  The strength and effect of Coulomb correlations in the (superconducting when
hydrated) x~1/3 and ``enhanced'' x~2/3 regimes of Na(x)CoO2 are evaluated using
the correlated band theory LDA+U method. Our results, neglecting quantum
fluctuations, are: (1) allowing only ferromagnetic order, there is a critical
U_c = 3 eV, above which charge disproportionation occurs for both x=1/3 and
x=2/3, (2) allowing antiferromagnetic order at x=1/3, U_c drops to 1 eV for
disproportionation, (3) disproportionation and gap opening occur
simultaneously, (4) in a Co(3+)-Co(4+) ordered state, antiferromagnetic
coupling is favored over ferromagnetic, while below U_c ferromagnetism is
favored. Comparison of the calculated Fermi level density of states compared to
reported linear specific heat coefficients indicates enhancement of the order
of five for x~0.7, but negligible enhancement for x~0.3. This trend is
consistent with strong magnetic behavior and local moments (Curie-Weiss
susceptibility) for x>0.5 while there no magnetic behavior or local moments
reported for x<0.5. We suggest that the phase diagram is characterized by a
crossover from effective single-band character with U >> W for x>0.5 into a
three-band regime for x<0.5, where U --> U_eff <= U/\sqrt(3) ~ W and
correlation effects are substantially reduced.
