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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)
3
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+
4
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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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)
22
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)
23
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