Generalized gradient approximation (GGA) John Perdew Temple University From the beginning, LDA was more successful in solid state physics (with its emphasis on geometric structures, lattice vibration frequencies, surface energies and work functions) than in quantum chemistry (with its emphasis on reaction energies including atomization energies). It was natural to look for improvement from the second-order gradient expansion of the exchange-correlation energy, valid for slowly-varying densities. In fact, this was tried by Ma and Brueckner, but with dismally bad results (e.g., positive correlation energies for atoms). Langreth and Perdew analyzed the gradient expansion, showing that it violated several of the exact constraints on the hole that LDA satisfied. This analysis led to approximations in which the exchange-correlation energy density at depends upon and , but in a way that is proportional to only for slowly-varying densities. The PW91 GGA based upon satisfying exact constraints on the hole agreed closely with the PBE GGA based on satisfaction of selected exact constraints on the energy functional, an early but misleading success While these GGAs greatly reduced the errors of atomization energies, they were found over time to show serious limitations. They violated many known exact constraints, and over-corrected LDA lattice constants, vibration frequencies, and surface energies. They also lost the approximate description of intermediate-range van der Waals interaction that LDA had shown.