Your SlideShare is downloading. ×
  • Like
Intro to DFT+U
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×

Now you can save presentations on your phone or tablet

Available for both IPhone and Android

Text the download link to your phone

Standard text messaging rates apply
Published

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 …

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.

Published in Education , Technology
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Be the first to comment
No Downloads

Views

Total Views
11,652
On SlideShare
0
From Embeds
0
Number of Embeds
6

Actions

Shares
Downloads
224
Comments
0
Likes
3

Embeds 0

No embeds

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
    No notes for slide
  • 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.

Transcript

  • 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).