LDAUTYPE: Difference between revisions

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:and is determined by the PAW on-site occupancies
:and is determined by the PAW on-site occupancies
::<math>
::<span id="occmat"><math>
{\hat n}_{\gamma_1\gamma_2} = \langle \Psi^{s_2} \mid m_2 \rangle
{\hat n}_{\gamma_1\gamma_2} = \langle \Psi^{s_2} \mid m_2 \rangle
\langle m_1 \mid \Psi^{s_1} \rangle
\langle m_1 \mid \Psi^{s_1} \rangle
</math>
</math></span>
:and the (unscreened) on-site electron-electron interaction
:and the (unscreened) on-site electron-electron interaction
::<math>
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: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,
: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,
::<math>\hat n^{\sigma} = \hat n^{\sigma} \hat n^{\sigma}</math>.
::<math>\hat n^{\sigma} = \hat n^{\sigma} \hat n^{\sigma}</math>.



Revision as of 20:49, 1 March 2011

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

Description: LDAUTYPE specifies which type of L(S)DA+U approach will be used.


  • LDAUTYPE=1: The rotationally invariant LSDA+U introduced by Liechtenstein et al.[1]
This particular flavour of LSDA+U is of the form
and is determined by the PAW on-site occupancies
and the (unscreened) on-site electron-electron interaction
where |m⟩ are real spherical harmonics of angular momentum L=LDAUL.
The unscreened e-e interaction Uγ1γ3γ2γ4 can be written in terms of the Slater integrals , , , and (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 ).
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 (LDAUU and LDAUJ, respectively). 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):
- -
-
The essence of the LSDA+U method consists of the assumption that one may now write the total energy as:
where the Hartree-Fock like interaction replaces the LSDA on site due to the fact that one subtracts a double counting energy , which supposedly equals the on-site LSDA contribution to the total energy,
  • LDAUTYPE=2: The simplified (rotationally invariant) approach to the LSDA+U, introduced by Dudarev et al.[2]
This flavour of LSDA+U is of the following form:
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,
.
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,

Related Tags and Sections

LDAU, LDAUL, LDAUU, LDAUJ, LDAUPRINT

References


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