theory and applications lecture 2: different approximations for the exchange...

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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

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 exchangeexchange--correlation functional in DFTcorrelation functional in DFT

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http://alps.comp-phys.org/mediawiki-1.9.3/index.php/DFT-short-course

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DFT: Short summary from previous lecture

Hohenberg-Kohn:

Thomas-Fermi-Dirac: For non-interacting electron gas:

For any electronic system:

Kinetic energy

Coulomb repulsion

Exchange interaction

Kohn-Sham: Real, interacting system

Reference, non-interacting system

But with some strange term

+




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DFT: Kohn-Sham equations (1965)

Kohn-Shamequations

XC energy:

XC potential: It acts locally, but may know about density distribution

in other points




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LDA: Local density approximation for Exc[n]

The key point in DFT is an explicit form The key point in DFT is an explicit form EExcxc[n[n]. ].

LDA: exchange-correlation energy density equals toexchange-correlation energy density of homogeneous electronic gas in given point

LDALDALSDAGGA

LDA+U

LDA+DMFT

due to Dirac (see TFD theory)

It is local in sense that it knows only density in given point




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was obtained from QMC simulation for varying densities [PRL 45, 566 (1980)]

And latter interpolated by Vosko, Wilk and NusairCan. J. Phys. 58, 1200 (1980):

LDA: Local density approximation for Exc[n]

LDALDALSDAGGA

LDA+U

LDA+DMFT

,




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ExtensionExtension ofof LDA: LDA: LocalLocal SpinSpin DensityDensity ApproximationApproximation (LSDA)(LSDA)

LDALSDALSDAGGA

LDA+U

LDA+DMFT




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Advantages and disadvantages of L(S)DAAdvantages and disadvantages of L(S)DA

LDALSDALSDAGGA

LDA+U

LDA+DMFT

• Structural properties of solids are often goodè usually underestimates bulk lattice constants by a small amount:




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Advantages and disadvantages of L(S)DAAdvantages and disadvantages of L(S)DA

LDALSDALSDAGGA

LDA+U

LDA+DMFT

• Structural properties of solids are often goodè usually underestimates bulk lattice constants by a small amount è phonons too stiff LSDA

EXP




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AdvantagesAdvantages andand disadvantagesdisadvantages ofof L(S)DAL(S)DA

LDALSDALSDAGGA

LDA+U

LDA+DMFT

• Structural properties of solids are often goodè usually underestimates bulk lattice constants by a small amount è phonons too stiff

• As a result binding energies are too large

This is due to the fact, that L(S)DA favors electronic densities that are more homogeneous than they should be;




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LDALDA and LSDAand LSDA: : Where they should and should not work ?Where they should and should not work ?

• will work good for isotropic and homogeneous system such as metals.

a) Isolated molecules

+-

LDALSDALSDAGGA

LDA+U

LDA+DMFT

Generally one may expect that

• will be problematic for the description of inhomogeneous systems such as e.g.

b) Polarized insulators

c) Strongly correlated materials

++

--




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GGA: GGA: Generalized Gradient ApproximationGeneralized Gradient Approximation

LDALSDAGGAGGA

LDA+U

LDA+DMFT

Idea: Taylor expansion of the density near “homogeneous gas point”

GGA generally improves magnetic energies (with respect to LSDA)

Total energy calculations for FeTotal energy calculations for Fe

LSDA GGA

J. Phys.: Cond. Matter10, 5081 (1998)

one of contrary instances: PRB 65, 132104 (2002)

Experiment: Experiment: FM, bccFM, bcc




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GGA: GGA: GeneralizedGeneralized GradientGradient ApproximationApproximation

LDALSDAGGAGGA

LDA+U

LDA+DMFT

Lattice constants:

GGA sometimes gives better agreement in structural constants, than LDAPDF created with pdfFactory Pro trial version www.pdffactory.com



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AbAb--initioinitio HH22O O treatmenttreatment

LDALSDAGGAGGA

LDA+U

LDA+DMFT

metaGGA

GGA

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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L(S)DA L(S)DA oror GGA: GGA: bandband gapgap problemproblem

LDALSDAGGAGGA

LDA+U

LDA+DMFT

Band gap for correlated oxides

Important: Kohn-Sham orbital energies have NO explicit physical meaning!

In general they are no more than orbital energies of some auxiliary Kohn-Sham system.

Thus are even not owed to give correct gap !

But, this is not only uncomfortable...PDF created with pdfFactory Pro trial version www.pdffactory.com



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Example of inappropriate use of LDA:

LDALSDAGGAGGA

LDA+U

LDA+DMFT

TiTi--Ti Ti dimerdimers s –– has triplet ground state, has triplet ground state,

i.e. FM !?i.e. FM !?

Analysis of GGA results:

Ti3+: d1




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L(S)DA L(S)DA oror GGA: GGA: bandband gapgap problemproblem

LDALSDAGGAGGA

LDA+U

LDA+DMFT

Experiment:

NiO: CT insulator

Band gap: ~ 4 eV

Crystal fieldsplitting




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LDA+U: Different treatment of LDA+U: Different treatment of physically physically different electrons:different electrons:

LDALSDAGGA

LDA+ULDA+U

LDA+DMFT

s,p – electrons are considered on LDA level

d – electron part of functional is corrected,as it is done in model approaches (Hubbard)‏

double counting term

occupied states :

unoccupied states :

Anisimov et al., J. Phys.: Condens. Matter 9 (1997) 767PDF created with pdfFactory Pro trial version www.pdffactory.com



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Very simple LDA+U example:

LDALSDAGGA

LDA+ULDA+U

LDA+DMFT

Eexact(H) = 1.0 Ry

ELDA(H) = 0.957 Ry ~ Eexact(H)

eLDA (H) = 0.538 Ry << Eexact (H)

Ionization energy of Hydrogen atom:

Calculated U = 0.9448 Ry

Occupied (H) state:eLDA+U = eLDA (H) + U/2 = 1.0104 Ry = 1 Ry

Unoccupied (H+) state:eLDA+U(H+) = eLDA (H) - U/2 = 0.0656 Ry ~0 Ry




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LDA+U LDA+U resultsresults forfor TMOTMO

LDALSDAGGA

LDA+ULDA+U

LDA+DMFT

Problems of LDA+U:

• How to chose U ?• How to chose DC term ?

LDA+U for NiO:U=8 eV, J=0.85 eV

Correct gap,NiO: CT - insulator

Anisimov et al.,J. Phys.: Condens. Matter 9 (1997) 767




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LDA+U: How to chose U ? Calculate it !

LDALSDAGGA

LDA+ULDA+U

LDA+DMFT

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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Conduction band

LDA+U: How to chose U ? Calculate it !LDA+U: How to chose U ? Calculate it !

LDALSDAGGA

LDA+ULDA+U

LDA+DMFT

dn-1 dn+1

ee

Thus, since U can be calculated, Thus, since U can be calculated, LDA+U can be considered as LDA+U can be considered as

fully fully abab--initioinitio..

Important: this type of calculation takes into account screening

Drawback: unfortunately calculated values of U strongly depends on the details of calculation (method, RMT, allowed screening channels)!

ee ee ee ee ee ee ee ee ee




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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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LDA+DMFT LDA+DMFT calculationcalculation exampleexample: : gammagamma--alphaalpha CeCe

LDALSDAGGA

LDA+U

LDA+LDA+DMFTDMFT

LDA+DMFT

LDA

PES: occ. part IPES: unocc. partPRB 28, 7354 (1983)

PRB 55, 2056 (1997)

Experiments:




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KohnKohn--ShamShamequationsequations

•• 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).

Treatment of potential:

•• MTMT--potential, Atomic sphere potential, Atomic sphere approximation (ASA)approximation (ASA) •• Full potentialFull potential

•• PseudopotentialPseudopotential

Choice of wave functions (Method):•• Plane waves (PW)Plane waves (PW) •• Augmented plane waves (APW, LAPW)Augmented plane waves (APW, LAPW) •• MTMT--orbitals (MTO, LMTO)orbitals (MTO, LMTO) •• Lin. Comb. of atomic orbitals (LCAO)Lin. Comb. of atomic orbitals (LCAO) •• Gaussians Gaussians

Approximations for exchange correlation part:

LDALDALSDALSDA

GGAGGA LDA+ULDA+ULDA+DMFTLDA+DMFTmeta GGAmeta GGA

What is the difference between What is the difference between different band structure calculations ?different band structure calculations ?




theory and applications lecture 2: different approximations for the exchange...

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