Supercell band calculations and correlation for high-$T_C$ copper oxide superconductors

First principle band calculations based on local versions of density functional theory (DFT), together with results from nearly free-electron models, can describe many typical but unusual properties of the high-$T_C$ copper oxides. The methods and a few of the most important results are reviewed. Some additional calculations are presented and the problems with the commonly used approximate versions of DFT for oxides are discussed with a few ideas for corrections. It is concluded that rather modest corrections to the approximate DFT, without particular assumptions about strong correlation, can push the ground state towards anti-ferro magnetic (AFM) order. Spin fluctuations interacting with phonons are crucial for the mechanism of superconductivity in this scenario.