Formation of the one-electron structure of CoO$_{2}$ layers

We use the N-order muffin tin orbital method to reduce the LDA band structure of a CoO$_{2}$ layer, the principal element of the novel superconductor Na$_{x}$CoO$_{2}\cdot y$H$_2$O, to a three-band model and to understand it in terms of simple tight binding. We show how to introduce correctly the trigonal distortion on top of the approximately cubic local symmetry. Some of the conclusions are: (i) no adequate description of the electronic structure is possible in a one-band model and (ii) third nearest neighbor interaction appears to be important, which can be understood in terms of a superexchange via intermediate Co $e_g$ states. We also present a set of parameters for what we believe is the minimal tight-binding model.