Local-density approximation

The local-density approximation (LDA) is an approximation of the exchange-correlation (XC) energy functional in density functional theory (DFT) by taking the XC energy of an electron in a homogeneous electron gas of a density equal to the density at the electron in the system being calculated (which in general is inhomogeneous). This approximation was applied to DFT by Kohn and Sham in an early paper.cite journal | author = W. Kohn and L. J. Sham | title = Self-Consistent Equations Including Exchange and Correlation Effects | journal = Phys. Rev. | volume = 140 | pages = A1133–A1138 | year = 1965 | url = http://link.aps.org/abstract/PR/v140/pA1133 | doi = 10.1103/PhysRev.140.A1133 | format = abstract ]

The Hohenberg-Kohn theorem states that the energy of the ground state of a system of electrons is a functional of the electronic density, in particular the exchange and correlation energy is also a functional of the density (this energy can be seen as the quantum part of the electron-electron interaction). This XC functional is not known exactly and must be approximated. [cite journal | author = P. Hohenberg and W. Kohn | title = Inhomogeneous Electron Gas | journal = Phys. Rev. | volume = 136 | pages = B864–B871 | year = 1964 | url = http://link.aps.org/abstract/PR/v136/pB864 | doi = 10.1103/PhysRev.136.B864 | format = abstract ]

LDA is the simplest approximation for this functional, it is "local" in the sense that the electron exchange and correlation energy at any point in space is a function of the electron density at that point only. [cite journal | author = John R. Smith | title = Beyond the Local-Density Approximation: Surface Properties of (110) W | journal = Phys. Rev. Lett. | volume = 25 | issue = 15 | pages = 1023–1026 | year = 1970 | url = http://link.aps.org/abstract/PRL/v25/p1023 | doi = 10.1103/PhysRevLett.25.1023 | format = abstract]

The XC energy functional is the sum of an exchange functional $E_x,;$ and a correlation functional $E_c,;$

: $E_{xc} = E_x + E_c ,;$

Exchange

LDA uses the exchange for the uniform electron gas of a density equal to the density at the point where the exchange is to be evaluated,

: $E_x = int d^3r , n(vec{r}) left($ -3e^2}over{4pi ight) left(3 pi^2 n(vec{r}) ight)^{1 over 3}

in SI units where $n(vec{r})$ is the electron density at the point $vec{r} ,;$ and $e,;$ is the charge of an electron. [cite journal | title = Nonempirical Construction of Current-Density Functionals from Conventional Density-Functional Approximations | author = Jianmin Tao and John P. Perdew | journal = Phys. Rev. Lett. | volume = 95 | pages = 196403 | year = 2005 | url = http://link.aps.org/abstract/PRL/v95/p196403 | doi = 10.1103/PhysRevLett.95.196403 | format = abstract ]

Correlation

There are several forms of correlation:

Wigner correlation is obtained by using equally spaced electrons and applying perturbation theory.

References

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