Density functional theory
Density functional theory (DFT) is a theory of electronic structure that is written in terms of the electron density distribution. This contrasts with traditional methods in quantum mechanics which are based on the more complicated many-electron wavefunction.
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2 Early Models 3 Kohn-Sham Theory 4 Solids and Gases 5 References |
Modern DFT is in principle an exact theory, although in practice various approximations have to be made. In many cases DFT gives quite satisfactory results (in comparison to experimental data, or to more precise quantum methods) at a relatively low computational cost.
DFT has been very popular for calculations in solid state physics since the 1970s. However, it was not considered accurate enough for calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined. DFT is now the leading method for electronic structure calculations in both fields.
The first true density functional theory was developed by Thomas and Fermi in the 1920's. They calculated the energy of an atom by representing its kinetic energy as a functional of the electron density, combining this with the classical expressions for the nuclear-electron and electron-electron interactions (which can both also be represented in terms of the electron density).
Although this was an important first step, the Thomas-Fermi equation's accuracy was limited because it did not attempt to represent the exchange energy of an atom predicted by Hartree-Fock theory. An exchange energy functional was added by Dirac in 1928.
However, the Thomas-Fermi-Dirac theory remained rather inaccurate for most applications because it is difficult to represent kinetic energy with a density functional, and it neglects electron correlation entirely.
The DFT method was placed on an intellectually rigorous footing by Hohenberg and Kohn [1] in 1964, when they proved that in principle it is possible to calculate all molecular properties exactly with only a knowledge of the electron density. Kohn and Sham then presented a clever approximation to the kinetic energy functional in 1965 [2] which greatly improved the accuracy of DFT. However, this step forward was made at the cost of reintroducing the concept of 'orbitals' from wavefunction methods. The Kohn-Sham DFT is no longer constructed entirely out of explicit functionals of the density, although it has the great benefit of being in principle an exact theory.
The major problem with Kohn-Sham theory is that the exact functionals for exchange and correlation are not known. But approximations exist which permit the calculation of certain phyiscal quantities quite accurately. The most widely used approximation is the local density approximation, where the functional depends only on the density at the coordinate where the functional is evaluated. Generalized Gradient approximations are still local but also take into account the gradient of the density at the same coordinate. Using the latter (GGA) very good results for molecular geometries and ground state energies have been achieved. Many further incremental improvements have been made to DFT by developing better representations of the functionals.
In practice, Kohn-Sham theory can be applied in two distinct ways depending on what is being investigated. In the solid state, plane wave basis sets are used with periodic boundary conditions. Moreover, great emphasis is placed upon remaining consistent with the idealised model of a 'uniform electron gas', which exhibits similar behaviour to an infinite solid. In the gas and liquid phases, this emphasis is relaxed somewhat, as the uniform electron gas is a poor model for the behaviour of discrete atoms and molecules. Because of the relaxed constraints, a huge variety of exchange-correlation functionals have been developed for chemical applications. The most famous and popular of these is known as B3LYP [3-5]. The adjustable parameters of these functionals are generally fitted to a 'training set' of molecules.Description of the Theory
Early Models
Kohn-Sham Theory
Solids and Gases