Generalized gradient approximation

The generalized gradient approximation (GGA from English Generalized Gradient Approximation ) is an approximation method in solid-state physics that is used for calculations of the electronic band structure with the help of density functional theory .

The generalized gradient approximation is distinguished from the local spin density approximation in that in addition to the local spin densities , their gradients are also taken into account in the term for the exchange energy . ${\displaystyle E_{\mathrm {XC} }}$ ${\displaystyle (n_{\uparrow },n_{\downarrow })}$ ${\displaystyle (\nabla n_{\uparrow },\nabla n_{\downarrow })}$

${\displaystyle E_{\mathrm {XC} }^{\mathrm {GGA} }\left[n_{\uparrow },n_{\downarrow }\right]=\int \varepsilon _{\mathrm {XC} }\left(n_{\uparrow },n_{\downarrow },\nabla n_{\uparrow },\nabla n_{\downarrow }\right)n({\vec {r}})\,{\mathrm {d} }^{3}r.}$

rating

Calculations within the framework of the local spin density approximation have shown that typically the contribution of the exchange energy is underestimated and the contribution of the correlation energy is overestimated. While these errors often largely compensate each other with regard to the energy, they often lead to significant errors with regard to the spin densities, which result in faulty ground-state energies and crystal geometries as well as incorrect parameters of the lattice vibrations ( phonons ). When using the GGA, these errors are much less apparent.

Types

• PW91 named after JP P erdew and Y. W ang who proposed this function in 19 91 .
• PBE named after JP P erdew, S. B urke and M. E rnzerhof, who in 1996 proposed a functional equivalent to the PW91 that gets by with fewer parameters.

Individual evidence

1. ↑ John P. Perdew, JA Chevary, SH Vosko, Koblar A. Jackson, Mark R. Pederson, DJ Singh, Carlos Fiolhais: Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation . In: Physical Review B . 46, No. 11, 1992, p. 6671. bibcode : 1992PhRvB..46.6671P . doi : 10.1103 / physrevb.46.6671 .
2. ^ Axel D. Becke: Density-functional exchange-energy approximation with correct asymptotic behavior . In: Physical Review A . 38, No. 6, 1988, p. 3098. bibcode : 1988PhRvA..38.3098B . doi : 10.1103 / physreva.38.3098 . PMID 9900728 .
3. ↑ David C Langreth, MJ Mehl: Beyond the local-density approximation in calculations of ground-state electronic properties . In: Physical Review B . 28, No. 4, 1983, p. 1809. bibcode : 1983PhRvB..28.1809L . doi : 10.1103 / physrevb.28.1809 .
4. ↑ Axel D. Becke: Perspective: Fifty years of density-functional theory in chemical physics . In: The Journal of Chemical Physics . 140, No. 18, May 14, 2014, ISSN  0021-9606 , p. A301. bibcode : 2014JChPh.140rA301B . doi : 10.1063 / 1.4869598 . PMID 24832308 .
5. ↑ JP Perdew, Y. Wang: - . In: Phys. Rev. B . tape 45 , 1992, pp. 13244 . and references therein.
6. ↑ JP Perdew: - . In: P. Ziesche and H. Eschrig (Eds.): Electronic Structure of Solids . Akademie Verlag, Berlin 1991, p. 11 .
7. ↑ JP Perdew, S. Burke and M. Ernzerhof: eneralized Gradient Approximation Made Simple . In: Phys. Rev. Lett. tape 77 , 1996, pp. 3865 ff . ( uci.edu [PDF]).