Climbing the ladder of density functional approximations
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uction 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 calculation in materials science, condensed matter physics, and quantum 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 applications 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 interacting 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 quantum 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 G G density n ( r ), which is a function of position r in the material, G G and in the electron spin densities n↑ ( r ) and 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 provide us with answers to all of these questions. Fundamental quantum mechanics provides answers that are more rigorous and precise, but those answers are computationally very difficult to find. Using the direct quantum mechanical approach with correlated wave functions, it is necessary 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 electrons and the z-components of the spins of all N electrons, a very complicated object.8 We are usually more interested in
John P. Perdew, Department of Physics, Temple University; [email protected] DOI: 10.1557/mrs.2013.178
© 2013 Materials Research Society
MRS BULLETIN • VOLUME 38 • SEPTEMBER 2013 • www.mrs.org/bulletin
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CLIMBING THE LADDER OF DENSITY FUNCTIONAL APPROXIMATIONS
a simpler object such as the density (number of electrons per unit volume at each point in space), which we can obtain in principle if not always in practice by squaring the wave function and integrating over all the coordinates but one (the one representing the chosen point in space) and summing over all of the spins. In the Schrödinger equation, the motion of each electron is coupled to
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