Local-density approximation: Difference between revisions

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{{Short description|Approximations in density functional theory}}

{{distinguish|linear discriminant analysis}}

'''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).

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== Homogeneous electron gas ==

Approximation for ''є''_xc depending only upon the density can be developed in numerous ways. The most successful approach is based on the homogeneous electron gas. This is constructed by placing ''N'' interacting electrons in to a volume, ''V'', with a positive background charge keeping the system neutral. ''N'' and ''V'' are then taken to infinity in the manner that keeps the density (''ρ'' = ''N'' / ''V'') finite. This is a useful approximation, as the total energy consists of contributions only from the kinetic energy, electrostatic interaction energy and exchange-correlation energy, and that the wavefunction is expressible in terms of planewavesplane waves. In particular, for a constant density ''ρ'', the exchange energy density is proportional to ''ρ''^⅓.

== 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 is not homogeneous, is obtained by applying the HEG results pointwise, yielding the expression<ref name="parryang">{{cite book|last=Parr|first=Robert G|author2=Yang, Weitao |title=Density-Functional Theory of Atoms and Molecules|publisher=Oxford University Press|___location=Oxford |year=1994|isbn=978-0-19-509276-9}}</ref><ref>{{cite journal
|last last1=Dirac |first first1=P. A. M. |year authorlink1=Paul Dirac
| date=1930
| title=Note on exchange phenomena in the Thomas-Fermi atom
| journal=Proc.[[Mathematical Camb.Proceedings Phil.of Soc.the Cambridge Philosophical Society]]
| volume=26
| pages=376–385
| doi=10.1017/S0305004100016108
| issue=3
| bibcode = 1930PCPS...26..376D |doi-access=free}}</ref>

| doi-access=free}}</ref>

:<math>E_{\rm x}^{\mathrm{LDA}}[\rho]
= - \frac{33e^2}{416\pi\varepsilon_0}\left( \frac{3}{\pi} \right)^{1/3}\int\rho(\mathbf{r})^{4/3}\ \mathrm{d}\mathbf{r}\ .</math>

= - \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 ==

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:<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. The Wigner-Seitz parameter ''a''<mathsub>r_{\rm s}0</mathsub>. isIn relatedterms toof the density as''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.

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<math>\zeta = 0\,</math> corresponds to the diamagnetic spin-unpolarized situation with equal

<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, ''є''_c(''ρ'',''ςζ''), is constructed so to interpolate the extreme values. Several forms have been developed in conjunction with LDA correlation functionals.<ref>{{cite journal|last=von Barth|first=U.|author2=Hedin, L. |year=1972|title=A local exchange-correlation potential for the spin polarized case|journal=J. Phys. C: Solid State Phys.|volume=5|pages=1629–1642|doi=10.1088/0022-3719/5/13/012|bibcode = 1972JPhC....5.1629V|issue=13 |s2cid=120985795 }}</ref>

== Exchange-correlation potential ==

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:<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 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 highest occupied molecular orbital ([[HOMO]]) energy being too high in energy, so that any predictions for the [[ionization potential]] based on [[Koopmans' 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|author2=Nogueira, Fernando |author3=Marques Miguel |title=A Primer in Density Functional Theory|publisher=Springer|year=2003|isbn=978-3-540-03083-6|page=60}}</ref> In the case of spin polarization, the exchange-correlation potential acquires spin indices. 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):<ref>{{Cite book |last=Giustino |first=Feliciano |title=Materials Modelling Using Density Functional Theory: Properties and Predictions |publisher=Oxford University Press |year=2014 |pages=229}}</ref>

<math>v_{\rm xc, \alpha \beta}^{\mathrm{LDA}}(\mathbf{r}) = \frac{\delta E^{\mathrm{LDA}}}{\delta\rho_{\alpha \beta}(\mathbf{r})} =