However, overwhelmingly successful local approximations are those that have been derived from the homogeneous electron gas (HEG) model.
In this regard, LDA is generally synonymous with functionals based on the HEG approximation, which are then applied to realistic systems (molecules and solids).
In general, for a spin-unpolarized system, a local-density approximation for the exchange-correlation energy is written as where ρ is the electronic density and єxc is the exchange-correlation energy per particle of a homogeneous electron gas of charge density ρ.
The exchange-correlation energy is decomposed into exchange and correlation terms linearly, so that separate expressions for Ex and Ec are sought.
Only limiting expressions for the correlation density are known exactly, leading to numerous different approximations for єc.
The local-density approximation was first introduced by Walter Kohn and Lu Jeu Sham in 1965.
[1] Local density approximations, as with GGAs are employed extensively by solid state physicists in ab-initio DFT studies to interpret electronic and magnetic interactions in semiconductor materials including semiconducting oxides and spintronics.
The importance of these computational studies stems from the system complexities which bring about high sensitivity to synthesis parameters necessitating first-principles based analysis.
The prediction of Fermi level and band structure in doped semiconducting oxides is often carried out using LDA incorporated into simulation packages such as CASTEP and DMol3.
[3] Starting in 1998, the application of the Rayleigh theorem for eigenvalues has led to mostly accurate, calculated band gaps of materials, using LDA potentials.
[4][1] A misunderstanding of the second theorem of DFT appears to explain most of the underestimation of band gap by LDA and GGA calculations, as explained in the description of density functional theory, in connection with the statements of the two theorems of DFT.
This is constructed by placing N interacting electrons in to a volume, V, with a positive background charge keeping the system neutral.
The LDA for exchange employs this expression under the approximation that the exchange-energy in a system where the density is not homogeneous, is obtained by applying the HEG results pointwise, yielding the expression[5][6] where the second formulation applies in atomic units.
The calculated correlation energies are in agreement with the results from quantum Monte Carlo simulation to within 2 milli-Hartree.
Accurate quantum Monte Carlo simulations for the energy of the HEG have been performed for several intermediate values of the density, in turn providing accurate values of the correlation energy density.
[8] The extension of density functionals to spin-polarized systems is straightforward for exchange, where the exact spin-scaling is known, but for correlation further approximations must be employed.
A spin polarized system in DFT employs two spin-densities, ρα and ρβ with ρ = ρα + ρβ, and the form of the local-spin-density approximation (LSDA) is For the exchange energy, the exact result (not just for local density approximations) is known in terms of the spin-unpolarized functional:[9] The spin-dependence of the correlation energy density is approached by introducing the relative spin-polarization:
This result is in error; the true exchange-correlation potential decays much slower in a Coulombic manner.
The LDA potential can not support a Rydberg series and those states it does bind are too high in energy.
However, if one only considers the exchange part of the exchange-correlation, one obtains a potential that is diagonal in spin indices (in atomic units):[12]