Density functional theory/Introduction

Density functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function, which in this case is the spatially dependent electron density. Hence the name density functional theory comes from the use of functionals of the electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry. DFT has been very popular for calculations in solid-state physics since the 1970s. However, DFT was not considered accurate enough for calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better model the exchange and correlation interactions. In many cases the results of DFT calculations for solid-state systems agree quite satisfactorily with experimental data. Computational costs are relatively low when compared to traditional methods, such as Hartree–Fock theory and its descendants based on the complex many-electron wavefunction. Despite recent improvements, there are still difficulties in using density functional theory to properly describe intermolecular interactions]], especially van der Waals forces (dispersion); charge transfer excitations; transition states, global potential energy surfaces, dopant interactions and some other strongly correlated systems; and in calculations of the band gap and ferromagnetism in semiconductors.Assadi M.H.N et al. Theoretical study on copper's energetics and magnetism in TiO2 polymorphs Journal of Applied Physics 2013 113 23 233913 http://arxiv.org/ftp/arxiv/papers/1304/1304.1854.pdf 10.1063/1.4811539 Its incomplete treatment of dispersion can adversely affect the accuracy of DFT (at least when used alone and uncorrected) in the treatment of systems which are dominated by dispersion (e.g. interacting noble gas atoms)Tanja Van Mourik Gdanitz, Robert J. 2002 A critical note on density functional theory studies on rare-gas dimers Journal of Chemical Physics 116 22 9620–9623 10.1063/1.1476010 2002JChPh.116.9620V or where dispersion competes significantly with other effects (e.g. in biomolecules).Jiří Vondrášek Bendová, Lada; Klusák, Vojtěch; and Hobza, Pavel 2005 Unexpectedly strong energy stabilization inside the hydrophobic core of small protein rubredoxin mediated by aromatic residues: correlated ab initio quantum chemical calculations Journal of the American Chemical Society 15725017 127 8 2615–2619 10.1021/ja044607h The development of new DFT methods designed to overcome this problem, by alterations to the functional and inclusion of additional terms to account for both core and valence electrons Stefan Grimme 2006 Semiempirical hybrid density functional with perturbative second-order correlation Journal of Chemical Physics 16438568 124 3 034108 10.1063/1.2148954 2006JChPh.124c4108G or by the inclusion of additive terms,Urs Zimmerli Parrinello, Michele; and Koumoutsakos, Petros 2004 Dispersion corrections to density functionals for water aromatic interactions Journal of Chemical Physics 15268413 120 6 2693–2699 10.1063/1.1637034 2004JChPh.120.2693Z Stefan Grimme 2004 Accurate description of van der Waals complexes by density functional theory including empirical corrections Journal of Computational Chemistry 15224390 25 12 1463–1473 10.1002/jcc.20078O. Anatole Von Lilienfeld Tavernelli, Ivano; Rothlisberger, Ursula; and Sebastiani, Daniel 2004 Optimization of effective atom centered potentials for London dispersion forces in density functional theory Physical Review Letters 15524874 93 15 153004 10.1103/PhysRevLett.93.153004 2004PhRvL..93o3004VAlexandre Tkatchenko Scheffler, Matthias 2009 Accurate Molecular Van Der Waals Interactions from Ground-State Electron Density and Free-Atom Reference Data Physical Review Letters 102 073005 10.1103/PhysRevLett.102.073005 7 19257665 2009PhRvL.102g3005T is a current research topic. Overview of method Although density functional theory has its conceptual roots in the Thomas–Fermi model, DFT was put on a firm theoretical footing by the two Hohenberg–Kohn theorems (H–K).Inhomogeneous electron gas Physical Review 1964 Pierre Hohenberg Walter Kohn 136 3B B864–B871 10.1103/PhysRev.136.B864 1964PhRv..136..864H The original H–K theorems held only for non-degenerate ground states in the absence of a magnetic field, although they have since been generalized to encompass these.Levy Mel 1979 Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem Proceedings of the National Academy of Sciences United States National Academy of Sciences 76 12 6062–6065 10.1073/pnas.76.12.6062 1979PNAS...76.6062L Vignale G. Mark Rasolt Density-functional theory in strong magnetic fields 1987 Physical Review Letters 59 20 2360–2363 American Physical Society 10.1103/PhysRevLett.59.2360 10035523 1987PhRvL..59.2360V The first H–K theorem demonstrates that the ground state properties of a many-electron system are uniquely determined by an electron density that depends on only 3 spatial coordinates. It lays the groundwork for reducing the many-body problem of N electrons with 3N spatial coordinates to 3 spatial coordinates, through the use of functionals of the electron density. This theorem can be extended to the time-dependent domain to develop time-dependent density functional theory (TDDFT), which can be used to describe excited states. The second H–K theorem defines an energy functional for the system and proves that the correct ground state electron density minimizes this energy functional. Within the framework of Kohn–Sham DFT (KS DFT), the intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of non-interacting electrons moving in an effective potential. The effective potential includes the external potential and the effects of the Coulomb interactions between the electrons, e.g., the exchange and correlation interactions. Modeling the latter two interactions becomes the difficulty within KS DFT. The simplest approximation is the local-density approximation (LDA), which is based upon exact exchange energy for a uniform electron gas, which can be obtained from the Thomas–Fermi model, and from fits to the correlation energy for a uniform electron gas. Non-interacting systems are relatively easy to solve as the wavefunction can be represented as a Slater determinant of orbitals. Further, the kinetic energy functional of such a system is known exactly. The exchange-correlation part of the total-energy functional remains unknown and must be approximated. Another approach, less popular than KS DFT but arguably more closely related to the spirit of the original H-K theorems, is orbital-free density functional theory (OFDFT), in which approximate functionals are also used for the kinetic energy of the non-interacting system. Note: Recently, another foundation to construct the DFT without the Hohenberg–Kohn theorems is getting popular, that is, as a Legendre transformation from external potential to electron density. See, e.g., Density Functional Theory – an introduction, Rev. Mod. Phys. 78, 865–951 (2006), and references therein. A book, 'The Fundamentals of Density Functional Theory' written by H. Eschrig, contains detailed mathematical discussions on the DFT; there is a difficulty for N-particle system with infinite volume; however, we have no mathematical problems in finite periodic system (torus).