1 Band calculations: Band calculations: Theory and Applications Theory and Applications • Local density approximation (LDA) • Generalized gradient approximation (GGA) • LDA+U • LDA+DMFT http://alps.comp-phys.org/mediawiki-1.9.3/index.php/DFT-short-course Lecture 2: Lecture 2: Different approximations for the Different approximations for the exchange exchange - - correlation functional in DFT correlation functional in DFT PDF created with pdfFactory Pro trial version www.pdffactory.com
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Theory and Applications Lecture 2: Different approximations for the exchange …alps.comp-phys.org/mediawiki/images/8/83/Lecture2.pdf · 2009. 3. 9. · GGA: Generalized Gradient
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Band calculations:Band calculations:Theory and ApplicationsTheory and Applications
• Local density approximation (LDA) • Generalized gradient approximation (GGA)• LDA+U• LDA+DMFT
Lecture 2:Lecture 2: Different approximations for the Different approximations for the exchangeexchange--correlation functional in DFTcorrelation functional in DFT
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For the description of isolated molecules nor For the description of isolated molecules nor L(S)DA neither GGA cannot be used !L(S)DA neither GGA cannot be used !
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We want to compute U, from LDA,We want to compute U, from LDA,but how much Coulomb interaction is in the LDA? but how much Coulomb interaction is in the LDA?
LetLet’’s think that:s think that:
In atomic limit:In atomic limit:
So in LDA:So in LDA:
This derivative can be estimatedThis derivative can be estimatednumerically in supernumerically in super--cell calculation:cell calculation:
orbital energy
center of gravity
occupancy
DC termDC term
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LDA+U: LDA+U: HowHow to to chosechose DC ? DC ? NoNo definitedefinite answeranswer..
LDALSDAGGA
LDA+ULDA+U
LDA+DMFT
In simple LDA:
Let's think that correlated electrons can be described, as
,
NiO in LDA+Uwith differentDC terms:
(1) Fully localized limit: works for strongly correlated materials
(2) Near MF limit: works for materials with intermediatecorrelation strength
PRB 48, 16929 (1993) PRB 67, 153106 (2003)
Due to hybridization and non-liniarity Ec, it's not easy to extract the part of e-e Coulomb repulsion taken into account on LDA level for TM-d orbitals.
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LDA+DMFT: The way to include dynamical LDA+DMFT: The way to include dynamical correlations on the top correlations on the top realreal band structure.band structure.
LDALSDAGGA
LDA+U
LDA+LDA+DMFTDMFT
LDA: Reality:
T≠0
nstart n VH(k)
Hscf(k)
G(τ)Imp. solver
LDA loop
DMFT loop
FULLSCF loop
,,Pay attention: most of the LDA+DMFT calculations presently available are not self-consistent !
There are very few exceptions where authors repeat LDA part after scf DMFT solution:
Phys. Rev. Lett. 101, 096405 (2008)
Phys. Rev. B 71, 125119 (2005)
cond-mat/0106308
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LDA+DMFT: The way to include dynamical LDA+DMFT: The way to include dynamical correlations on the top correlations on the top realreal band structure.band structure.
LDALSDAGGA
LDA+U
LDA+LDA+DMFTDMFT
nstart n VH(k)
Hscf(k)
G(τ)Imp. solver
LDA loop
DMFT loop
FULLSCF loop
,,
Q: When full self consistent LDA+DMFT needed ?A: When the number of electrons is changed significantly
Ce-alpha
LDA: n=1.19LDA+DMFT: n=1.06
Other problems of LDA+DMFT:
1) Inherited from LDA+U, DC problem and choice of U,J
2) How to retrieve correlated only part of hamiltonian from LDA ?
3) Specific DMFT problems like choice of solver, temperature etc.
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•• Relativistic treatment Relativistic treatment of electrons. of electrons.
•• SemiSemi--relativistic treatment relativistic treatment of electrons (SO only for of electrons (SO only for core level electrons). core level electrons).