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.
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