intro to dft+u
DESCRIPTION
This is a series of slides prepared by Heather Kulik (http://www.stanford.edu/~hkulik or email hkulik at stanford dot edu) for a talk given at the University of Pennsylvania in February 2012. It covers a basic introduction to DFT+U and related approaches for improving descriptions of transition metals and other systems with localized electrons.TRANSCRIPT
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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
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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?
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
http://www.stanford.edu/~hkulik
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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.
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Density functional theory
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.
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Density functional theory
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.
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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.
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
DFT+U
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
DFT+U DFT+U+V
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
DFT+U DFT+U+V
DFT+U(R)
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
DFT+U DFT+U+V
DFT+U(R)in practice
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
DFT+U
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Basic Hubbard model Hamiltonian
Conductor to insulator transition
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Basic Hubbard model Hamiltonian
Conductor to insulator transition
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Basic Hubbard model Hamiltonian
Conductor to insulator transition
DFT conductors to DFT+U insulators
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Basic Hubbard model Hamiltonian
Conductor to insulator transition
DFT conductors to DFT+U insulators
DFT
E
conductors
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Basic Hubbard model Hamiltonian
Conductor to insulator transition
DFT conductors to DFT+U insulators
DFT
E
DFT+U
E
conductors
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Basic Hubbard model Hamiltonian
Conductor to insulator transition
DFT conductors to DFT+U insulators
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).
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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).
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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).
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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).
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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).
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Physical meaning of DFT+U
N-1 N N+1
Energ
y
# of Electrons
exact
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).
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Physical meaning of DFT+U
N-1 N N+1
Energ
y
# of Electrons
exact
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).
LDA/GGA
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Physical meaning of DFT+U
N-1 N N+1
Energ
y
# of Electrons
exact
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).
LDA/GGA
+U
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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).
The “+U” contribution to standard DFT:exact
LDA+U
+U
U is the extent of curvature: we calculate this uniquely for each system.
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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)
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
DFT+U+V
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info. Extending the Hubbard model
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info. Extending the Hubbard model
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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).
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Functional formExtended Hubbard Model
Campo and Cococcioni, J. Phys. Cond. Matt. (2010).
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Functional formExtended Hubbard Model Generalized FLL double counting
Campo and Cococcioni, J. Phys. Cond. Matt. (2010).
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Functional formExtended Hubbard Model Generalized FLL double counting
Campo and Cococcioni, J. Phys. Cond. Matt. (2010).
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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.
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What happens to states
nII nIJ
nJI nJJ
Internal competition
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
What happens to states
nII nIJ
nJI nJJ
Standard U: Favors integer occupations in block diagonals, weak off-site blocks.
Internal competition
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
What happens to states
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.
Internal competition
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MO2 bent linear
Gong, Chem. Rev. 2009 and references therein.
Experiments:
Can theory predict transition?
180
100
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MnO2: Single or double well?
r
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MnO2: Single or double well?
r
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MnO2 hybridization
r
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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).
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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).
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FeO2 Splitting and Angle
+U +V
Expt GS
GS ∠
U = 0 V = 0
U = 5 V = 0
U = 5 V = 2
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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.
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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.
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Solid state applicationsNiOCubic rock-salt structure
Si and GaAs
Campo and Cococcioni, J. Phys. Cond. Matt. (2010).
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
DFT+U(R)
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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).
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Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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).
In DFT+U, we average U over all points. Works well most of the time!!
HJK and N. Marzari, J. Chem. Phys. (2011).
![Page 56: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/56.jpg)
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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).
Electronic structure in differing bonding regimes
In DFT+U, we average U over all points. Works well most of the time!!
HJK and N. Marzari, J. Chem. Phys. (2011).
![Page 57: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/57.jpg)
Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
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).
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).
![Page 58: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/58.jpg)
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Even better with DFT+U(R)
![Page 59: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/59.jpg)
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Even better with DFT+U(R)
DFT+U Forces
Interpolated
![Page 60: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/60.jpg)
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Even better with DFT+U(R)
DFT+U Forces
Interpolated
CC value
![Page 61: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/61.jpg)
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Even better with DFT+U(R)
DFT+U Forces
Interpolated
CC value
In practice, interpolate over forces or interpolate over energies with a common physical reference.
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U variation from occupations
![Page 63: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/63.jpg)
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U variation from occupations
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U variation from occupations
![Page 65: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/65.jpg)
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U variation from occupations
Component of forces gradient
![Page 66: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/66.jpg)
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U variation from occupations
From linear response
Component of forces gradient
![Page 67: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/67.jpg)
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U variation from occupations
4F FeO+: U vs. R
1.6 2.6R (Å)
U (eV
)
6
0
ActualPredicted
![Page 68: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/68.jpg)
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U variation from occupations
4F FeO+: U vs. R
1.6 2.6R (Å)
U (eV
)
6
0
ActualPredicted
![Page 69: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/69.jpg)
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Predicting U variation from forces
![Page 70: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/70.jpg)
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Predicting U variation from forces
![Page 71: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/71.jpg)
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Predicting U variation from forces
![Page 72: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/72.jpg)
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Predicting U variation from forces
![Page 73: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/73.jpg)
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Predicting U variation from forces
Exiting linear regime for derivatives of forces is a numerical challenge.
![Page 74: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/74.jpg)
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Numerical noise in practice
Predicted U trends for 4F FeO+
![Page 75: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/75.jpg)
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Numerical noise in practice
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Numerical noise in practice
In principle, the force-based approach is more exact. In practice, it suffers from a greater degree of numerical noise.
![Page 77: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/77.jpg)
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When U(R) matters
A metric: when is U ½ of lin.resp. U?
![Page 78: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/78.jpg)
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When U(R) matters
A metric: when is U ½ of lin.resp. U?
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
![Page 79: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/79.jpg)
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When U(R) matters
A metric: when is U ½ of lin.resp. U?
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
Including more variables
![Page 80: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/80.jpg)
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When U(R) matters
A metric: when is U ½ of lin.resp. U?
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
Including more variables
Some matter more than others
![Page 81: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/81.jpg)
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Expt.
![Page 82: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/82.jpg)
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Aligned at the effectiveunited atom limit
Expt.
![Page 83: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/83.jpg)
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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.
![Page 84: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/84.jpg)
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in practice
![Page 85: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/85.jpg)
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Numerical instabilities
Full manifolds or integer occupations
Unperturbed or rigid occupations
a
Example:
![Page 86: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/86.jpg)
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Numerical instabilities
c
c-1
![Page 87: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/87.jpg)
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Numerical instabilities
c
c-1
![Page 88: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/88.jpg)
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Projection dependence
![Page 89: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/89.jpg)
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Projection dependence
DFT: significant PSP dependence
![Page 90: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/90.jpg)
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Projection dependence
DFT: significant PSP dependence
+U: Different Us, less PSP dependence
![Page 91: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/91.jpg)
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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?
![Page 92: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/92.jpg)
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linear
bent
![Page 93: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/93.jpg)
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linear
bent
![Page 94: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/94.jpg)
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![Page 95: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/95.jpg)
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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!
![Page 96: Intro to DFT+U](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5466f125af7959f1238b7126/html5/thumbnails/96.jpg)
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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).