Episode 234
Localdensity approximation Sat, 2017Dec23 02:59 UTC Length  3:24
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With 209,360 views on Friday, 22 December 2017 our article of the day is Localdensity approximation.
Localdensity approximations (LDA) are a class of approximations to the exchange–correlation (XC) energy functional in density functional theory (DFT) that depend solely upon the value of the electronic density at each point in space (and not, for example, derivatives of the density or the Kohn–Sham orbitals). Many approaches can yield local approximations to the XC energy. 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 spinunpolarized system, a localdensity approximation for the exchangecorrelation energy is written as
E
x
c
L
D
A
[
ρ
]
=
∫
ρ
(
r
)
ϵ
x
c
(
ρ
)
d
r
,
{\displaystyle E_{xc}^{\mathrm {LDA} }[\rho ]=\int \rho (\mathbf {r} )\epsilon _{xc}(\rho )\ \mathrm {d} \mathbf {r} \ ,}
where ρ is the electronic density and εxc is the exchangecorrelation energy per particle of a homogeneous electron gas of charge density ρ. The exchangecorrelation energy is decomposed into exchange and correlation terms linearly,
E
x
c
=
E
x
+
E
c
,
{\displaystyle E_{xc}=E_{x}+E_{c}\ ,}
so that separate expressions for Ex and Ec are sought. The exchange term takes on a simple analytic form for the HEG. Only limiting expressions for the correlation density are known exactly, leading to numerous different approximations for εc.
Localdensity approximations are important in the construction of more sophisticated approximations to the exchangecorrelation energy, such as generalized gradient approximations or hybrid functionals, as a desirable property of any approximate exchangecorrelation functional is that it reproduce the exact results of the HEG for nonvarying densities. As such, LDA's are often an explicit component of such functionals.
This recording reflects the Wikipedia text as of 02:59 UTC on Saturday, 23 December 2017.
For the full current version of the article, go to http://en.wikipedia.org/wiki/Localdensity_approximation.
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