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# 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 non-interacting homogeneous electron gas of a density equivalent to the density at the electron in the system being calculated. This approximation was applied to DFT by Kohn and Sham in an early paper.

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.

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.

The LDA functional assumes that the per-electron exchange-correlation energy at every point in space is equal to the per-electron exchange-correlation energy of a homogeneous electron gas.

The XC correlation functional is the sum of a correlation functional and an exchange functional $E_{xc} = E_x + E_c \,\;$

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

in SI units where $n(\vec{r})$ is the electron density per unit volume at the point $\vec{r} \,\;$and $e\,\;$ is the charge of an electron.

## Correlation

There are several forms of correlation:

• Vosko-Wilk-Nusair (VWN) 
• Perdew-Zunger (PZ) 
• Cole-Perdew (CP) 
• Lee-Yang-Parr (LYP) 
• Perdew-Wang (PW92) 

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

VWN, PZ and PW92 are fit to a quantum Monte Carlo calculation of the electron gas at varying densities.

LYP is based on data fit to the helium atom.

## References

1. ^ a b c W. Kohn and L. J. Sham (1965). "Self-Consistent Equations Including Exchange and Correlation Effects". Phys. Rev. 140: A1133 - A1138. doi:10.1103/PhysRev.140.A1133.
2. ^ P. Hohenberg and W. Kohn (1964). "Inhomogeneous Electron Gas". Phys. Rev. 136: B864 - B871. doi:10.1103/PhysRev.136.B864.
3. ^ John R. Smith (1970). "Beyond the Local-Density Approximation: Surface Properties of (110) W". Phys. Rev. Lett. 25 (15): 1023 - 1026. doi:10.1103/PhysRevLett.25.1023.
4. ^ Jianmin Tao and John P. Perdew (2005). "Nonempirical Construction of Current-Censity Functionals from Conventional Density-Functional Approximations". Phys. Rev. Lett. 95: 196403. doi:10.1103/PhysRevLett.95.196403.
5. ^ a b E. Wigner (1934). "On the Interaction of Electrons in Metals". Phys. Rev. 46: 1002 - 1011. doi:10.1103/PhysRev.46.1002.
6. ^ N. D. Lang and W. Kohn (1970). "Theory of Metal Surfaces: Charge Density and Surface Energy". Phys. Rev. B 1: 4555 - 4568. doi:10.1103/PhysRevB.1.4555.
7. ^ S. H. Vosko, L. Wilk and M. Nusair (1980). "Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis". Can. J. Phys. 58: 1200.
8. ^ J. P. Perdew and A. Zunger (1981). "Self-interaction correction to density-functional approximations for many-electron systems". Phys. Rev. B 23: 5048. doi:10.1103/PhysRevB.23.5048.
9. ^ L. A. Cole and J. P. Perdew (1982). "Calculated electron affinities of the elements". Phys. Rev. A 25: 1265. doi:10.1103/PhysRevA.25.1265.
10. ^ a b Chengteh Lee, Weitao Yang, and Robert G. Parr (1988). "Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density". Phys. Rev. B 37: 785 - 789. doi:10.1103/PhysRevB.37.785.
11. ^ a b John P. Perdew and Yue Wang (1992). "Accurate and simple analytic representation of the electron-gas correlation energy". Phys. Rev. B 45: 13244 - 13249. doi:10.1103/PhysRevB.45.13244.
12. ^ D. M. Ceperley and B. J. Alder (1980). "Ground State of the Electron Gas by a Stochastic Method". Phys. Rev. Lett. 45: 566 - 569. doi:10.1103/PhysRevLett.45.566.