Local-density approximation: Difference between revisions

Revision as of 20:44, 1 December 2010

Revision as of 17:10, 23 April 2011

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== 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 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 |dateyear=1994|isbn=978-0-19-509276-9}}</ref><ref>{{cite journal|last=Dirac|first=P. A. M.|dateyear=1930|title=Note on exchange phenomena in the Thomas-Fermi atom|journal=Proc. Cambridge Phil. Roy. Soc.|volume=26|pages=376–385|doi=10.1017/S0305004100016108}}</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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:<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 | bibcode=1980PhRvL..45..566C}}</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

* Vosko-Wilk-Nusair (VWN) <ref name="vwn">{{cite journal | title = Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis | author = S. H. Vosko, L. Wilk and M. Nusair | journal = Can. J. Phys. | volume = 58 | pages = 1200 | year = 1980 | doi = 10.1139/p80-159 }}</ref>

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:<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. |dateyear=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>

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<math>\zeta = 0\,</math> corresponds to the paramagnetic 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, ''ε''_c(''ρ'',''ς''), is constructed so to interpolate the extreme values. Several forms have been developed in conjunction with LDA correlation functionals.<ref name="vwn"/><ref>{{cite journal|last=von Barth|first=U.|coauthors=Hedin, L.|dateyear=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}}</ref>

== Exchange-correlation potential ==

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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>{{cite book|last=Fiolhais|first=Carlos|coauthors=Nogueira, Fernando; Marques Miguel|title=A Primer in Density Functional Theory|publisher=Springer|dateyear=2003|isbn=978-3-540-03083-6|page=60}}</ref><ref>{{cite journal|last=Perdew|first=J. P. |coauthors=Zunger, Alex |dateyear=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>

== References ==