Local-density approximation: Difference between revisions

Revision as of 02:22, 28 February 2014

Revision as of 23:24, 8 March 2014

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== Applications ==

Local density approximations, as with Generalised Gradient Approximations (GGA) are employed extensively by [[solid-state physics |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_theoryDensity functional theory#Approximations_Approximations .28exchange-correlation_functionalscorrelation 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 TiO₂ 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>

== Homogeneous electron gas ==

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* Perdew-Wang (PW92) <ref name=pw92>{{cite journal | title = Accurate and simple analytic representation of the electron-gas correlation energy | author = John P. Perdew and Yue Wang | journal = Phys. Rev. B | volume = 45 | pages = 13244–13249 | year = 1992 | doi = 10.1103/PhysRevB.45.13244 |bibcode = 1992PhRvB..4513244P | issue = 23 }}</ref>

Predating these, and even the formal foundations of DFT itself, is the Wigner correlation functional obtained [[Møller-Plesset_perturbation_theoryPlesset perturbation theory#Rayleigh-Schr.C3.B6dinger_perturbation_theoryB6dinger perturbation theory|perturbatively]] from the HEG model.<ref name=wigner>{{cite journal | title = On the Interaction of Electrons in Metals | author = E. Wigner | journal = Phys. Rev. | volume = 46 | pages = 1002–1011 | year = 1934 | url = http://link.aps.org/abstract/PR/v46/p1002 | doi = 10.1103/PhysRev.46.1002 | format = abstract |bibcode = 1934PhRv...46.1002W | issue = 11 }}</ref>

== Spin polarization ==

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:<math>v_{xc}^{\mathrm{LDA}}(\mathbf{r}) = \frac{\delta E^{\mathrm{LDA}}}{\delta\rho(\mathbf{r})} = \epsilon_{xc}(\rho(\mathbf{r})) + \rho(\mathbf{r})\frac{\partial \epsilon_{xc}(\rho(\mathbf{r}))}{\partial\rho(\mathbf{r})}\ .</math>

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 name="pz81"/><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"/>

== References ==