Local-density approximation: Difference between revisions

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The '''local-density approximation''' (LDA) is an approximation of the [[Exchange interaction|exchange]]-[[Electron correlation|correlation]] (XC) energy functional in [[Density functional theory|density functional theory]] (DFT) by taking the XC energy of an electron in an interacting homogeneous [[Free electron model|electron gas]] of a density equivalent to the density at the electron in the system being calculated. This approximation was applied to DFT by [[Walter Kohn|Kohn]] and [[Lu Jeu Sham|Sham]] in an early paper.<ref name=kohn-sham>{{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 }}</ref>

The Hohenberg-Kohn theorem states that the energy of the [[Stationary state|ground state]] of a system of electrons is a [[Functional (mathematics)|functional]] of the [[Electronic density|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.<ref>{{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 }}</ref>

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.<ref>{{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}}</ref>

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Wigner correlation is gotten by using equally spaced electrons and applying perturbation theory.<ref name=wigner />

VWN, PZ and PW92 are fit to a [[Quantum Monte Carlo|quantum Monte Carlo]] calculation<ref>{{cite journal | title = Ground State of the Electron Gas by a Stochastic Method | author = D. M. Ceperley and B. J. Alder | journal = Phys. Rev. Lett. | volume = 45 | pages = 566 - 569 | year = 1980 | url = http://link.aps.org/abstract/PRL/v45/p566 | doi = 10.1103/PhysRevLett.45.566}}</ref> of the electron gas at varying densities.<ref name=pw92 />

LYP is based on data fit to the helium atom.<ref name=lyp />

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<references />

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[[Category:Density functional theory]]

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