  It has often been suggested that correlation effects suppress the small e_g'
Fermi surface pockets of NaxCoO_2 that are predicted by LDA, but absent in
ARPES measurements. It appears that within the dynamical mean field theory
(DMFT) the ARPES can be reproduced only if the on-site energy of the eg'
complex is lower than that of the a1g complex at the one-electron level, prior
to the addition of local correlation effects. Current estimates regarding the
order of the two orbital complexes range from -200 meV to 315 meV in therms of
the energy difference. In this work, we perform density functional theory
calculations of this one-electron splitting \Delta= \epsilon_a1g-\epsilon_e_g'
for the full two-layer compound, Na2xCo2O4, accounting for the effects of Na
ordering, interplanar interactions and octahedral distortion. We find that
\epsilon a_1g-\epsilon e_g' is negative for all Na fillings and that this is
primarily due to the strongly positive Coulomb field created by Na+ ions in the
intercalant plane. This field disproportionately affects the a_1g orbital which
protrudes farther upward from the Co plane than the e_g' orbitals. We discuss
also the secondary effects of octahedral compression and multi-orbital filling
on the value of \Delta as a function of Na content. Our results indicate that
if the e_g' pockets are indeed suppressed that can only be due to nonlocal
correlation effects beyond the standard DMFT.
