Local-density approximation: Difference between revisions

Revision as of 23:25, 16 November 2008

Revision as of 16:45, 17 November 2008

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In general, for a spin-unpolarized system, a local-density approximation for the exchange-correlation energy is written as

:<math>E_{xc}^{\mathrm{LDA}}[\rho] = \int\mathrm{d}\mathbf{r}\ \rho(\mathbf{r})\epsilon_{xc}(\rho)\ \mathrm{d}\mathbf{r}\ ,</math>

where ''ρ'' is the [[electronic density]] and ''ε''_xc, the exchange-correlation energy density, is a function of the density alone. The exchange-correlation energy is decomposed in to exchange and correlation terms linearly,

:<math>E_{xc} = E_x + E_c, \ ,</math>

so that separate expressions for ''ε''_x and ''ε''_c 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 ''ε''_c.

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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 in not homogeneous, is obtained by applying the HEG results pointwise, yielding the expression<ref name="parryang">{{cite book|last=Parr|first=Robert G|coauthors=Yang, Weitao|title=Density-Functional Theory of Atoms and Molecules|publisher=Oxford University Press|___location=Oxford |date=1994|isbn=978-0-19-509276-9}}</ref><ref>{{cite journal|last=Dirac|first=P. A. M.|date=1930|title=Note on exchange phenomena in the Thomas-Fermi atom|journal=Proc. Cambridge Phil. Roy. Soc.|volume=26|pages=376–385}}</ref>

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

== Correlation functional ==

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Analytic expressions for the correlation energy of the HEG are not known except in the high- and low-density limits corresponding to infinitely-weak and infinitely-strong correlation. For a HEG with density ''ρ'', the high-density limit of the correlation energy density is<ref name="parryang"/>

:<math>\epsilon_{c} = A\ln(r_{s}) + B + r_{s}(C\ln(r_{s}) + D),\ ,</math>

and the low limit

:<math>\epsilon_{c} = \frac{1}{2}\left(\frac{g_{0}}{r_{s}} + \frac{g_{1}}{r_{s}^{3/2}} + \dots\right)\ ,</math>

where the Wigner-Seitz radius is related to the density as

:<math>\frac{4}{3}\pi r_{s}^{3} = \frac{1}{\rho}\ .</math>

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.<ref>{{cite journal | title = Ground State of the Electron Gas by a Stochastic Method | author = D. M. Ceperley and B. J. Alder | journal = Phys. Rev. Lett. | volume = 45 | pages = 566–569 | year = 1980 | doi = 10.1103/PhysRevLett.45.566 }}</ref> The most popular LDA's to the correlation energy density interpolate these accurate values obtained from simulation while reproducing the exactly known limiting behavior. Various approaches, using different analytic forms for ''ε''_c, have generated several LDA's for the correlation functional, including

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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''_α and ''ρ''_β with ''ρ''&nbsp;=&nbsp;''ρ''_α&nbsp;+&nbsp;''ρ''_β, and the form of the local-spin-density approximation (LSDA) is

:<math>E_{xc}^{\mathrm{LSDA}}[\rho_{\alpha},\rho_{\beta}] = \int\mathrm{d}\mathbf{r}\ \rho(\mathbf{r})\epsilon_{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.|coauthors=Perdew, J. P. |date=1979|title=Spin-density gradient expansion for the kinetic energy|journal=Phys. Rev. A|volume=20|pages=397-403|doi=10.1103/PhysRevA.20.397}}</ref>:

:<math>E_{x}[\rho_{\alpha},\rho_{\beta}] = \frac{1}{2}\bigg( E_{x}[2\rho_{\alpha}] + E_{x}[2\rho_{\beta}] \bigg)\ .</math>

The spin-dependence of the correlation energy density is approached by introducing the relative spin-polarization:

:<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 paramagnetic spin-unpolarized situation with equal

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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_{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>{{cite book|last=Fiolhais|first=Carlos|coauthors=Nogueira, Fernando; Marques Miguel|title=A Primer in Density Functional Theory|publisher=Springer|date=2003|pages=60|isbn=978-3-540-03083-6}}</ref><ref>{{cite journal|last=Perdew|first=J. P. |coauthors=Zunger, Alex |date=1981|title=Self-interaction correction to density-functional approximations for many-electron systems|journal=Phys. Rev. B|volume=23|issue=10|pages=5048–5079|doi=10.1103/PhysRevB.23.5048 }}</ref>