Intro to DFT+U

Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.

Recent developments in Hubbard-augmented DFT

Heather Kulik02/03/12

http://www.stanford.edu/~hkulik


Nicola MarzariMIT/EPFL

Matteo CococcioniU Minnesota

Quantum-ESPRESSO

http://www.quantum-espresso.orgOpen source plane-wave, pseudopotential codeOther codes with similar implementations: VASP, ONETEP, Qbox, others?Coming soon: TeraChem, GPAW?






Density functional theory

Exact…in theory One-to-one mapping of many-body interacting system onto a non-interacting one.Quantum mechanis becomes computationally tractable.




Exact…in theory

Approximations in practice

One-to-one mapping of many-body interacting system onto a non-interacting one.Quantum mechanis becomes computationally tractable.




Exact…in theory

Approximations in practiceCharge transfer (short or long range)Electron delocalizationWrong dissociations…all some form of self-interaction error.

One-to-one mapping of many-body interacting system onto a non-interacting one.Quantum mechanis becomes computationally tractable.



Electronic structure methods

A density worldview

higher derivatives of the densityadding in Hartree-Fock exchangeparameterizing until the end of time

A “sophisticated” condensed matter electronic structure worldview

Density matrix renormalization groupDynamical mean field theoryGW approximationQuantum Monte Carlo

A wavefunction worldview

Hartree-Fock/MCSCFPerturbative theories + RAS/CAS/etc.

Coupled cluster methods(Some approximation to) Full CI



But I just want results…

My (slightly different) density worldview

Physics-based, parameter free methods to alleviate self-interaction

For 1-1000 atoms (or more with GPUs), approaches that balance accuracy with computational efficiency.



DFT+U



DFT+U DFT+U+V



DFT+U DFT+U+V

DFT+U(R)



DFT+U DFT+U+V

DFT+U(R)in practice



DFT+U



Basic Hubbard model Hamiltonian

Conductor to insulator transition









DFT conductors to DFT+U insulators






DFT

E

conductors






DFT

E

DFT+U

E

conductors






DFT

E

DFT+U

E

conductorsconductors insulators

V.I. Anisimov, J. Zaanen and O.K. Andersen. Phys. Rev. B, (1991).M. Cococcioni and S. de Gironcoli. Phys. Rev. B, (2005).



DFT+U for moleculesUGE Perera, HJK et al Phys. Rev. Lett. (2010).

HJK et al J. Am. Chem. Soc. (2009).

HJK et al Phys. Rev. Lett. (2006). HJK et al J. Chem. Phys. (2008).

HJK et al Phys. Rev. Lett. (2006). HJK et al Fuel Cell Science (2010).



Physical meaning of DFT+U

N-1 N N+1

Energ

y

# of Electrons

M. Cococcioni and S. de Gironcoli. PRB, 71: 2005.

Energy of an atom

J.P. Perdew, R.G. Parr, M. Levy, and J. L. Balduz, Jr. Phys. Rev. Lett. (1982).




N-1 N N+1

Energ

y

# of Electrons


Energy of an atom





N-1 N N+1

Energ

y

# of Electrons


Energy of an atom





N-1 N N+1

Energ

y

# of Electrons

exact


Energy of an atom





N-1 N N+1

Energ

y

# of Electrons

exact


Energy of an atom


LDA/GGA




N-1 N N+1

Energ

y

# of Electrons

exact


Energy of an atom


LDA/GGA

+U




N-1 N N+1

Energ

y

# of Electrons


Energy of an atom


The “+U” contribution to standard DFT:exact

LDA+U

+U

U is the extent of curvature: we calculate this uniquely for each system.



Choosing occupations1) Select the localized manifold or manifolds for each atom “site”

2) Choose the projections

Results in this talk: Other options:Wannier/Boys functionsPopulation schemes



Linear response U

6+ MXae

n

n’

U is the curvature: We calculate it from linear response:

In lieu of constrained occupations

Bare response due to rigid potential shift on localized manifold

Converged response (from an SCF calculation)



U is a system-dependent property

MXU

(eV)

FeO+ 5.50FeN 4.38MnO 3.41CrO- 2.85CrF 2.00Isoelectronic

Series

Less co

vale

nt

HJK and N. Marzari, J. Chem. Phys. (2010).

6+ MXA property that should be calculated

Electron configurationCovalency/ionicitySpin states/charge statesElement identityCoordination numbers



A self-consistent U

HJK et al., Phys. Rev. Lett. (2006).

Calculate U self-consistently on the DFT+U system:

x7

Most key for when DFT and DFT+U ground states differ



DFT+U+V


Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info. Extending the Hubbard model





IJ K

UIIVIJ VIK

V favors intersite interactions

J. Hubbard Proc. R. Soc. A 285 (1965).J. Hubbard Proc. R. Soc. A 296 (1967).

V. I. Anisimov, I. S. Elfimov, N. Hamada, and K. Terakura Phys. Rev. B 54 (1996).



Functional formExtended Hubbard Model

Campo and Cococcioni, J. Phys. Cond. Matt. (2010).



Functional formExtended Hubbard Model Generalized FLL double counting




Functional formExtended Hubbard Model Generalized FLL double counting




Generalized occupations

m and m’ defined by interacting manifolds

Connection to atomic projections is clear. Wannier basis less so (already bond-centered?)

nII nIJ

nJI nJJ

Block diagonals: on-site standard occupations.



What happens to states

nII nIJ

nJI nJJ

Internal competition




nII nIJ

nJI nJJ

Standard U: Favors integer occupations in block diagonals, weak off-site blocks.





nII nIJ

nJI nJJ

Standard U: Favors integer occupations in block diagonals, weak off-site blocks.

New V term: strong intersite occupations in off diagonal.




MO2 bent linear

Gong, Chem. Rev. 2009 and references therein.

Experiments:

Can theory predict transition?

180

100



MnO2: Single or double well?

r



MnO2: Single or double well?

r



MnO2 hybridization

r



O-M-O Structures

2

2

2

DFT+U+U|r0

+U+VExpt.

DFT +U +U+V

MnO2 1.61 1.70 1.59FeO2 1.59 1.67 1.58CoO2 1.55 1.63 1.56

Angles Bonds

+U|r0: angle from M-O bond fixed to DFT value.

HJK and N. Marzari, J. Chem. Phys. 134, 094103 (2011).



O-M-O Structures

2

2

2

DFT+U+U|r0

+U+VExpt.

DFT +U +U+V

MnO2 1.61 1.70 1.59FeO2 1.59 1.67 1.58CoO2 1.55 1.63 1.56

Angles Bonds

+U|r0: angle from M-O bond fixed to DFT value.

HJK and N. Marzari, J. Chem. Phys. 134, 094103 (2011).



FeO2 Splitting and Angle

+U +V

Expt GS

GS ∠

U = 0 V = 0

U = 5 V = 0

U = 5 V = 2



Solid state applicationsLDA+DMFT+V for VO2

A. S. Belozerov, et al. PRB (2012).

Monoclinic M1

Magnetic susceptibilities

Cheaper than cluster DMFT but yields similar results.



Solid state applicationsLDA+DMFT+V for VO2

A. S. Belozerov, et al. PRB (2012).

Monoclinic M1

Magnetic susceptibilities

Cheaper than cluster DMFT but yields similar results.



Solid state applicationsNiOCubic rock-salt structure

Si and GaAs




DFT+U(R)



Inspiration for a variable U

re we De DE

Erro

r

GGAGGA+U

Errors for 22 MX (X=H,C,N,O,F)

(eV)(eV)(cm-

1/100)

(Åx10)

HJK and N. Marzari. J. Chem. Phys. (2010).HJK and N. Marzari, J. Chem. Phys. (2011).




re we De DE

Erro

r

GGAGGA+U


(eV)(eV)(cm-

1/100)

(Åx10)

HJK and N. Marzari. J. Chem. Phys. (2010).

In DFT+U, we average U over all points. Works well most of the time!!





re we De DE

Erro

r

GGAGGA+U


(eV)(eV)(cm-

1/100)

(Åx10)


Electronic structure in differing bonding regimes

In DFT+U, we average U over all points. Works well most of the time!!





re we De DE

Erro

r

GGAGGA+U


(eV)(eV)(cm-

1/100)

(Åx10)


Electronic structure in differing bonding regimes

In DFT+U, we average U over all points. Works well most of the time!! DFT+U(R), changes

in U incorporated directly for key cases.

!HJK and N. Marzari, J. Chem. Phys. (2011).



Even better with DFT+U(R)




DFT+U Forces

Interpolated




DFT+U Forces

Interpolated

CC value




DFT+U Forces

Interpolated

CC value

In practice, interpolate over forces or interpolate over energies with a common physical reference.



U variation from occupations










Component of forces gradient




From linear response

Component of forces gradient




4F FeO+: U vs. R

1.6 2.6R (Å)

U (eV

)

6

0

ActualPredicted




4F FeO+: U vs. R

1.6 2.6R (Å)

U (eV

)

6

0

ActualPredicted



Predicting U variation from forces













Exiting linear regime for derivatives of forces is a numerical challenge.



Numerical noise in practice

Predicted U trends for 4F FeO+







In principle, the force-based approach is more exact. In practice, it suffers from a greater degree of numerical noise.



When U(R) matters

A metric: when is U ½ of lin.resp. U?



When U(R) matters


Molecule U dU/dR DrU½

2S+ CoC 4.8 -4.0 0.62S- CrN 4.3 -2.3 0.94F+ FeO+ 6.3 -5.0 0.65S+ MnF 2.4 -4.8 0.26S+ CrF 2.0 -0.1 9.0



When U(R) matters




Including more variables



When U(R) matters




Including more variables

Some matter more than others


Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info. Ordering multiple U(R) surfaces

Expt.


Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info. Ordering multiple U(R) surfaces

Aligned at the effectiveunited atom limit

Expt.



DFT+U(R) Improvements

re (Å)

we (cm-1)

De(eV)

1) Binding curves: Errors on worst case subset from MX DFT+U

2) Reaction coordinates: H2 on FeO+

CC value

3) Work in progress: Molecular adsorbates on TM surfaces. Preliminary evidence: U(R) improves binding energies.



in practice



Numerical instabilities

Full manifolds or integer occupations

Unperturbed or rigid occupations

a

Example:




c

c-1




c

c-1



Projection dependence




DFT: significant PSP dependence




DFT: significant PSP dependence

+U: Different Us, less PSP dependence



Multiple manifoldsStrong hybridization between 3d and 4s in TM hydrides

dd ds

sd ss

U3d=(0-1--1)dd

U4s=(0-1--1)ss

In the solid state: Ce 4f/5d/6s, MOFs?


Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info. Angle dependence of n and c

linear

bent



linear

bent





A renormalized U

An equivalent U along a coordinate:

Redefining response functions:

All dependence of U on O-Mn-O angle is from filling/emptying d states!


Heather Kulik

Maybe change colors of this


Conclusions

For transition metals and materials with localized electrons:

DFT+U-works well in most casesDFT+U+V-a balance of localization/delocalization, more general cases like semiconductorsDFT+U(R)-bond breaking for chemical applicationsIn practice, things don’t always go according to plan (method is still not a black box).


Education

Intro to DFT+U