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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.[1]

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.[2]

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.[3]

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.[1]

The XC correlation functional is the sum of a correlation functional and an exchange functional[1]

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

## 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.[4]

## Correlation

There are several forms of correlation:

• Vosko-Wilk-Nusair (VWN) [7]
• Perdew-Zunger (PZ) [8]
• Cole-Perdew (CP) [9]
• Lee-Yang-Parr (LYP) [10]
• Perdew-Wang (PW92) [11]

Wigner correlation is gotten by using equally spaced electrons and applying perturbation theory.[5]

VWN, PZ and PW92 are fit to a quantum Monte Carlo calculation[12] of the electron gas at varying densities.[11]

LYP is based on data fit to the helium atom.[10]

## 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.