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{{Short description|Approximations in density functional theory}}
'''Local-density approximations''' ('''LDA''') are a class of approximations to the [[Exchange interaction|exchange]]-[[Electron correlation|correlation]] (XC) energy [[Functional (mathematics)|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 equations|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).▼
{{distinguish|linear discriminant analysis}}
▲'''Local-density approximations''' ('''LDA''') are a class of approximations to the [[Exchange interaction|exchange]]
In general, for a spin-unpolarized system, a local-density approximation for the exchange-correlation energy is written as
:<math>E_{\rm xc}^{\mathrm{LDA}}[\rho] = \int \rho(\mathbf{r})\epsilon_{\rm xc}(\rho(\mathbf{r}))\ \mathrm{d}\mathbf{r}\ ,</math>
where ''ρ'' is the [[electronic density]] and ''
:<math>E_{\rm xc} =
so that separate expressions for ''E''<sub>x</sub> and ''E''<sub>c</sub> 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 ''
Local-density approximations are important in the construction of more sophisticated approximations to the exchange-correlation energy, such as [[generalized gradient approximation]]s (GGA) or [[hybrid functional]]s, as a desirable property of any approximate exchange-correlation functional is that it reproduce the exact results of the HEG for non-varying densities. As such, LDA's are often an explicit component of such functionals.
The local-density approximation was first introduced by [[Walter Kohn]] and [[Lu Jeu Sham]] in 1965.<ref name=":0" />
== Applications ==▼
▲== Applications ==
Local density approximations, as with Generalised Gradient Approximations (GGA) are employed extensively by [[solid-state physiscs |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 <ref>{{cite journal| last1=Segall| first1=M.D.| last2=Lindan| first2=P.J | title= First-principles simulation: ideas, illustrations and the CASTEP code | journal= Journal of Physics: Condensed Matter | year= 2002| volume=14| issue=11| pages=2717}}</ref>. However an underestimation in [[Band gap]] values often associated with LDA and [[Density_functional_theory#Approximations_.28exchange-correlation_functionals.29|GGA]] approximations may lead to false predictions of impurity mediated conductivity and/or carrier mediated magnetism in such systems. <ref>{{cite journal| last1=Assadi| first1=M.H.N| last2=et al.| title= Theoretical study on copper's energetics and magnetism in TiO<sub>2</sub> polymorphs| journal= Journal of Applied Physics | year=2013| volume=113| issue=23| pages= 233913| url=http://arxiv.org/ftp/arxiv/papers/1304/1304.1854.pdf| doi=10.1063/1.4811539}}</ref>▼
▲Local density approximations, as with
== Homogeneous electron gas ==
Approximation for ''
== Exchange functional ==
The exchange-energy density of a HEG is known analytically. The LDA for exchange employs this expression under the approximation that the exchange-energy in a system where the density
| | date=1930 | title=Note on exchange phenomena in the Thomas-Fermi atom | journal= | volume=26 | pages=376–385 | doi=10.1017/S0305004100016108 | issue=3 | bibcode = 1930PCPS...26..376D | doi-access=free}}</ref>
:<math>E_{\rm x}^{\mathrm{LDA}}[\rho]
= - \frac{ = - \frac{3}{4}\left( \frac{3}{\pi} \right)^{1/3}\int\rho(\mathbf{r})^{4/3}\ \mathrm{d}\mathbf{r}\,,</math>
where the second formulation applies in [[Atomic units|atomic units]].
== Correlation functional ==
Analytic expressions for the correlation energy of the HEG are
:<math>\epsilon_{\rm c} = A\ln(r_{\rm s}) + B + r_{\rm s}(C\ln(r_{\rm s}) + D)\ ,</math>
and the low limit
:<math>\epsilon_{\rm c} = \frac{1}{2}\left(\frac{g_{0}}{r_{\rm s}} + \frac{g_{1}}{r_{\rm s}^{3/2}} + \dots\right)\ ,</math>
where the [[Wigner–Seitz cell|Wigner-Seitz parameter]] <math>r_{\rm s}</math> is dimensionless.<ref name="Murray Gell-Mann and Keith A. Brueckner 1957 364">{{cite journal | title = Correlation Energy of an Electron Gas at High Density | author = Murray Gell-Mann and Keith A. Brueckner | journal = Phys. Rev. | volume = 106 | pages = 364–368 | year = 1957 | doi = 10.1103/PhysRev.106.364 | issue = 2| bibcode = 1957PhRv..106..364G | s2cid = 120701027 | url = https://authors.library.caltech.edu/3713/1/GELpr57b.pdf }}</ref> It is defined as the radius of a sphere which encompasses exactly one electron, divided by the Bohr radius ''a''<sub>0</sub>. In terms of the density ''ρ'', this means
:<math>\frac{4}{3}\pi r_{\rm s}^{3} = \frac{1}{\rho \, a_0^3}\ .</math>
An analytical expression for the full range of densities has been proposed based on the many-body perturbation theory. The calculated correlation energies are in agreement with the results from [[quantum Monte Carlo]] simulation to within 2 milli-Hartree.
== Spin polarization ==
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The extension of density functionals to [[Spin polarization|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, ''ρ''<sub>α</sub> and ''ρ''<sub>β</sub> with ''ρ'' = ''ρ''<sub>α</sub> + ''ρ''<sub>β</sub>, and the form of the local-spin-density approximation (LSDA) is
:<math>E_{\rm xc}^{\mathrm{LSDA}}[\rho_{\alpha},\rho_{\beta}] = \int\mathrm{d}\mathbf{r}\ \rho(\mathbf{r})\epsilon_{\rm xc}(\rho_{\alpha},\rho_{\beta})\ .</math>
For the exchange energy, the exact result (not just for local density approximations) is known in terms of the spin-unpolarized functional:<ref>{{cite journal|last=Oliver|first=G. L.|
:<math>E_{\rm x}[\rho_{\alpha},\rho_{\beta}] = \frac{1}{2}\bigg( E_{\rm x}[2\rho_{\alpha}] + E_{\rm x}[2\rho_{\beta}] \bigg)\ .</math>
The spin-dependence of the correlation energy density is approached by introducing the relative spin-polarization:
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:<math>\zeta(\mathbf{r}) = \frac{\rho_{\alpha}(\mathbf{r})-\rho_{\beta}(\mathbf{r})}{\rho_{\alpha}(\mathbf{r})+\rho_{\beta}(\mathbf{r})}\ .</math>
<math>\zeta = 0\,</math> corresponds to the
<math>\alpha\,</math> and <math>\beta\,</math> spin densities whereas <math>\zeta = \pm 1</math> corresponds to the ferromagnetic situation where one spin density vanishes. The spin correlation energy density for a given values of the total density and relative polarization, ''
== Exchange-correlation potential ==
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The exchange-correlation potential corresponding to the exchange-correlation energy for a local density approximation is given by<ref name="parryang"/>
:<math>v_{\rm xc}^{\mathrm{LDA}}(\mathbf{r}) = \frac{\delta E^{\mathrm{LDA}}}{\delta\rho(\mathbf{r})} = \epsilon_{\rm xc}(\rho(\mathbf{r})) + \rho(\mathbf{r})\frac{\partial \epsilon_{\rm xc}(\rho(\mathbf{r}))}{\partial\rho(\mathbf{r})}\ .</math>
In finite systems, the LDA potential decays asymptotically with an exponential form. This result is in error; the true exchange-correlation potential decays much slower in a Coulombic manner. The artificially rapid decay manifests itself in the number of
<math>v_{\rm xc, \alpha \beta}^{\mathrm{LDA}}(\mathbf{r}) = \frac{\delta E^{\mathrm{LDA}}}{\delta\rho_{\alpha \beta}(\mathbf{r})} =
▲In finite systems, the LDA potential decays asymptotically with an exponential form. This is in error; the true exchange-correlation potential decays much slower in a Coulombic manner. The artificially rapid decay manifests itself in the number of Kohn-Sham orbitals the potential can bind (that is, how many orbitals have energy less than zero). The LDA potential can not support a Rydberg series and those states it does bind are too high in energy. This results in the [[HOMO]] energy being too high in energy, so that any predictions for the [[ionization potential]] based on [[Koopman's theorem]] are poor. Further, the LDA provides a poor description of electron-rich species such as [[anion]]s where it is often unable to bind an additional electron, erroneously predicating species to be unstable.<ref>{{cite book|last=Fiolhais|first=Carlos|coauthors=Nogueira, Fernando; Marques Miguel|title=A Primer in Density Functional Theory|publisher=Springer|year=2003|isbn=978-3-540-03083-6|page=60}}</ref><ref name="pz81"/>
\frac{1}{2}\delta_{\alpha\beta}\frac{\delta E^{\mathrm{LDA}}[2\rho_{\alpha}]}{\delta \rho_{\alpha}} = - \delta_{\alpha\beta}\Big(\frac{3}{\pi}\Big)^{1/3}2^{1/3}\rho_{\alpha}^{1/3}</math>
== References ==
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