DFT+U: formalism

DFT+U is a method that was proposed to improve the description of systems with strongly correlated $d$ or $f$ electrons, like antiferromagnetic NiO for instance, that are usually inaccurately described with the standard LDA and GGA functionals. Several variants of the DFT+U method exist (see Refs. for reviews) that differ for instance in the way the double counting term $E_{\text{dc}}({\hat {n}})$ is calculated. Three variants of them are implemented in VASP, whose formalism is briefly summarized below.

• LDAUTYPE=1: The rotationally invariant DFT+U introduced by Liechtenstein et al.

This particular flavour of DFT+U is of the form

$E_{\rm {HF}}({\hat {n}})={\frac {1}{2}}\sum _{\{\gamma \}}(U_{\gamma _{1}\gamma _{3}\gamma _{2}\gamma _{4}}-U_{\gamma _{1}\gamma _{3}\gamma _{4}\gamma _{2}}){\hat {n}}_{\gamma _{1}\gamma _{2}}{\hat {n}}_{\gamma _{3}\gamma _{4}}$

and is determined by the PAW on-site occupancies

${\hat {n}}_{\gamma _{1}\gamma _{2}}=\langle \Psi ^{s_{2}}\mid m_{2}\rangle \langle m_{1}\mid \Psi ^{s_{1}}\rangle$

and the (unscreened) on-site electron-electron interaction

$U_{\gamma _{1}\gamma _{3}\gamma _{2}\gamma _{4}}=\langle m_{1}m_{3}\mid {\frac {1}{|\mathbf {r} -\mathbf {r} ^{\prime }|}}\mid m_{2}m_{4}\rangle \delta _{s_{1}s_{2}}\delta _{s_{3}s_{4}}$

where $|m\rangle$ represents a real spherical harmonics of angular momentum $l$=LDAUL.

The unscreened electron-electron interaction $U_{\gamma _{1}\gamma _{3}\gamma _{2}\gamma _{4}}$ can be written in terms of the Slater integrals $F^{0}$, $F^{2}$, $F^{4}$, 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 electron-electron 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 for instance 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-DFT calculations.

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

$L\;$	$F^{0}\;$	$F^{2}\;$	$F^{4}\;$	$F^{6}\;$
$1\;$	$U\;$	$5J\;$	-	-
$2\;$	$U\;$	${\frac {14}{1+0.625}}J$	$0.625F^{2}\;$	-
$3\;$	$U\;$	${\frac {6435}{286+195\cdot 0.668+250\cdot 0.494}}J$	$0.668F^{2}\;$	$0.494F^{2}\;$

The essence of the DFT+U method consists of the assumption that one may now write the total energy as:

$E_{\mathrm {tot} }(n,{\hat {n}})=E_{\mathrm {DFT} }(n)+E_{\mathrm {HF} }({\hat {n}})-E_{\mathrm {dc} }({\hat {n}})$

where the Hartree-Fock-like interaction replaces the semilocal on-site due to the fact that one subtracts a double-counting energy $E_{\mathrm {dc} }$, which supposedly equals the on-site semilocal contribution to the total energy,

$E_{\mathrm {dc} }({\hat {n}})={\frac {U}{2}}{\hat {n}}_{\mathrm {tot} }({\hat {n}}_{\mathrm {tot} }-1)-{\frac {J}{2}}\sum _{\sigma }{\hat {n}}_{\mathrm {tot} }^{\sigma }({\hat {n}}_{\mathrm {tot} }^{\sigma }-1).$

• LDAUTYPE=2: The simplified (rotationally invariant) approach to the DFT+U, introduced by Dudarev et al.

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

$E_{\mathrm {DFT+U} }=E_{\mathrm {LSDA} }+{\frac {(U-J)}{2}}\sum _{\sigma }\left[\left(\sum _{m_{1}}n_{m_{1},m_{1}}^{\sigma }\right)-\left(\sum _{m_{1},m_{2}}{\hat {n}}_{m_{1},m_{2}}^{\sigma }{\hat {n}}_{m_{2},m_{1}}^{\sigma }\right)\right].$

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

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

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 $U-J$ is meaningful.

• LDAUTYPE=3: This option is for the calculation of the parameter $U$ using the linear response approach from Ref. . The steps to use this method are shown for the example of NiO.

• LDAUTYPE=4: same as LDAUTYPE=1, but without exchange splitting (i.e., the total spin-up plus spin-down occupancy matrix is used). The double-counting term is given by

$E_{\mathrm {dc} }({\hat {n}})={\frac {U}{2}}{\hat {n}}_{\mathrm {tot} }({\hat {n}}_{\mathrm {tot} }-1)-{\frac {J}{2}}\sum _{\sigma }{\hat {n}}_{\mathrm {tot} }^{\sigma }({\hat {n}}_{\mathrm {tot} }^{\sigma }-1).$

Related Tags and Sections

LDAU, LDAUTYPE, LDAUL, LDAUU, LDAUJ, LDAUPRINT, LMAXMIX,

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