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- 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 insulatorsV.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 interactionsJ. Hubbard Proc. R. Soc. A 285 (1965). V. I. Anisimov, I. S. Elfimov, N. Hamada, andJ. 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 linearExperiments: 180 100Can theory predict transition? EGong, 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 0.25Error 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 0.25Error 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. WorksHJK 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 0.25Error 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. WorksHJK 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 0.25Error 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. WorksHJK and N. Marzari. J. Chem. Phys. (2010). in U incorporatedHJK 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 0dE/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 3.0dE/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 3.0dE/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 practiceIn principle, theforce-basedapproach is moreexact. In practice, itsuffers from agreater degree ofnumerical 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
- 87. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info. Numerical instabilities
- 88. Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info. Projection dependence
- 89. 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
- 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 URedefining responsefunctions:An equivalent U along acoordinate: 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).

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