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| {{TAGDEF|LDAUTYPE|1 {{!}} 2 {{!}} 4|2}} | | {{TAGDEF|LDAUTYPE|1 {{!}} 2 {{!}} 4|2}} |
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| Description: {{TAG|LDAUTYPE}} specifies which type of DFT+U approach will be used. | | Description: {{TAG|LDAUTYPE}} specifies the DFT+U variant that will be used. |
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| The semilocal LDA and GGA functionals often fail to describe systems with localized (strongly correlated) ''d'' and ''f''-electrons (this manifests itself primarily in the form of unrealistic one-electron energies and too small magnetic moments). In some cases this can be remedied by introducing a strong intra-atomic interaction in a (screened) Hartree-Fock like manner, as an on-site replacement of the semilocal functional. This approach is commonly known as the DFT+U method (traditionally called L(S)DA+U). Setting {{TAG|LDAU}}=.TRUE. in the {{FILE|INCAR}} file switches on DFT+U. | | The following variants of the [[DFT+U: formalism|DFT+U approach]] are available: |
| The first VASP DFT+U calculations, including some additional technical details on the VASP implementation, can be found in Ref. <ref name="rohrbach:jcp:03"/> (the original implementation was done by Olivier Bengone <ref name="Bengone:prb:00"/> and Georg Kresse).
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| *{{TAG|LDAUTYPE}}=1: The rotationally invariant DFT+U introduced by Liechtenstein ''et al.''<ref name="liechtenstein:prb:95"/> | | *{{TAG|LDAUTYPE}}=1: The rotationally invariant DFT+U introduced by Liechtenstein ''et al.''{{cite|liechtenstein:prb:95}} |
| :This particular flavour of DFT+U is of the form
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| ::<math>
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| E_{\rm HF}=\frac{1}{2} \sum_{\{\gamma\}}
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| (U_{\gamma_1\gamma_3\gamma_2\gamma_4} -
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| U_{\gamma_1\gamma_3\gamma_4\gamma_2}){ \hat
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| n}_{\gamma_1\gamma_2}{\hat n}_{\gamma_3\gamma_4}
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| </math>
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| :and is determined by the <span id="occmat">PAW on-site occupancies
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| ::<math>
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| {\hat n}_{\gamma_1\gamma_2} = \langle \Psi^{s_2} \mid m_2 \rangle
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| \langle m_1 \mid \Psi^{s_1} \rangle
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| </math></span>
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| :and the (unscreened) on-site electron-electron interaction
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| ::<math>
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| U_{\gamma_1\gamma_3\gamma_2\gamma_4}= \langle m_1 m_3 \mid
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| \frac{1}{|\mathbf{r}-\mathbf{r}^\prime|} \mid m_2 m_4 \rangle
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| \delta_{s_1 s_2} \delta_{s_3 s_4}
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| </math>
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| :where |''m''⟩ are real spherical harmonics of angular momentum <math>\ell</math>={{TAG|LDAUL}}.
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| :The unscreened electron-electron interaction <math>U_{\gamma_{1}\gamma_{3}\gamma_{2}\gamma_{4}}</math> 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>).
| | *{{TAG|LDAUTYPE}}=2: The simplified (rotationally invariant) approach to DFT+U, introduced by Dudarev ''et al.''{{cite|dudarev:prb:98}} |
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| :In practice these integrals are often treated as parameters, ''i.e.'', adjusted to reach agreement with experiment for a property like the equilibrium volume, the magnetic moment or the band gap. They are normally specified in terms of the effective on-site Coulomb- and exchange parameters, <math>U</math> and <math>J</math> ({{TAG|LDAUU}} and {{TAG|LDAUJ}}, respectively). <math>U</math> and <math>J</math> can also be extracted from constrained-LSDA calculations. | | *{{TAG|LDAUTYPE}}=3: Linear response ansatz of Cococcioni et al. {{cite|cococcioni:2005}} to compute U. See [[Calculate U for LSDA+U|how to calculate U]]. |
| | {{NB|mind|For {{TAG|LDAUTYPE}}{{=}}3, the {{TAG|LDAUU}} and {{TAG|LDAUJ}} tags specify the strength of the spherical potential acting on the spin-up and spin-down manifolds, respectively.|:}} |
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| :These translate into values for the Slater integrals in the following way (as implemented in VASP at the moment): | | *{{TAG|LDAUTYPE}}=4: Same as {{TAG|LDAUTYPE}}=1, but without exchange splitting. |
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| ::{| cellpadding="5" cellspacing="0" border="1"
| | A method to estimate the parameters for DFT+U is the [[Constrained-random-phase approximation|constrained-random-phase approximation]]. Another method is the linear response ansatz with {{TAG|LDAUTYPE}}=3, mentioned above. On the other hand, in many applications, the DFT+U parameters are used as tuning parameters to fit experimental data. |
| | <math>L\;</math> || <math>F^0\;</math> || <math>F^2\;</math> || <math>F^4\;</math> || <math>F^6\;</math> | | {{NB|tip|For band-structure calculations, increase {{TAG|LMAXMIX}} to 4 (<math>d</math> elements) or 6 (<math>f</math> elements).}} |
| |- | | This is because the {{FILE|CHGCAR}} file contains only information up to angular momentum quantum number set by {{TAG|LMAXMIX}} for the [[LDAUTYPE#occmat|on-site PAW occupancy matrices]]. When the {{FILE|CHGCAR}} file is read and kept fixed in the course of the calculations ({{TAG|ICHARG}}=11), the results will necessarily not be identical to a self-consistent run. The deviations are often large for DFT+U calculations. |
| | <math>1\;</math> || <math>U\;</math> || <math>5J\;</math> || - || - | | {{NB|warning|The total energy will depend on the parameters <math>U</math> ({{TAG|LDAUU}}) and <math>J</math> ({{TAG|LDAUJ}}). It is, therefore, not meaningful to compare the total energies resulting from calculations with different <math>U</math> and/or <math>J</math>; or <math>U-J</math> in the case of Dudarev's approach ({{TAG|LDAUTYPE}}{{=}}2).}} |
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| | <math>2\;</math> || <math>U\;</math> || <math>\frac{14}{1+0.625}J</math> || <math>0.625 F^2\;</math> || -
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| |-
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| | <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>
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| |}
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| :The essence of the DFT+U method consists of the assumption that one may now write the total energy as:
| | It is possible to use {{TAG|LDAUTYPE}}=1, 2, and 3 for a non–spin-polarized calculation with {{TAG|ISPIN}}=1. |
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| ::<math>
| | == Related tags and articles == |
| E_{\mathrm{tot}}(n,\hat n)=E_{\mathrm{DFT}}(n)+E_{\mathrm{HF}}(\hat n)-E_{\mathrm{dc}}(\hat n)
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| </math>
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| :where the Hartree-Fock like interaction replaces the LSDA on site due to the fact that one subtracts a double counting energy <math>E_{\mathrm{dc}}</math>, which supposedly equals the on-site LSDA contribution to the total energy,
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| ::<math>
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| E_{\mathrm{dc}}(\hat n) = \frac{U}{2} {\hat n}_{\mathrm{tot}}({\hat n}_{\mathrm{tot}}-1) -
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| \frac{J}{2} \sum_\sigma {\hat n}^\sigma_{\mathrm{tot}}({\hat n}^\sigma_{\mathrm{tot}}-1).
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| </math>
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| *{{TAG|LDAUTYPE}}=2: The simplified (rotationally invariant) approach to the DFT+U, introduced by Dudarev ''et al.''<ref name="dudarev:prb:98"/>
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| :This flavour of DFT+U is of the following form:
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| ::<math>
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| E_{\mathrm{DFT+U}}=E_{\mathrm{LSDA}}+\frac{(U-J)}{2}\sum_\sigma \left[
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| \left(\sum_{m_1} n_{m_1,m_1}^{\sigma}\right) - \left(\sum_{m_1,m_2}
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| \hat n_{m_1,m_2}^{\sigma} \hat n_{m_2,m_1}^{\sigma} \right) \right].
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| </math>
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| :This can be understood as adding a penalty functional to the LSDA total energy expression that forces the [[#occmat|on-site occupancy matrix]] in the direction of idempotency,
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| ::<math>\hat n^{\sigma} = \hat n^{\sigma} \hat n^{\sigma}</math>.
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| :Real matrices are only idempotent when their eigenvalues are either 1 or 0, which for an occupancy matrix translates to either fully occupied or fully unoccupied levels.
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| :'''Note''': in Dudarev's approach the parameters <math>U</math> and <math>J</math> do not enter seperately, only the difference <math>U-J</math> is meaningful.
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| *{{TAG|LDAUTYPE}}=4: same as {{TAG|LDAUTYPE}}=1, but LDA+U instead of LSDA+U (i.e. no LSDA exchange splitting).
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| :In the LDA+U case the double counting energy is given by,
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| ::<math>
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| E_{\mathrm{dc}}(\hat n) = \frac{U}{2} {\hat n}_{\mathrm{tot}}({\hat n}_{\mathrm{tot}}-1) -
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| \frac{J}{2} \sum_\sigma {\hat n}^\sigma_{\mathrm{tot}}({\hat n}^\sigma_{\mathrm{tot}}-1).
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| </math>
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| ----
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| '''Warning''': it is important to be aware of the fact that when using the L(S)DA+U, in general the total energy will depend on the parameters <math>U</math> and <math>J</math> ({{TAG|LDAUU}} and {{TAG|LDAUJ}}, respectively). It is therefore not meaningful to compare the total energies resulting from calculations with different <math>U</math> and/or <math>J</math>, or <math>U-J</math> and in case of Dudarev's approach ({{TAG|LDAUTYPE}}=2).
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| '''Note on bandstructure calculation''': the {{FILE|CHGCAR}} file contains only information up to angular momentum quantum number <math>\ell</math>={{TAG|LMAXMIX}} for the [[LDAUTYPE#occmat|on-site PAW occupancy matrices]]. When the {{FILE|CHGCAR}} file is read and kept fixed in the course of the calculations ({{TAG|ICHARG}}=11), the results will be necessarily not identical to a selfconsistent run. The deviations are often large for L(S)DA+U calculations. For the calculation of band structures within the L(S)DA+U approach, it is hence strictly required to increase {{TAG|LMAXMIX}} to 4 (<math>d</math> elements) and 6 (<math>f</math> elements).
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| == Related Tags and Sections == | |
| {{TAG|LDAU}}, | | {{TAG|LDAU}}, |
| {{TAG|LDAUL}}, | | {{TAG|LDAUL}}, |
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| {{TAG|LDAUJ}}, | | {{TAG|LDAUJ}}, |
| {{TAG|LDAUPRINT}}, | | {{TAG|LDAUPRINT}}, |
| {{TAG|LMAXMIX}} | | {{TAG|LMAXMIX}}, |
| | {{TAG|DFT+U: formalism}} |
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| {{sc|LDAUTYPE|Examples|Examples that use this tag}} | | {{sc|LDAUTYPE|Examples|Examples that use this tag}} |
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| == References == | | == References == |
| <references> | | <references/> |
| <ref name="rohrbach:jcp:03"> [https://doi.org/10.1088/0953-8984/15/6/325 A. Rohrbach, J. Hafner, and G. Kresse J. Phys.: Condens. Matter '''15''', 979 (2003).]</ref>
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| <ref name="Bengone:prb:00"> [https://doi.org/10.1103/PhysRevB.62.16392 O. Bengone, M. Alouani, P. Blöchl, and J. Hugel, Phys. Rev. B '''62''', 16392 (2000).]</ref>
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| <ref name="liechtenstein:prb:95">[https://doi.org/10.1103/PhysRevB.52.R5467 A. I. Liechtenstein, V. I. Anisimov, and J. Zaane, Phys. Rev. B '''52''', R5467 (1995).]</ref>
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| <ref name="dudarev:prb:98">[https://doi.org/10.1103/PhysRevB.57.1505 S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B '''57''', 1505 (1998).]</ref>
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| </references>
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| ---- | | ---- |
| [[The_VASP_Manual|Contents]]
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| [[Category:INCAR]][[Category:Exchange-correlation functionals]][[Category:LSDA+U]] | | [[Category:INCAR tag]][[Category:Exchange-correlation functionals]][[Category:DFT+U]] |
