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| The local and semilocal [[exchange-correlation functionals]] depend locally on quantities like the electron density <math>n</math> or the kinetic-energy density <math>\tau</math>. Most of them can be classified into one of the three subcategories listed below, depending on the variables on which <math>E_{\mathrm{xc}}</math> depends.
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| === Local density approximation (LDA) ===
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| The LDA functionals are purely local in the sense that they depend solely on <math>n</math>:
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| :<math>
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| E_{\mathrm{xc}}^{\mathrm{LDA}}=\int\epsilon_{\mathrm{xc}}^{\mathrm{LDA}}(n)d^{3}r
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| </math>
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| with a corresponding exchange-correlation potential calculated as
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| :<math>
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| v_{\mathrm{xc}}^{\mathrm{LDA}} = \frac{\partial\epsilon_{\mathrm{xc}}^{\mathrm{LDA}}}{\partial n}
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| </math>
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| The most common LDA functionals, e.g,. {{TAG|GGA}}=CA {{cite|dirac:mpcps:1930}}{{cite|ceperley1980}}{{cite|perdewzunger1981}}, provide the (nearly) exact exchange-correlation energy for the homogeneous electron gas.
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| === Generalized-gradient approximation (GGA) ===
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| In the GGA, there is an additional dependency on the gradient of <math>n</math>:
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| :<math>
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| E_{\mathrm{xc}}^{\mathrm{GGA}}=\int\epsilon_{\mathrm{xc}}^{\mathrm{GGA}}(n,\nabla n)d^{3}r
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| </math>
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| leading to an additional term in the potential compared to LDA:
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| :<math>
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| v_{\mathrm{xc}}^{\mathrm{GGA}} = \frac{\partial\epsilon_{\mathrm{xc}}^{\mathrm{GGA}}}{\partial n} -
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| \nabla\cdot\frac{\partial\epsilon_{\mathrm{xc}}^{\mathrm{GGA}}}{\partial\nabla n}
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| </math>
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| The GGA that has been the most commonly used in solid-state physics is PBE.{{cite|perdew:prl:1996}}
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| === Meta-GGA ===
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| Compared to the GGAs, the meta-GGA functionals depend additionally on the kinetic-energy density <math>\tau</math> and/or the Laplacian of the electron density <math>\nabla^{2}n</math>:
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| :<math>
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| E_{\mathrm{xc}}^{\mathrm{MGGA}}=\int\epsilon_{\mathrm{xc}}^{\mathrm{MGGA}}(n,\nabla n,\nabla^{2}n,\tau)d^{3}r
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| </math>
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| leading to
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| :<math>
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| \hat{v}_{\mathrm{xc}}\psi_{i} =
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| \frac{\delta E_{\mathrm{xc}}^{\mathrm{MGGA}}}{\delta\psi_{i}^{*}} =
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| \left(\frac{\partial\epsilon_{\mathrm{xc}}^{\mathrm{MGGA}}}{\partial n} -
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| \nabla\cdot\frac{\partial\epsilon_{\mathrm{xc}}^{\mathrm{MGGA}}}{\partial\nabla n} +
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| \nabla^2\frac{\partial\epsilon_{\mathrm{xc}}^{\mathrm{MGGA}}}{\partial\nabla^2 n}
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| \right)\psi_{i} -
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| \frac{1}{2}\nabla\cdot\left(\frac{\partial\epsilon_{\mathrm{xc}}^{\mathrm{MGGA}}}{\partial \tau}
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| \nabla\psi_{i}\right)
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| </math>
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| where the last term is of non-multiplicative nature and arises due to the dependency of the functional on <math>\tau</math>.{{cite|neumann:mp:1996}}{{cite|sun:prb:11}} With such a non-multiplicative potential the method lies outside the traditional Kohn-Sham scheme,{{cite|kohn:pr:1965}} but rather belongs to the generalized Kohn-Sham scheme.{{cite|seidl:prb:96}}
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| Although '''meta-GGAs''' are slightly more expensive than GGAs, they are still fast to evaluate and appropriate for very large systems. Furthermore, meta-GGAs can be more accurate than GGAs and more broadly applicable.
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| == How to ==
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| *LDA and GGA: {{TAG|GGA}}
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| *Meta-GGA: {{TAG|METAGGA}}
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| How to do a [[band-structure calculation using meta-GGA functionals]].
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| ----
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| [[Category:VASP|PAW]][[Category:Exchange-correlation functionals]]
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