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Charge disproportionation (CD) and spin differentiation in Na$_{1/2}$CoO$_2$are studied using the correlated band theory approach. The simultaneous CD andgap opening seen previously is followed through a first order chargedisproportionation transition 2Co$^{3.5+} \to$ Co$^{3+}$+Co$^{4+}$, whose ionicidentities are connected more closely to spin (S=0, S=1/2 respectively) than toreal charge. Disproportionation in the Co $a_g$ orbital is compensated byopposing charge rearrangement in other 3d orbitals. At the transition large andopposing discontinuities in the (all-electron) kinetic and potential energiesare slightly more than balanced by a gain in correlation energy. The CD stateis compared to characteristics of the observed charge-ordered insulating phasein Na$_{1/2}$CoO$_2$, suggesting the Coulomb repulsion value $U$ isconcentration-dependent, with $U(x=1/2)\simeq$3.5 eV.Comment: 4 pages and 4 embedded figure

physical review letters

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