) that has significant dependence on the carrier
concentration $x$, for which we obtain a phase diagram. The connection between
the oxygen height and effects of on-site correlation is also reported.
