Local-density approximation: Difference between revisions

Revision as of 09:29, 11 December 2023

Revision as of 09:30, 11 December 2023

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:<math>E_{\rm xc} = E_{\rm x} + E_{\rm c}\ ,</math>

so that separate expressions for ''E''_x and ''E''_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 ''εє''_c.

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.

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== Homogeneous electron gas ==

Approximation for ''εє''_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''&nbsp;=&nbsp;''N''&nbsp;/&nbsp;''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 planewaves. In particular, for a constant density ''ρ'', the exchange energy density is proportional to ''ρ''^⅓.

== Exchange functional ==

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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, ''εє''_c(''ρ'',''ς''), 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 }}</ref>

== Exchange-correlation potential ==