LDAUTYPE: Difference between revisions
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*{{TAG|LDAUTYPE}}=1: The rotationally invariant LSDA+U introduced by Liechtenstein ''et al.''<ref name="liechtenstein:prb:95"/> | *{{TAG|LDAUTYPE}}=1: The rotationally invariant LSDA+U introduced by Liechtenstein ''et al.''<ref name="liechtenstein:prb:95"/> | ||
::<math> | |||
E_{\rm HF}=\frac{1}{2} \sum_{\{\gamma\}} | |||
(U_{\gamma_1\gamma_3\gamma_2\gamma_4} - | |||
U_{\gamma_1\gamma_3\gamma_4\gamma_2}){ \hat | |||
n}_{\gamma_1\gamma_2}{\hat n}_{\gamma_3\gamma_4} | |||
</math> | |||
:and is determined by the PAW on site occupancies | |||
::<math> | |||
{\hat n}_{\gamma_1\gamma_2} = \langle \Psi^{s_2} \mid m_2 \rangle | |||
\langle m_1 \mid \Psi^{s_1} \rangle | |||
</math> | |||
:and the (unscreened) on site electron-electron interaction | |||
::<math> | |||
U_{\gamma_1\gamma_3\gamma_2\gamma_4}= \langle m_1 m_3 \mid | |||
\frac{1}{|\mathbf{r}-\mathbf{r}^\prime|} \mid m_2 m_4 \rangle | |||
\delta_{s_1 s_2} \delta_{s_3 s_4} | |||
</math> | |||
:where |''m''⟩ are real spherical harmonics of angular momentum ''L''={{TAG|LDAUL}}. | |||
:The unscreened e-e interaction ''U''<sub>γ<sub>1</sub></sub><sub>γ<sub>3</sub></sub><sub>γ<sub>2</sub></sub><sub>γ<sub>4</sub></sub> can be written in terms of the Slater integrals <math>F^0</math>, <math>F^2</math>, <math>F^4</math>, and <math>F^6</math> (f-electrons). Using values for the Slater integrals calculated from atomic orbitals, however, would lead to a large overestimation of the true e-e interaction, since in solids the Coulomb interaction is screened (especially <math>F^0</math>). | |||
:In practice these integrals are therefore often treated as parameters, i.e., adjusted to reach agreement with experiment in some sense: equilibrium volume, magnetic moment, band gap, structure. They are normally specified in terms of the effective on site Coulomb- and exchange parameters, ''U'' and ''J'' ({{TAG|LDAUU}} and {{TAG|LDAUJ}}, respectively). ''U'' and ''J'' are sometimes extracted from constrained-LSDA calculations. | |||
:These translate into values for the Slater integrals in the following way (as implemented in VASP at the moment): | |||
::{| cellpadding="5" cellspacing="0" border="1" | |||
| <math>L\;</math> || <math>F^0\;</math> || <math>F^2\;</math> || <math>F^4\;</math> || <math>F^6\;</math> | |||
|- | |||
| <math>1\;</math> || <math>U\;</math> || <math>5J\;</math> || - || - | |||
|- | |||
| <math>2\;</math> || <math>U\;</math> || <math>\frac{14}{1+0.625}J</math> || <math>0.625 F^2\;</math> || - | |||
|- | |||
| <math>3\;</math> || <math>U\;</math> || <math>\frac{6435}{286+195 \cdot 0.668+250 \cdot 0.494}J</math> || <math>0.668 F^2\;</math> || <math>0.494 F^2\;</math> | |||
|} | |||
:The essence of the L(S)DA+U method consists of the assumption that one may now write the total energy as: | |||
::<math> | |||
E_{\mathrm{tot}}(n,\hat n)=E_{\mathrm{DFT}}(n)+E_{\mathrm{HF}}(\hat n)-E_{\mathrm{dc}}(\hat n) | |||
</math> | |||
:where the Hartree-Fock like interaction replaces the L(S)DA on site due to the fact that one subtracts a double counting energy (<math>E_{\mathrm{dc}}</math>) which supposedly equals the on site L(S)DA contribution to the total energy, | |||
::<math> | |||
E_{\mathrm{dc}}(\hat n) = \frac{U}{2} {\hat n}_{\mathrm{tot}}({\hat n}_{\mathrm{tot}}-1) - | |||
\frac{J}{2} \sum_\sigma {\hat n}^\sigma_{\mathrm{tot}}({\hat n}^\sigma_{\mathrm{tot}}-1). | |||
</math> | |||
*{{TAG|LDAUTYPE}}=2: The simplified (rotationally invariant) approach to the LSDA+U, introduced by Dudarev ''et al.''<ref name="dudarev:prb:98"/> | *{{TAG|LDAUTYPE}}=2: The simplified (rotationally invariant) approach to the LSDA+U, introduced by Dudarev ''et al.''<ref name="dudarev:prb:98"/> |
Revision as of 20:15, 1 March 2011
LDAUTYPE = 1 | 2 | 4
Default: LDAUTYPE = 2
Description: LDAUTYPE specifies which type of L(S)DA+U approach will be used.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): E_{{{\rm {HF}}}}={\frac {1}{2}}\sum _{{\{\gamma \}}}(U_{{\gamma _{1}\gamma _{3}\gamma _{2}\gamma _{4}}}-U_{{\gamma _{1}\gamma _{3}\gamma _{4}\gamma _{2}}}){{\hat n}}_{{\gamma _{1}\gamma _{2}}}{{\hat n}}_{{\gamma _{3}\gamma _{4}}}
- and is determined by the PAW on site occupancies
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {{\hat n}}_{{\gamma _{1}\gamma _{2}}}=\langle \Psi ^{{s_{2}}}\mid m_{2}\rangle \langle m_{1}\mid \Psi ^{{s_{1}}}\rangle
- and the (unscreened) on site electron-electron interaction
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): U_{{\gamma _{1}\gamma _{3}\gamma _{2}\gamma _{4}}}=\langle m_{1}m_{3}\mid {\frac {1}{|{\mathbf {r}}-{\mathbf {r}}^{\prime }|}}\mid m_{2}m_{4}\rangle \delta _{{s_{1}s_{2}}}\delta _{{s_{3}s_{4}}}
- where |m⟩ are real spherical harmonics of angular momentum L=LDAUL.
- The unscreened e-e interaction Uγ1γ3γ2γ4 can be written in terms of the Slater integrals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): F^{0} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): F^{2} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): F^{4} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): F^{6} (f-electrons). Using values for the Slater integrals calculated from atomic orbitals, however, would lead to a large overestimation of the true e-e interaction, since in solids the Coulomb interaction is screened (especially Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): F^{0} ).
- In practice these integrals are therefore often treated as parameters, i.e., adjusted to reach agreement with experiment in some sense: equilibrium volume, magnetic moment, band gap, structure. They are normally specified in terms of the effective on site Coulomb- and exchange parameters, U and J (LDAUU and LDAUJ, respectively). U and J are sometimes extracted from constrained-LSDA calculations.
- These translate into values for the Slater integrals in the following way (as implemented in VASP at the moment):
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): L\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): F^{0}\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): F^{2}\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): F^{4}\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): F^{6}\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 1\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): U\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 5J\; - - Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 2\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): U\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\frac {14}{1+0.625}}J Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 0.625F^{2}\; - Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 3\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): U\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\frac {6435}{286+195\cdot 0.668+250\cdot 0.494}}J Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 0.668F^{2}\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 0.494F^{2}\;
- The essence of the L(S)DA+U method consists of the assumption that one may now write the total energy as:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): E_{{{\mathrm {tot}}}}(n,{\hat n})=E_{{{\mathrm {DFT}}}}(n)+E_{{{\mathrm {HF}}}}({\hat n})-E_{{{\mathrm {dc}}}}({\hat n})
- where the Hartree-Fock like interaction replaces the L(S)DA on site due to the fact that one subtracts a double counting energy (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): E_{{{\mathrm {dc}}}} ) which supposedly equals the on site L(S)DA contribution to the total energy,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): E_{{{\mathrm {dc}}}}({\hat n})={\frac {U}{2}}{{\hat n}}_{{{\mathrm {tot}}}}({{\hat n}}_{{{\mathrm {tot}}}}-1)-{\frac {J}{2}}\sum _{\sigma }{{\hat n}}_{{{\mathrm {tot}}}}^{\sigma }({{\hat n}}_{{{\mathrm {tot}}}}^{\sigma }-1).
- LDAUTYPE=2: The simplified (rotationally invariant) approach to the LSDA+U, introduced by Dudarev et al.[2]
Related Tags and Sections
LDAU, LDAUL, LDAUU, LDAUJ, LDAUPRINT