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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 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
Examples • Exact exchange • Hybrids (B3LYP, PBE0, etc.) • Meta-GGAs (TPSS, M06, etc.) same expression as in the Hartree‒Focktheory
The challenge Kohn‒Sham potentials corresponding to orbital-dependent functionals cannot be evaluated in closed form
Optimized effective potential (OEP) method Find as the solution to the minimization problem OEP = functional derivative of the functional
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 {}
I. First approximation to the OEP: An orbital-averaged potential (OAP) Define operator such that The OAP is a weighted average:
Example: Slater potential Fock exchange operator: Slater potential:
Calculation of orbital-averaged potentials • by definition (hard, functional specific) • by inverting the Kohn‒Sham equations (easy, general)
Kohn‒Sham inversion Kohn‒Sham equations: multiply by , sum over i, divide by
Removal of oscillations A. P. Gaiduk, I. G. Ryabinkin, VNS, JCTC9, 3959 (2013)
Kohn‒Sham inversion for orbital-specific potentials Generalized Kohn‒Sham equations: same manipulations
Kohn‒Sham inversion for a fixed set of Hartree‒Fock orbitals Slater potential: But if, then
II. Assumption that the OEP and HF orbitals are the same The assumption leads to the eigenvalue-consistent orbital-averaged potential (ECOAP)
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)
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
Inverting the Kohn–Sham equations Kohn‒Sham equations: multiply by , sum over i, divide by local ionization potential
Inverting the Hartree–Fock equations Hartree‒Fockequations: same manipulations Slater potential built with HF orbitals
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)
HFXC potentials are practically exact OEPs! Numerical OEP: Engel et al.
HFXC potentials can be easily computed for molecules Numerical OEP: Makmalet al.
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)
Virial energy discrepancies For exact OEPs, where
Hierarchy of approximations to the EXX potential OAP ECOAP HFXC
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
Acknowledgments • Eberhard Engel • LeeorKronik for OEP benchmarks