743 © 2013 Materials Research Society MRS BULLETIN • VOLUME 38 • SEPTEMBER 2013 • www.mrs.org/bulletin Introduction The subject of my talk is the Kohn–Sham density functional theory (DFT). 1–6 This DFT uses auxiliary occupied orbitals. It is the most widely used method of electronic structure calcula- tion in materials science, condensed matter physics, and quan- tum chemistry. As evidence of this, the 18 most-cited physics papers published in the period from 1981 to 2010 have been identified. 7 Ten of these papers are concerned with DFT: either fundamental theory, development of approximations, or appli- cations and computational methodologies for DFT. Kohn–Sham density functional theory Materials consist of atoms, molecules, nanostructures, solids, and surfaces. They are all systems constructed from many inter- acting electrons and nuclei. Compared to the electrons, the nuclei are heavy and almost classical, and we often treat them as classical particles that obey Newton’s laws of motion. On the other hand, the electrons must be described using quan- tum mechanics. We are usually interested in the measurable ground-state properties of a system, such as the total energy, E, and changes of the total energy due to adding or removing an electron, stretching or compressing a bond, removing an atom, and other processes. We are also interested in the electron density ( ) G nr , which is a function of position G r in the material, and in the electron spin densities () ↑ G n r and () ↓ G n r . We further want to know the positions of the nuclei in equilibrium and the vibration frequencies for the nuclei. Kohn–Sham DFT can pro- vide us with answers to all of these questions. Fundamental quantum mechanics provides answers that are more rigorous and precise, but those answers are com- putationally very difficult to find. Using the direct quantum mechanical approach with correlated wave functions, it is nec- essary to solve the many-electron Schrödinger equation. The Hamiltonian or quantum-mechanical energy operator in this equation must include the kinetic energy of all the electrons, the interaction of each electron with an external potential (which is actually the Coulomb attraction between that electron and all of the nuclei), the Coulomb repulsion among the nuclei, and the Coulomb repulsion among the electrons. The wave function is therefore a function of the positions of all N elec- trons and the z-components of the spins of all N electrons, a very complicated object. 8 We are usually more interested in Climbing the ladder of density functional approximations John P. Perdew The following article is an edited transcript of the MRS Materials Theory Award presentation by John P. Perdew on November 26, 2012, at the Materials Research Society Fall Meeting in Boston. The Award “recognizes exceptional advances made by materials theory to the fundamental understanding of the structure and behavior of materials.” Kohn–Sham density functional theory is the most widely used method of electronic-structure calculation in materials physics and chemistry because it reduces the many-electron ground- state problem to a computationally tractable self-consistent one-electron problem. Exact in principle for the ground-state energy and electron density, it requires in practice an approximation to the density functional for the exchange-correlation energy. Common approximations fall on the rungs of a ladder, with higher rungs being more complicated to construct and use but potentially more accurate. Each rung of the ladder introduces an additional ingredient to the energy density. From bottom to top, the rungs are (1) the local spin density approximation, (2) the generalized gradient approximation (GGA), (3) the meta-GGA, (4) the hybrid functional, and (5) the generalized random phase approximation. The semi-local rungs (1–3) are important, because they are computationally eficient, they can be constructed non- empirically, they can serve as input to fourth-rung functionals, and the meta-GGA by itself can be accurate for equilibrium properties. Recent and continuing improvements to the meta-GGA are emphasized here. John P. Perdew, Department of Physics, Temple University; perdew@temple.edu DOI: 10.1557/mrs.2013.178