  The ground states of Na$_x$CoO$_2$ ($0.0<x<1.0$) is studied by the
LDA+Gutzwiller approach, where charge transfer and orbital fluctuations are all
self-consistently treated {\it ab-initio}. In contrast to previous studies,
which are parameter-dependent, we characterized the phase diagram as: (1)
Stoner magnetic metal for $x>0.6$ due to $a_{1g}$ van-Hove singularity near
band top; (2) correlated non-magnetic metal without $e_g^{\prime}$ pockets for
$0.3<x<0.6$; (3) $e_g^{\prime}$ pockets appear for $x<0.3$, and additional
magnetic instability involves. Experimental quasi-particle properties is well
explained, and the $a_{1g}$-$e_g^{\prime}$ anti-crossing is attributed to
spin-orbital coupling.
