Efficient methods for computing exchangecorrelation potentials for orbitaldependent

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

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

Examples • Exact exchange same expression as in the Hartree‒Fock theory • Hybrids (B 3 LYP, PBE 0, etc. ) • Meta-GGAs (TPSS, M 06, etc. ) 3

The challenge Kohn‒Sham potentials corresponding to orbitaldependent functionals cannot be evaluated in closed form 4

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

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

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

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

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

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

Kohn‒Sham inversion Kohn‒Sham equations: 11

LDA-X potential via Kohn-Sham inversion 12

PBE-XC potential via Kohn‒Sham inversion 13

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

Kohn‒Sham inversion for orbitalspecific potentials Generalized Kohn‒Sham equations: same manipulations 15

Example: Slater potential through Kohn‒Sham inversion where 16

Slater potential via Kohn‒Sham inversion 17

OAPs constructed by Kohn‒Sham inversion 18

Correlation potentials via Kohn‒Sham inversion 19

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

Dependence of KS inversion on orbital energies 21

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

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Calculated exact-exchange (EXX) energies m. a. v. KLI ELP=LHF=CEDA ECOAP 2. 88 2. 84 2. 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) 24

III. Hartree‒Fock exchangecorrelation (HFXC) potential 25

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

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

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

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

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

Energies from exchange potentials m. a. v. KLI LHF BJ 1. 74 1. 66 5. 30 Basisset OEP 0. 12 HFXC 0. 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) 33

Virial energy discrepancies For exact OEPs, where KLI m. a. v. 438. 0 LHF BJ 629. 2 1234. 1 Basis-set HFXC OEP 1. 76 2. 76 34

HFXC potentials in finite basis sets 35

Hierarchy of approximations to the EXX potential OAP ECOAP HFXC 36

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 37

Acknowledgments • Eberhard Engel • Leeor Kronik for OEP benchmarks 38