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

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

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

5. Optimized effective potential (OEP) method 5 OEP = functional derivative of the functional

6. Computing the OEP Expand the Kohn ‒ Sham orbitals: Expand the OEP: orbital basis functions auxiliary basis functions 6

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

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

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

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

11. 11 Kohn ‒ Sham inversion Kohn ‒ Sham equations:

12. 12 LDA-X potential via Kohn-Sham inversion

13. 13 PBE-XC potential via Kohn ‒ Sham inversion

14. 14 A. P. Gaiduk, I. G. Ryabinkin, VNS, JCTC 9, 3959 (2013) Removal of oscillations

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

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

17. 17 Slater potential via Kohn ‒ Sham inversion

18. 18 OAPs constructed by Kohn ‒ Sham inversion

19. 19 Correlation potentials via Kohn ‒ Sham inversion

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

21. 21 Dependence of KS inversion on orbital energies

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

23. 23

24. 24 Calculated exact-exchange (EXX) energies KLIELP=LHF=CEDAECOAP m.a.v. 2.882.842.47 Sample: 12 atoms from He to Ba Basis set: UGBS A. A. Kananenka, S. V. Kohut, A. P. Gaiduk, I. G. Ryabinkin, VNS, JCP 139, 074112 (2013)

25. III. Hartree ‒ Fock exchange- correlation (HFXC) potential 25

26. 26 Inverting the Kohn–Sham equations Kohn ‒ Sham equations: local ionization potential

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

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

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

30. 30

31. 31

32. 32 HFXC potentials can be easily computed for molecules Numerical OEP: Makmal et al.

33. 33 Energies from exchange potentials KLILHFBJ Basis- set OEP HFXC m.a.v.1.741.665.300.120.05 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, PRL 139, 013001 (2013)

34. 34 Virial energy discrepancies KLILHFBJ Basis-set OEP HFXC m.a.v.438.0629.21234.11.762.76 where For exact OEPs,

35. 35 HFXC potentials in finite basis sets

36. 36 Hierarchy of approximations to the EXX potential OAP ECOAP HFXC

37. 37 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

38. 38 Acknowledgments Eberhard Engel Leeor Kronik for OEP benchmarks