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.
1. 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 Kulik
02/03/12
2. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Nicola Marzari
MIT/EPFL Quantum-ESPRESSO
Matteo Cococcioni
U Minnesota
http://www.quantum-espresso.org
Open source plane-wave, pseudopotential code
Other codes with similar implementations:
VASP, ONETEP, Qbox, others?
Coming soon: TeraChem, GPAW?
3. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
http://www.stanford.edu/~hkulik
4. 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.
5. 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.
Approximations in practice
6. 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.
Approximations in practice
Charge transfer (short or long range)
Electron delocalization
Wrong dissociations
…all some form of self-interaction error.
7. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Electronic structure methods
A wavefunction worldview
A density worldview
Hartree-Fock/MCSCF
higher derivatives of the density Perturbative theories + RAS/CAS/etc.
adding in Hartree-Fock exchange Coupled cluster methods
parameterizing until the (Some approximation to) Full CI
end of time
A “sophisticated” condensed matter
electronic structure worldview
Density matrix renormalization group
Dynamical mean field theory
GW approximation
Quantum Monte Carlo
8. 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.
9. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
DFT+U
10. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
DFT+U
DFT+U+V
11. 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)
12. 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
13. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
DFT+U
14. 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
15. 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
16. 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
17. 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 E
DFT+U insulators
DFT
conductors
18. 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 E E
DFT+U insulators
DFT DFT+U
conductors
19. 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 E E
DFT+U insulators
DFT DFT+U
conductors insulators
V.I. Anisimov, J. Zaanen and O.K. Andersen. Phys. Rev. B, (1991).
M. Cococcioni and S. de Gironcoli. Phys. Rev. B, (2005).
20. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
DFT+U for molecules
UGE Perera, HJK et al
Phys. Rev. Lett. (2010).
HJK et al J. Am. Chem.
Soc. (2009).
1.0
_
_
_ _ MRCI
6 4
_ DFT+U+
FeOH +CH3
Relative Energy (eV)
0.0
_ _
_
-1.0
_
_ _
_ _
_
_
-2.0
_
_
HJK et al Phys. Rev. Lett. (2006).
-3.0 HJK et al Phys. Rev. Lett. (2006).
HJK et al/CH Chem. Phys. (2008). Fe /CH OH
FeO J. 1 TS1 2 TS2 +
3
4
+
3
HJK et al Fuel Cell Science (2010).
21. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Physical meaning of DFT+U
Energy of an atom
Energy
N-1 N N+1
# of Electrons
J.P. Perdew, R.G. Parr, M. Levy, and J. L. Balduz, Jr. Phys. Rev. Lett. (1982).
M. Cococcioni and S. de Gironcoli. PRB, 71: 2005.
22. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Physical meaning of DFT+U
Energy of an atom
Energy
N-1 N N+1
# of Electrons
J.P. Perdew, R.G. Parr, M. Levy, and J. L. Balduz, Jr. Phys. Rev. Lett. (1982).
M. Cococcioni and S. de Gironcoli. PRB, 71: 2005.
23. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Physical meaning of DFT+U
Energy of an atom
Energy
N-1 N N+1
# of Electrons
J.P. Perdew, R.G. Parr, M. Levy, and J. L. Balduz, Jr. Phys. Rev. Lett. (1982).
M. Cococcioni and S. de Gironcoli. PRB, 71: 2005.
24. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Physical meaning of DFT+U
Energy of an atom
exact
Energy
N-1 N N+1
# of Electrons
J.P. Perdew, R.G. Parr, M. Levy, and J. L. Balduz, Jr. Phys. Rev. Lett. (1982).
M. Cococcioni and S. de Gironcoli. PRB, 71: 2005.
25. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Physical meaning of DFT+U
Energy of an atom
exact
LDA/GGA
Energy
N-1 N N+1
# of Electrons
J.P. Perdew, R.G. Parr, M. Levy, and J. L. Balduz, Jr. Phys. Rev. Lett. (1982).
M. Cococcioni and S. de Gironcoli. PRB, 71: 2005.
26. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Physical meaning of DFT+U
Energy of an atom
exact
LDA/GGA
Energy
+U
N-1 N N+1
# of Electrons
J.P. Perdew, R.G. Parr, M. Levy, and J. L. Balduz, Jr. Phys. Rev. Lett. (1982).
M. Cococcioni and S. de Gironcoli. PRB, 71: 2005.
27. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Physical meaning of DFT+U
Energy of an atom
The “+U” contribution to standard DFT:
exact
LDA+U
Energy
+U
U is the extent of curvature: we
calculate this uniquely for each system.
N-1 N N+1
# of Electrons
J.P. Perdew, R.G. Parr, M. Levy, and J. L. Balduz, Jr. Phys. Rev. Lett. (1982).
M. Cococcioni and S. de Gironcoli. PRB, 71: 2005.
28. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Choosing occupations
1) Select the localized manifold or manifolds for each
atom “site”
2) Choose the projections
Results in this talk: Other options:
Wannier/Boys functions
Population schemes
29. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Linear response U
U is the curvature: We calculate it from linear response:
In lieu of constrained occupations
n’
6 + MX
Converged response
(from an SCF calculation)
n
Bare response due to
rigid potential shift on
localized manifold
30. 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
A property that should be calculated
6 + MX
MX U (eV)
FeO+ 5.50 Electron configuration
Covalency/ionicity
Less covalent
FeN 4.38 Spin states/charge states
MnO 3.41 Element identity
Coordination numbers
CrO- 2.85
CrF 2.00
Isoelectronic
Series
HJK and N. Marzari, J. Chem. Phys. (2010).
31. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
A self-consistent U
Calculate U self-consistently
Most key for when
on the DFT+U system:
DFT and DFT+U
ground states differ
HJK et al., Phys. Rev. Lett. (2006).
32. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
DFT+U+V
33. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Extending the Hubbard model
34. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Extending the Hubbard model
35. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Extending the Hubbard model
J I K
VIJ UII VIK
V favors intersite interactions
J. Hubbard Proc. R. Soc. A 285 (1965). V. I. Anisimov, I. S. Elfimov, N. Hamada, and
J. Hubbard Proc. R. Soc. A 296 (1967). K. Terakura Phys. Rev. B 54 (1996).
36. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Functional form
Extended Hubbard Model
Campo and Cococcioni, J. Phys. Cond. Matt. (2010).
37. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Functional form
Extended Hubbard Model Generalized FLL double counting
Campo and Cococcioni, J. Phys. Cond. Matt. (2010).
38. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Functional form
Extended Hubbard Model Generalized FLL double counting
Campo and Cococcioni, J. Phys. Cond. Matt. (2010).
39. 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
nII nIJ
Connection to atomic projections
is clear. Wannier basis less so
nJI nJJ
(already bond-centered?)
Block diagonals: on-site
standard occupations.
40. 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
Internal competition
41. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
What happens to states
Standard U: Favors integer occupations in
block diagonals, weak off-site blocks. nII nIJ
nJI nJJ
Internal competition
42. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
What happens to states
Standard U: Favors integer occupations in
block diagonals, weak off-site blocks. nII nIJ
New V term: strong intersite occupations
in off diagonal.
nJI nJJ
Internal competition
43. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
MO2 bent linear
Experiments:
180
100
Can theory predict transition? E
Gong, Chem. Rev. 2009 and references therein. q
44. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
MnO2: Single or double well?
0.8
rMn-O=1.55Å rMn-O=1.70Å rMn-O=1.85Å
0.7
Relative energy (eV)
0.6
U=6
0.5 U=4
U=0
0.4
0.3
0.2
0.1
0.0
110 130 150 170 110 130 150 170 110 130 150 170
O-Mn-O Angle (o) O-Mn-O Angle (o) O-Mn-O Angle (o)
r
45. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
MnO2: Single or double well?
0.8
rMn-O=1.55Å rMn-O=1.70Å rMn-O=1.85Å
0.7
Relative energy (eV)
0.6
U=6
0.5 U=4
U=0
0.4
0.3
0.2
0.1
0.0
110 130 150 170 110 130 150 170 110 130 150 170
O-Mn-O Angle (o) O-Mn-O Angle (o) O-Mn-O Angle (o)
r
46. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
MnO2 hybridization
r
47. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
O-M-O Structures
Angles Bonds
DFT +U +U+V
MnO2 1.61 1.70 1.59
2
FeO2 1.59 1.67 1.58
CoO2 1.55 1.63 1.56
2
DFT
+U +U|r0: angle from
+U|r0 M-O bond fixed
+U+V to DFT value.
2 Expt.
HJK and N. Marzari, J. Chem. Phys. 134, 094103 (2011).
48. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
O-M-O Structures
Angles Bonds
DFT +U +U+V
MnO2 1.61 1.70 1.59
2
FeO2 1.59 1.67 1.58
CoO2 1.55 1.63 1.56
2
DFT
+U +U|r0: angle from
+U|r0 M-O bond fixed
+U+V to DFT value.
2 Expt.
HJK and N. Marzari, J. Chem. Phys. 134, 094103 (2011).
49. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
FeO2 Splitting and Angle
Expt GS
GS
U=
0V=
0
U=
5V=
0
U=
5V=
2
+U +V
50. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Solid state applications
LDA+DMFT+V for VO2 Monoclinic
M1
Cheaper than cluster DMFT but yields
similar results.
Magnetic susceptibilities
A. S. Belozerov, et al. PRB (2012).
51. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Solid state applications
LDA+DMFT+V for VO2 Monoclinic
M1
Cheaper than cluster DMFT but yields
similar results.
Magnetic susceptibilities
A. S. Belozerov, et al. PRB (2012).
52. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Solid state applications
NiO
Cubic rock-salt
structure
Si and
GaAs
Campo and Cococcioni, J. Phys. Cond. Matt. (2010).
53. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
DFT+U(R)
54. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Inspiration for a variable U
Errors for 22 MX (X=H,C,N,O,F)
0.40 GGA
0.35 GGA+U
0.30
0.25
Error
0.20
0.15
0.10
0.05
0.00
re e De E
(cm-
(Åx10) (eV) (eV)
1/100
)
HJK and N. Marzari. J. Chem. Phys. (2010).
HJK and N. Marzari, J. Chem. Phys. (2011).
55. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Inspiration for a variable U
Errors for 22 MX (X=H,C,N,O,F)
0.40 GGA
0.35 GGA+U
0.30
0.25
Error
0.20
0.15
0.10
0.05
0.00
re e De E
(cm-
(Åx10) (eV) (eV)
1/100 In DFT+U, we average U
) over all points. Works
HJK and N. Marzari. J. Chem. Phys. (2010).
HJK and N. Marzari, J. Chem. Phys. (2011). well most of the time!
56. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Inspiration for a variable U
Electronic structure in
Errors for 22 MX (X=H,C,N,O,F) differing bonding regimes
0.40 GGA
0.35 GGA+U
0.30
0.25
Error
0.20
0.15
0.10
0.05
0.00
re e De E
(cm-
(Åx10) (eV) (eV)
1/100 In DFT+U, we average U
) over all points. Works
HJK and N. Marzari. J. Chem. Phys. (2010).
HJK and N. Marzari, J. Chem. Phys. (2011). well most of the time!
57. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Inspiration for a variable U
Electronic structure in
Errors for 22 MX (X=H,C,N,O,F) differing bonding regimes
0.40 GGA
0.35 GGA+U
0.30
0.25
Error
0.20
0.15
0.10
0.05
0.00
re e De E
(cm-
(Åx10) (eV) (eV)
1/100 In DFT+U, we average U
DFT+U(R), changes
) over all points. Works
HJK and N. Marzari. J. Chem. Phys. (2010). in U incorporated
HJK and N. Marzari, J. Chem. Phys. (2011). well most ofkey cases.
directly for the time!
58. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Even better with DFT+U(R)
59. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Even better with DFT+U(R)
4
2
0
dE/dR (eV/Å)
-2 Interpolated
-4
-6
-8
DFT+U Forces
-10
1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
Fe-O Distance (Å)
60. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Even better with DFT+U(R)
4 4.0
2
CC value
Relative Energy (eV)
0 3.0
dE/dR (eV/Å)
-2 Interpolated
-4 2.0
-6
1.0
-8
DFT+U Forces 0 U 6
-10
0.0
1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
Fe-O Distance (Å) Fe-O Distance (Å)
61. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Even better with DFT+U(R)
4 4.0
2
CC value
Relative Energy (eV)
0 3.0
dE/dR (eV/Å)
-2 Interpolated
-4 2.0
-6
1.0
-8
DFT+U Forces 0 U 6
-10
0.0
1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
Fe-O Distance (Å) Fe-O Distance (Å)
In practice, interpolate over forces or interpolate over
energies with a common physical reference.
62. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
U variation from occupations
63. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
U variation from occupations
64. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
U variation from occupations
65. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
U variation from occupations
Component of
forces gradient
66. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
U variation from occupations
Component of From linear
forces gradient response
67. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
U variation from occupations
6
6
U
Actualfwd.diff.
U0
5
Predicted
Hubbard U (eV)
U (eV) 4
3
2
1 4 FeO+: U vs. R
00
1.6
1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6
2.6
R (Å)
Internuclear Separation (Å)
68. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
U variation from occupations
6
6
U
Actualfwd.diff.
U0
5
Predicted
Hubbard U (eV)
U (eV) 4
3
2
1 4 FeO+: U vs. R
00
1.6
1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6
2.6
R (Å)
Internuclear Separation (Å)
69. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Predicting U variation from forces
70. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Predicting U variation from forces
71. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Predicting U variation from forces
72. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Predicting U variation from forces
73. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Predicting U variation from forces
Exiting linear regime for
derivatives of forces is a
numerical challenge.
74. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Numerical noise in practice
Predicted U trends for 4 FeO+
75. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Numerical noise in practice
76. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Numerical noise in practice
In principle, the
force-based
approach is more
exact. In practice, it
suffers from a
greater degree of
numerical noise.
77. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
When U(R) matters
A metric: when is U ½ of lin.resp.
U?
78. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
When U(R) matters
A metric: when is U ½ of lin.resp.
U?
Molecule U dU/dR rU½
2 + CoC 4.8 -4.0 0.6
2 - CrN 4.3 -2.3 0.9
+ FeO+ 6.3 -5.0 0.6
5 + MnF 2.4 -4.8 0.2
6 + CrF 2.0 -0.1 9.0
79. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
When U(R) matters
A metric: when is U ½ of lin.resp.
U?
Molecule U dU/dR rU½ Including more variables
2 + CoC 4.8 -4.0 0.6
2 - CrN 4.3 -2.3 0.9
+ FeO+ 6.3 -5.0 0.6
5 + MnF 2.4 -4.8 0.2
6 + CrF 2.0 -0.1 9.0
80. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
When U(R) matters
A metric: when is U ½ of lin.resp.
U?
Molecule U dU/dR rU½ Including more variables
2 + CoC 4.8 -4.0 0.6
2 - CrN 4.3 -2.3 0.9
+ FeO+ 6.3 -5.0 0.6
5 + MnF 2.4 -4.8 0.2
6 +
Some matter
CrF 2.0 -0.1 9.0
more than
others
81. 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.
82. 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 effective
united atom limit
Expt.
83. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
DFT+U(R) Improvements
1) Binding curves: 2) Reaction coordinates:
Errors on worst case subset H2 on FeO+
from MX DFT+U
re (Å)
CC value
De(eV
)
e (cm-1)
3) Work in progress:
Molecular adsorbates on TM
surfaces. Preliminary evidence:
U(R) improves binding energies.
84. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
in practice
85. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Numerical instabilities
Example:
Full manifolds or
integer occupations
Unperturbed or
rigid occupations
86. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Numerical instabilities
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Numerical instabilities
88. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Projection dependence
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Projection dependence
DFT: significant PSP dependence
90. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Projection dependence
DFT: significant PSP dependence
+U: Different Us, less PSP dependence
91. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Multiple manifolds
Strong hybridization between 3d and 4s in TM hydrides
dd ds
sd ss
U3d=( -1 -1)
0 - dd
U4s=( -1- -1)
0 ss
In the solid state: Ce 4f/5d/6s, MOFs?
92. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Angle dependence of n and
5
4 bent
3
2
1
0
4.5 5.5 6.5 7.5
5
4 linea
3
2
r
1
0
4.5 5.5 6.5 7.5
93. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Angle dependence of n and
5
4 bent
3
2
1
0
4.5 5.5 6.5 7.5
5
4 linear
3
2
1
0
4.5 5.5 6.5 7.5
94. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Angle dependence of n and
95. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
A renormalized U
Redefining response
functions:
An equivalent U along a
coordinate:
All dependence of U on O-Mn-O angle is
from filling/emptying states!
96. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.
Conclusions
For transition metals and materials with
localized electrons:
DFT+U-works well in most cases
DFT+U+V-a balance of
localization/delocalization, more general cases
like semiconductors
DFT+U(R)-bond breaking for chemical
applications
In practice, things don’t always go according to
plan (method is still not a black box).
Hinweis der Redaktion
Merge something from old slide… maybe clean this up.
Merge something from old slide… maybe clean this up.
Merge something from old slide… maybe clean this up.
Merge something from old slide… maybe clean this up.