The geometrical frustration in the magnetic pyrochlore oxides (A 2 B 2 O 7 ) causes interesting properties at the low temperatures in these materials. Er 2 Ti 2 O 7 is a the magnetic pyrochlore oxides with space group Fd m. Er local magnetic moments confined to planes orthogonal to the local 111 axes, so we can describe the system in a model so-called XY-like model. we consider the system by using an ab intio method, Density Functional Theory (DFT) with Fleur code in LDA, LDA+U, LDA+U+SOC approximation. To take into account the f electrons coulomb interactions correctly we employed LDA+U, a Hubbard method correction. We gained energy gap equal to 2.99 eV from analysis of the density of states. The spin-orbit coupling (SOC) is very effective in these materials, so we added SOC to the LDA+U calculations (LDA+U+SOC). Using LDA+U+SOC, we consider the effect of non-collinear magnetism. The LDA+U+SOC calculations in the all-in spin configuration give rise to the magnetic moments 8.2 for Er 2 Ti 2 O 7 . This value deviates from the experimental results. The reason is that the DFT calculations are trapped in the local minima, so we need more sophisticated methods to find the global minimum in our electronic structure calculations. The analysis of the density of states of Erbium, titanium and oxygen in the crystal, show that Er 4f state hybridizes with O' 2p weakly and so the 4f orbitals remain well localized.
