**8. Projection-augmented plane wave method (PAW)**

We are always looking for the precision of the computation and thus to minimize the reasonable computation time as much as possible, we advise to use the least hard potentials possible, that is to say requiring the fewest plane waves to ensure convergence. The method of plane waves augmented by projection (PAW) (Projector Augmented Waves) [17] best meets this requirement, which explains their use in this thesis. The projection augmented plane wave method is an approach developed by P. Blochl, which models heart states from wave functions for an isolated atom; it assumes that these states are not modified when the atom is placed in a compound (approximation of frozen hearts). The PAW approach consists of using two kinds of basic functions: one uses partial atomic wave functions inside cores and the other uses a set of functions. Wave planes in the interstitial region. There is then a separation of space into two parts. The cause of this separation is the behavior of the effective potential seen by the valence shell electrons: in the interstitial zone, the total electron density is low. The wave functions oscillate rapidly which makes it possible to give strong variations in the amplitude of the total electron density. Because of these variations, the Fourier decomposition of the potential created by this charge distribution has components large wave vectors, which disadvantages the treatment in plane waves. We are therefore led to write in a different way the wave functions of the valence electrons in these two regions of space:

