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Efficient methods for computing exchange-correlation potentials for orbital-dependent functionals

Efficient methods for computing exchange-correlation potentials for orbital-dependent functionals. Viktor N. Staroverov Department of Chemistry, The University of Western Ontario, London, Ontario, Canada. IWCSE 2013 , Taiwan National University, Taipei , October 14 ‒17, 2013.

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Efficient methods for computing exchange-correlation potentials for orbital-dependent functionals

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  1. Efficient methods for computing exchange-correlation potentials for orbital-dependent functionals Viktor N. Staroverov Department of Chemistry, The University of Western Ontario, London, Ontario, Canada IWCSE 2013, Taiwan National University, Taipei, October 14‒17, 2013

  2. Orbital-dependent functionals Kohn-Sham orbitals • More flexible than LDA and GGAs (can satisfy more exact constraints) • Needed for accurate description of molecular properties

  3. Examples • Exact exchange • Hybrids (B3LYP, PBE0, etc.) • Meta-GGAs (TPSS, M06, etc.) same expression as in the Hartree‒Focktheory

  4. The challenge Kohn‒Sham potentials corresponding to orbital-dependent functionals cannot be evaluated in closed form

  5. Optimized effective potential (OEP) method Find as the solution to the minimization problem OEP = functional derivative of the functional

  6. Computing the OEP Expand the Kohn‒Sham orbitals: orbital basis functions Expand the OEP: auxiliary basis functions Minimize the total energy with respect to {} and {}

  7. Attempts to obtain OEP-X in finite basis sets size

  8. I. First approximation to the OEP: An orbital-averaged potential (OAP) Define operator such that The OAP is a weighted average:

  9. Example: Slater potential Fock exchange operator: Slater potential:

  10. Calculation of orbital-averaged potentials • by definition (hard, functional specific) • by inverting the Kohn‒Sham equations (easy, general)

  11. Kohn‒Sham inversion Kohn‒Sham equations: multiply by , sum over i, divide by

  12. LDA-X potential via Kohn-Sham inversion

  13. PBE-XC potential via Kohn‒Sham inversion

  14. Removal of oscillations A. P. Gaiduk, I. G. Ryabinkin, VNS, JCTC9, 3959 (2013)

  15. Kohn‒Sham inversion for orbital-specific potentials Generalized Kohn‒Sham equations: same manipulations

  16. Example: Slater potential through Kohn‒Sham inversion where

  17. Slater potential via Kohn‒Sham inversion

  18. OAPs constructed by Kohn‒Sham inversion

  19. Correlation potentials via Kohn‒Sham inversion

  20. Kohn‒Sham inversion for a fixed set of Hartree‒Fock orbitals Slater potential: But if, then

  21. Dependence of KS inversion on orbital energies

  22. II. Assumption that the OEP and HF orbitals are the same The assumption leads to the eigenvalue-consistent orbital-averaged potential (ECOAP)

  23. ECOAP KLI LHF

  24. Calculated exact-exchange (EXX) energies Sample: 12 atoms from He to Ba Basis set: UGBS A. A. Kananenka, S. V. Kohut, A. P. Gaiduk, I. G. Ryabinkin, VNS, JCP139, 074112 (2013)

  25. III. Hartree‒Fock exchange-correlation (HFXC) potential An HFXC potential is the which reproduces a HF density within the Kohn‒Sham scheme: That is, is such that

  26. Inverting the Kohn–Sham equations Kohn‒Sham equations: multiply by , sum over i, divide by local ionization potential

  27. Inverting the Hartree–Fock equations Hartree‒Fockequations: same manipulations Slater potential built with HF orbitals

  28. Closed-form expression for the HFXC potential Here , but,,and We treat this expression as a model potential within the Kohn‒Sham SCF scheme. Computational cost: same as KLI and Becke‒Johnson (BJ)

  29. HFXC potentials are practically exact OEPs! Numerical OEP: Engel et al.

  30. HFXC potentials can be easily computed for molecules Numerical OEP: Makmalet al.

  31. Energies from exchange potentials Sample: 12 atoms from Li to Cd Basis set for LHF, BJ, OEP (aux=orb), HFXC: UGBS KLI and true OEP values are from Engel et al. I. G. Ryabinkin, A. A. Kananenka,VNS, PRL139, 013001 (2013)

  32. Virial energy discrepancies For exact OEPs, where

  33. HFXC potentials in finite basis sets

  34. Hierarchy of approximations to the EXX potential OAP ECOAP HFXC

  35. Summary • Orbital-averaged potentials (e.g., Slater) can be constructed by Kohn‒Sham inversion • Hierarchy or approximations to the OEP:OAP (Slater) < ECOAP < HFXC • ECOAP Slater potential  KLI  LHF • HFXC potential  OEP • Same applies to all occupied-orbital functionals

  36. Acknowledgments • Eberhard Engel • LeeorKronik for OEP benchmarks

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