On site Coulomb interaction: L(S)DA+U

(Supported as of VASP.4.6.)

LDAU= .TRUE. | .FALSE.

LDAUTYPE= 1 | 2 | 4

LDAUL= [0 | 1 | 2 | 3 array] $\qquad$ LDAUU= [real array] $\qquad$ LDAUJ= [real array]

LDAUPRINT= 0 | 1 | 2

Defaults:
LDAU	= .FALSE.
LDAUTYPE	= 2
LDAUPRINT	= 0

The L(S)DA often fails to describe systems with localized (strongly correlated) $d$ and $f$ electrons (this manifests itself primarily in the form of unrealistic one-electron energies). 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 L(S)DA. This approach is commonly known as the L(S)DA+U method.
Setting LDAU=.TRUE. in the INCAR file switches on the L(S)DA+U.

By means of the LDAUTYPE-tag on specifies which type of L(S)DA+U approach will be used:

• LDAUTYPE=1: The rotationally invariant LSDA+U introduced by Liechtenstein et al. [90], which is of the form
  $$E_{\rm HF}=\frac{1}{2} \sum_{\{\gamma\}} (U_{\gamma_1\gamma_3\gam... ...mma_3\gamma_4\gamma_2}){ \hat n}_{\gamma_1\gamma_2}{\hat n}_{\gamma_3\gamma_4}$$
  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
  $$U_{\gamma_1\gamma_3\gamma_2\gamma_4}= \langle m_1 m_3 \mid \frac{... ...r}-{\bf r}^\prime\vert} \mid m_2 m_4 \rangle \delta_{s_1 s_2} \delta_{s_3 s_4}$$
  ( $\vert m \rangle$ are the spherical harmonics)

  The unscreened e-e interaction $U_{\gamma_1\gamma_3\gamma_2\gamma_4}$ can be written in terms of Slater's 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 e-e interaction, since in solids the Coulomb interaction is screened (especially $F^0$).

  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 $J$. ($U$ 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):

  • $p$-electrons: $F^0=U$, $F^2=5J$
  • $d$-electrons: $F^0=U$, $F^2=\frac{14}{1+0.625}J$, and $F^4=0.625 F^2$
  • $f$-electrons: $F^0=U$, $F^2=\frac{6435}{286+195 \cdot 0.668+250 \cdot 0.494}J$, $F^4=0.668 F^2$, and $F^6=0.494 F^2$

  The essence of the L(S)DA+U method consists of the assumption that one may now write the total energy as:

  $$E_{\rm tot}(n,\hat n)=E_{\rm DFT}(n)+E_{\rm HF}(\hat n)-E_{\rm dc}(\hat n)$$
  where the Hartree-Fock like interaction replaces the L(S)DA on site due to the fact that one subtracts a double counting energy ( $E_{\rm dc}$) which supposedly equals the on site L(S)DA contribution to the total energy,

  $$E_{\rm dc}(\hat n) = \frac{U}{2} {\hat n}_{\rm tot}({\hat n}_{\rm... ... \frac{J}{2} \sum_\sigma {\hat n}^\sigma_{\rm tot}({\hat n}^\sigma_{\rm tot}-1)$$ (6.57)

  
  

• LDAUTYPE=2 (Default): The simplified (rotationally invariant) approach to the LSDA+U, introduced by Dudarev et al. [91]. This flavour of LSDA+U is of the following form:
  $$E_{\rm LSDA+U}=E_{\rm LSDA}+\frac{(U-J)}{2}\sum_\sigma \left[ \l... ...{m_1,m_2} \hat n_{m_1,m_2}^{\sigma} \hat n_{m_2,m_1}^{\sigma} \right) \right]$$

  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, i.e., $\hat n^{\sigma} = \hat n^{\sigma} \hat n^{\sigma}$. (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 meaningfull.

• 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,

  $$E_{\rm dc}(\hat n) = \frac{U}{2} {\hat n}_{\rm tot}({\hat n}_{\rm... ... \frac{J}{2} \sum_\sigma {\hat n}^\sigma_{\rm tot}({\hat n}^\sigma_{\rm tot}-1)$$ (6.58)

  
  

LDAUL= $L_1$ $L_2$ ... specifies the $l$-quantum number (one number for each species) for which the on-site interaction is added.
(-1=no on-site terms added, 1= p, 2= d, 3= f, Default: LDAUL=2)

LDAUU= $U_1$ $U_2$ ... specifies the effective on-site Coulomb interaction parameters.

LDAUJ= $J_1$ $J_2$ ... specifies the effective on-site Exchange interaction parameters.

NB: LDAUL, LDAUU, and LDAUJ must be specified for all atomic species!

LDAUPRINT= 0 | 1 | 2 Controls the verbosity of the L(S)DA+U module.
(0: silent, 1: Write occupancy matrix to OUTCAR, 2: idem 1., plus potential matrix dumped to stdout, Default: LDAUPRINT=0)

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 $U$ and $J$. It is therefore not meaningful to compare the total energies resulting from calculations with different $U$ and/or $J$ (c.q. $U-J$ in case of Dudarev's approach).

Furthermore, since LDA+U usually results in aspherical charge densities at $d$ and $f$ atoms we recommend to set LASPH = .TRUE. in the INCAR file for gradient corrected functionals (see Sec. 6.44). For Ce$_2$O$_3$ for instance, identical results to the FLAPW methods can be only obtained setting LASPH = .TRUE.

Note on bandstructure calculation: The CHGCAR file also contains only information up to LMAXMIX (defaulted to 2) 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 can be (or actually are) 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). (see Sec. 6.63).