Category:DFT+U

The LDA and semilocal GGA functionals often fail to describe systems with localized (strongly correlated) ${\displaystyle d}$ or ${\displaystyle f}$ electrons (this manifests itself primarily in the form of unrealistic one-electron energies or too small magnetic moments). In some cases this can be remedied by introducing on the ${\displaystyle d}$ or ${\displaystyle f}$ atom a strong intra-atomic interaction in a simplified (screened) Hartree-Fock like manner (${\displaystyle E^{\text{HF}}[\hat{n}]}$), as an on-site replacement of the LDA/GGA functional:

${\displaystyle E_{\text{xc}}^{\text{DFT}+U}[n,\hat{n}]=E_{\text{xc}}^{\text{DFT}}[n]+E^{\text{HF}}[\hat{n}]-E_{\text{dc}}[\hat{n}]}$

where ${\displaystyle E_{\text{dc}}[\hat{n}]}$ is the double-counting term and ${\displaystyle \hat{n}}$ is the on-site occupancy matrix of the ${\displaystyle d}$ or ${\displaystyle f}$ electrons. This approach is known as the DFT+U method (traditionally called LSDA+U^[1]).

The first VASP DFT+U calculations, including some additional technical details on the VASP implementation, can be found in Ref. ^[2] (the original implementation was done by Olivier Bengone ^[3] and Georg Kresse).

More detail about the formalism is provided below.

Theory

DFT+U is a method that was proposed to improve the description of systems with strongly correlated ${\displaystyle d}$ or ${\displaystyle f}$ electrons, like antiferromagnetic NiO for instance, that are usually inaccurately described with the standard LDA and GGA functionals^[1]. Several variants of the DFT+U method exist (see Refs. ^[4][5] for reviews) that differ for instance in the way the double counting term ${\displaystyle E_{\text{dc}}[\hat{n}]}$ 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.^[6]

This particular flavour of DFT+U is of the form

${\displaystyle E^{\mathrm{H}\mathrm{F}}[\hat{n}]=\frac{1}{2}\sum _{\{\gamma \}}(U_{{\gamma }_1{\gamma }_3{\gamma }_2{\gamma }_4}-U_{{\gamma }_1{\gamma }_3{\gamma }_4{\gamma }_2}){\hat{n}}_{{\gamma }_1{\gamma }_2}{\hat{n}}_{{\gamma }_3{\gamma }_4}}$

and is determined by the PAW on-site occupancies

${\displaystyle {\hat{n}}_{{\gamma }_1{\gamma }_2}=⟨{\mathrm{\Psi }}^{s_2}\mid m_2⟩⟨m_1\mid {\mathrm{\Psi }}^{s_1}⟩}$

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

${\displaystyle U_{{\gamma }_1{\gamma }_3{\gamma }_2{\gamma }_4}=⟨m_1{m}_3\mid \frac{1}{|\mathbf{𝐫}-{\mathbf{𝐫}}^{\prime }|}\mid m_2{m}_4⟩{\delta }_{s_1{s}_2}{\delta }_{s_3{s}_4}}$

where ${\displaystyle |m⟩}$ represents a real spherical harmonics of angular momentum ${\displaystyle l}$=LDAUL.

The unscreened electron-electron interaction ${\displaystyle U_{{\gamma }_1{\gamma }_3{\gamma }_2{\gamma }_4}}$ can be written in terms of the Slater integrals ${\displaystyle F^0}$, ${\displaystyle F^2}$, ${\displaystyle F^4}$, and ${\displaystyle F^6}$ (${\displaystyle f}$ 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 ${\displaystyle F^0}$).

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, ${\displaystyle U}$ and ${\displaystyle J}$ (LDAUU and LDAUJ, respectively). ${\displaystyle U}$ and ${\displaystyle J}$ can also be extracted from constrained-DFT calculations^[7][8].

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

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

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

${\displaystyle E^{\mathrm{D}\mathrm{F}\mathrm{T}+U}[n,\hat{n}]=E^{\mathrm{D}\mathrm{F}\mathrm{T}}[n]+E^{\mathrm{H}\mathrm{F}}[\hat{n}]-E_{\mathrm{d}\mathrm{c}}[\hat{n}]}$

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

${\displaystyle E_{\mathrm{d}\mathrm{c}}[\hat{n}]=\frac{U}{2}{\hat{n}}_{\mathrm{t}\mathrm{o}\mathrm{t}}({\hat{n}}_{\mathrm{t}\mathrm{o}\mathrm{t}}-1)-\frac{J}{2}\sum _{\sigma }{\hat{n}}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{\sigma }({\hat{n}}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{\sigma }-1).}$

• LDAUTYPE=2: The simplified (rotationally invariant) approach to the DFT+U, introduced by Dudarev et al.^[9]

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

${\displaystyle E^{\mathrm{D}\mathrm{F}\mathrm{T}\mathrm{+}\mathrm{U}}=E^{\mathrm{D}\mathrm{F}\mathrm{T}}+\frac{(U-J)}{2}\sum _{\sigma }[\left(\sum _{m_1}{n}_{m_1,m_1}^{\sigma }\right)-\left(\sum _{m_1,m_2}{\hat{n}}_{m_1,m_2}^{\sigma }{\hat{n}}_{m_2,m_1}^{\sigma }\right)].}$

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,

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

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 ${\displaystyle U}$ and ${\displaystyle J}$ do not enter seperately, only the difference ${\displaystyle U-J}$ is meaningful.

• LDAUTYPE=3: This option is for the calculation of the parameter ${\displaystyle U}$ using the linear response approach from Ref. ^[10]. 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

${\displaystyle E_{\mathrm{d}\mathrm{c}}[\hat{n}]=\frac{U}{2}{\hat{n}}_{\mathrm{t}\mathrm{o}\mathrm{t}}({\hat{n}}_{\mathrm{t}\mathrm{o}\mathrm{t}}-1)-\frac{J}{2}\sum _{\sigma }{\hat{n}}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{\sigma }({\hat{n}}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{\sigma }-1).}$

Additional resources

How to

DFT+U can be switched on with the LDAU tag, while the LDAUTYPE tag determines the DFT+U flavor that is used. LDAUL specifies the ${\displaystyle l}$-quantum number for which the on-site interaction is added, and the effective on-site Coulomb and exchange interactions are set (in eV) with the LDAUU and LDAUJ tags, respectively. Note that it is recommended to increase LMAXMIX to 4 for d-electrons or 6 for f-elements.

Tutorials

• Tutorial for DFT+U on NiO.
• Tutorial for LSDA+U and pseudopotential selection of NiO and Heisenberg model for NiO using DFT+U.

Lectures

• Lecture on the optical gap, introduces DFT+U towards the end of the lecture.

References