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Advanced xc-functionals: DFT+U

Burak Himmetoglu

● Well known failures of LDA/GGA: transition metal oxides

● Importance of electronic correlations: Mott transition

● Introduction to Hubbard Model

● DFT+U: Formulation and implementation

● Calculation of U

● Some examples and applications

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Failures of LDA/GGA: Transition Metal Oxides

TM ion

Oxygen

● Anti-ferromagnetic (AFM) ground state rhombohedral symmetry

and possible structural distortions.

● Insulating (Mott/Charge transfer type)

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Example: GGA results on NiO

● Anti-ferromagnetic: OK

● Crystal structure cubic: OK

● Crystal field produces the band gap.

● Band gap is too small

● O-p states should be at the top of

the valence band.

Ni2+

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Example: GGA results on FeO

Fe2+

● Anti-ferromagnetic: NO → FM

● Crystal structure cubic: OK

● Ground state is metallic

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Importance of electronic correlationsConsider solid Na(2s22p63s1):

At equilibrium lattice constant a0:

Independent electrons: band theory Half filled band → metal

Consider very large a: ● Half-filled 3s orbital becomes

narrower, but it is still half-filled.● Band theory still gives a metal!

Isolated Na atoms still metallic; what has gone wrong?

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Importance of electronic correlations

● Hopping of electrons → kinetic

energy gain.

● Doubly occupied atomic site →

Coulomb energy cost.

● At small separations K.E. gain favors metallic behavior.

● At large separations, hopping of electrons are not energetically favorable.

● e-e interactions produce an insulating behavior.

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Introduction to Hubbard Model-1

t hopping matrix element→U on-site Coulomb repulsion →

Band term is easy to solve; introduce →

creation/annihilation operators

N: number of atoms

J.Hubbard, Proc. Roy. Soc. Lond. (1963-1967)

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Introduction to Hubbard Model-2

● Metallic when t >> U● Insulating when t << UMott transition

band-shape dependentconstant

Mott N.F.: Proc. Roy. Soc. A62, 416 (1949)

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Introduction to Hubbard Model-3Magnetic properties of the ground state:

2nd order perturbation theory:

Perturbation theory:virtual hopping

processes

● AFM ground state energetically favored.● Situation might change with the inclusion of more bands, bond

angles etc.

● Energy of FM and AFM configurations are the same at lowest order

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LDA/GGA Failures and DFT+U: ● LDA/GGA approximations over delocalize electrons serious problem →

for localized d and f states of transition metals.

● On-site e-e interactions (i.e. U in Hubbard model), are not well

accounted for.

● Energy functional contains self-interaction.

● GGA/LDA describes independent

electronic contributions well.

● Addition of Hubbard model based

corrections on top of LDA/GGA to

correct for localized electrons.

The idea:

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DFT+U functional-1:

LDA/GGAfunctional

Hubbardcorrection

subtract“double counting”

● Ehub

Contains an energy functional based on the Hubbard model →● E

dc Averaged e-e interaction energy to be subtracted (already contained in E→

DFT)

atomic orbital centered on I

KS states

occupation of KS states

occupation matrices:

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DFT+U functional-2:First approximation: → Ignore exchange type terms → Average over atomic orbitals

The double counting term is the sum of the averaged on-site interaction terms:

Collecting the contributions:

With these approximations, the Hubbard energy becomes:

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DFT+U functional-3:Use a representation where the occupation matrices are diagonal (i.e. linear combinations of atomic orbitals):

● EU is minimized for integer occupations of atomic orbitals (or their linear combinations on

the same site):

● Electron/Hole localization on atomic sites are encouraged.

U: spurious curvature of the xc-functional.

DFT+U recovers the difference between

electron affinity and ionization potential

(fundamental gap).

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DFT+U in action: NiO

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DFT+U in action: FeO

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Determining the U:● We have seen that U corresponds to the curvature of the DFT functional

● Then, we can compute energy derivatives to compute U from LDA/GGA

→ linear response

From self-consistent ground state

(screened response)

Kohn-Sham response: (due to re-hybridization

of orbitals)

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Determining the U, 2nd derivatives:● 2nd derivative of energy is not easily accessible.

● Instead, we use a Legendre transformed functional to compute first

derivatives:

Legendre transform

Potential shiftin order to perturb the number of states on site I

Cococcioni et.al. PRB 71, 035105 (2005)

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Determining the U, response matrices:● Treating as a perturbation, Kohn-Sham and screened response matrices are

computed in a supercell to isolate the perturbed atom:

Eg: 2d crystal with 2 atoms per unit cell:

● Create a 2x2 super cell (8 atoms)

● Response matrices will be 8x8

● Larger supercells convergence of the →

computed values of U

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1. An initial self-consistent calculation is

performed in a super-cell.

2. Starting from the saved wavefunction

and potential, perturbation to atomic

sites are added.

3. The response 0 χ at first iteration

4. The response χ is evaluated at self-

consistency.

5. Finally, the effective interaction is

obtained as:

Procedure:

Determining the U, response matrices:

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Importance of computing U● Consistency with the DFT+U functional and with the choice of the set

of localized orbitals, pseudo-potentials and the underlying

approximate xc-functional: the computed U is the one that is needed.

● Sensitivities to spin states, chemical/physical environments and

structural changes are captured by computing U in the relevant

phase.

Example: (MgxFe

1-x)O – HS to LS transition under pressure:

Tsuchiya et.al. PRL 96, 198501 (2006)

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Application: Structural properties of FeO

Broken symmetryphase

● DFT+U can contain multiple local energy minima. ● Correct structural properties of FeO obtained by identifying the right local minimum.

Rhombohedralangle

Cococcioni et.al. PRB 71, 035105 (2005)

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Application: Martensitic transition in Ni2MnGa

● Transition to tetragonal phase is driven by

magnetic (Heisenberg) energy.

● GGA overestimates inter-site exchange couplings,

leading to incorrect energy minima at both

stoichiometric and off stoichiometric compounds.

● GGA+U yields better agreement with experiments.

Himmetoglu et.al. JPCM 24, 185501 (2012)

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Further Developments● Inclusion of the exchange parameter: DFT+U+J

● Inter-site interactions: DFT+U+V

Himmetoglu et.al. PRB 84, 115108 (2011)

Application: e.g. Insulating cubic phase of CuO:

Campo Jr. et.al. JPCM 22, 055602 (2010)

Applications: 1. Unified description of Mott and band insulators 2. Molecules containing transition metals

NiO-GGA+U+V