LDAUTYPE: Difference between revisions

Revision as of 15:19, 6 April 2022

LDAUTYPE = 1 | 2 | 4
Default: LDAUTYPE = 2

Description: LDAUTYPE specifies which type of DFT+U approach will be used.

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. Setting LDAU=.TRUE. in the INCAR file switches on DFT+U. The first VASP DFT+U calculations, including some additional technical details on the VASP implementation, can be found in Ref. ^[1] (the original implementation was done by Olivie Bengone ^[2] and Georg Kresse).

LDAUTYPE=1: The rotationally invariant DFT+U introduced by Liechtenstein et al.^[3]

This particular flavour of DFT+U is of the form

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 U_γ₁_γ₃_γ₂_γ₄ 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} ,

F^{2}

,

F^{4}

, and

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

F^{0}

).

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,

U

and

J

(LDAUU and LDAUJ, respectively).

U

and

J

can also be extracted from constrained-LSDA calculations.

These translate into values for the Slater integrals in the following way (as implemented in VASP at the moment):

$L\;$	$F^{0}\;$	$F^{2}\;$	$F^{4}\;$	$F^{6}\;$
$1\;$	$U\;$	$5J\;$	-	-
$2\;$	$U\;$	${\frac {14}{1+0.625}}J$	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 DFT+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 LSDA 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 LSDA 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 DFT+U, introduced by Dudarev et al.^[4]

This flavour of DFT+U is of the following form:

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/":): {\displaystyle E_{\mathrm{DFT+U}}=E_{\mathrm{LSDA}}+\frac{(U-J)}{2}\sum_\sigma \left[ \left(\sum_{m_1} n_{m_1,m_1}^{\sigma}\right) - \left(\sum_{m_1,m_2} \hat n_{m_1,m_2}^{\sigma} \hat n_{m_2,m_1}^{\sigma} \right) \right]. }

This can be understood as adding a penalty functional to the LSDA total energy expression that forces the on-site occupancy matrix in the direction of idempotency,

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}^{{\sigma }}={\hat n}^{{\sigma }}{\hat n}^{{\sigma }} .

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.

Note: in Dudarev's approach the parameters U and J do not enter seperately, only the difference 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/":): {\displaystyle U-J} is meaningful.

LDAUTYPE=4: same as LDAUTYPE=1, but LDA+U instead of LSDA+U (i.e. no LSDA exchange splitting).

In the LDA+U case the double counting energy is given by,

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).

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 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 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/":): {\displaystyle J} (LDAUU and LDAUJ, respectively). It is therefore not meaningful to compare the total energies resulting from calculations with different 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 and/or 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/":): {\displaystyle J} , or 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/":): {\displaystyle U-J} and in case of Dudarev's approach (LDAUTYPE=2).

Note on bandstructure calculation: the CHGCAR file contains only information up to angular momentum quantum number L=LMAXMIX for the on-site PAW occupancy matrices. When the CHGCAR file is read and kept fixed in the course of the calculations (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 LMAXMIX to 4 (d elements) and 6 (f elements).

Related Tags and Sections

LDAU, LDAUL, LDAUU, LDAUJ, LDAUPRINT, LMAXMIX

Examples that use this tag

References

Contents

[rohrbach:jcp:03-1] A. Rohrbach, J. Hafner, and G. Kresse J. Phys.: Condens. Matter 15, 979 (2003).

[Bengone:prb:00-2] O. Bengone, M. Alouani, P. Blöchl, and J. Hugel, Phys. Rev. B 62, 16392 (2000).

[liechtenstein:prb:95-3] A. I. Liechtenstein, V. I. Anisimov, and J. Zaane, Phys. Rev. B 52, R5467 (1995).

[dudarev:prb:98-4] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).

[1]

[2]

[3]

[4]

@@ Line 70: / Line 70: @@
 :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.
-:'''Note''': in Dudarev's approach the parameters ''U'' and ''J'' do not enter seperately, only the difference (<math>U-J</math>) is meaningful.
+:'''Note''': in Dudarev's approach the parameters ''U'' and ''J'' do not enter seperately, only the difference <math>U-J</math> is meaningful.
 *{{TAG|LDAUTYPE}}=4: same as {{TAG|LDAUTYPE}}=1, but LDA+U instead of LSDA+U (i.e. no LSDA exchange splitting).