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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).
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>
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 ''є''<sub>c</sub>.
▲:<math>E_{xc} = E_x + E_c \,\;</math>
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.
== Exchange ==▼
The local-density approximation was first introduced by [[Walter Kohn]] and [[Lu Jeu Sham]] in 1965.<ref name=":0" />
== Applications ==
Local density approximations, as with GGAs 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|bibcode = 2002JPCM...14.2717S |doi = 10.1088/0953-8984/14/11/301 | s2cid=250828366}}</ref> However an underestimation in [[Band gap]] values often associated with LDA and [[Density functional theory#Approximations (exchange–correlation functionals)|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| 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= 233913–233913–5| doi=10.1063/1.4811539|arxiv = 1304.1854 |bibcode = 2013JAP...113w3913A | s2cid=94599250|display-authors=etal}}</ref> Starting in 1998, the application of the [[Rayleigh theorem for eigenvalues]] has led to mostly accurate, calculated band gaps of materials, using LDA potentials.<ref>{{Cite journal|last1=Zhao|first1=G. L.|last2=Bagayoko|first2=D.|last3=Williams|first3=T. D.|date=1999-07-15|title=Local-density-approximation prediction of electronic properties of GaN, Si, C, and RuO2|journal=Physical Review B|volume=60|issue=3|pages=1563–1572|doi=10.1103/physrevb.60.1563|bibcode=1999PhRvB..60.1563Z |issn=0163-1829}}</ref><ref name=":0">{{Cite journal|last=Bagayoko|first=Diola|date=December 2014|title=Understanding density functional theory (DFT) and completing it in practice|journal=AIP Advances|volume=4|issue=12|pages=127104|doi=10.1063/1.4903408|bibcode=2014AIPA....4l7104B |issn=2158-3226|doi-access=free}}</ref> A misunderstanding of the second theorem of DFT appears to explain most of the underestimation of band gap by LDA and GGA calculations, as explained in the description of [[density functional theory]], in connection with the statements of the two theorems of DFT.
== Homogeneous electron gas ==
== Correlation ==▼
Approximation for ''є''<sub>xc</sub> 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 plane waves. In particular, for a constant density ''ρ'', the exchange energy density is proportional to ''ρ''<sup>⅓</sup>.
▲== 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
| last1=Dirac | first1=P. A. M. | authorlink1=Paul Dirac
| date=1930
| title=Note on exchange phenomena in the Thomas-Fermi atom
| journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]
| 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{3e^2}{16\pi\varepsilon_0}\left( \frac{3}{\pi} \right)^{1/3}\int\rho(\mathbf{r})^{4/3}\ \mathrm{d}\mathbf{r}
= - \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 available 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_{\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.
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 | issue = 7| s2cid = 55620379 | url = https://digital.library.unt.edu/ark:/67531/metadc1059358/ }}</ref>
== Spin polarization ==
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.|author2=Perdew, J. P. |year=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|bibcode = 1979PhRvA..20..397O|issue=2 }}</ref>
:<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:
:<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 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''<sub>c</sub>(''ρ'',''ζ''), 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 ==
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 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})} =
\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 ==
{{reflist}}
[[Category:Density functional theory]]
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