This does not mean that the electron density is described uniformly throughout the whole chemical system, but instead at every point in space (usually determined on a 3D integration grid), the LDA energy functional is evaluated and summed up. This means that LDA is specifically good at describing slowly varying electron densities (e.g., simple metals) but not great for molecules and chemical bonding. Nowadays, it is mostly used in solid state physics for metals and should not be used for other chemical systems. The next rung are the DFs of the generalized gradient approximation (GGA) type. If the energy is expanded as functional of the electron density in a Taylor series, the step after LDA is to incorporate the first derivative of the electron density. After some removal of unphysical properties of the resulting gradient expansion approximation, the class of GGA functionals results. They are the most used class of DFs in material science due to the efficient algorithms for computing them paired with a good to very good accuracy for many systems. The GGA functional developed by Perdew, Burke and Ernzerhof (PBE) has become the quasi-standard in material science. This is partly due to the good performance but also partly due to early claims that it is "parameterfree". The only parameter-free DF thus is the exchange functional for the LDA by Dirac. But since the parameters in PBE are derived from physical boundary conditions to a (pretty unphysical) model system (the homogeneous electron gas) it was more appealing to many scientists compared to the fitting of parameters on accurate molecular data as is done, e.g., for BP86. In large benchmark studies, there is no notable performance difference. The third rungthe next step in the Taylor seriesleads to the class of meta-GGA functionals (e.g., TPSS, M06-L). Although they are used in molecular chemistry increasingly, they have not been applied to AS-ALD up to now. In contrast to the step from LDA to GGA, an increase in accuracy is not guaranteed. However, they form the basis for many modern DFs. The fourth rung leads to the hybrid functionals which include a fraction of "exact exchange" in the exchange functional. This term stems from using the HF expression for the exchange energy with the density from the DFT approach and leads to considerable improvement in accuracy, especially for main group chemistry, reaction barriers and optical properties. However, for the periodic boundary condition (PBC) calculations mostly used in AS-ALD, the large increase in computational cost (approx. 10x compared to GGA) often leads to these computations not being feasible.