Report

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 XC = 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 Xexact 1 =− 4 ,=1 ∗ ∗ ′ ′ ′ − ′ same expression as in the Hartree‒Fock theory • Hybrids (B3LYP, PBE0, etc.) • Meta-GGAs (TPSS, M06, etc.) 3 The challenge Kohn‒Sham potentials corresponding to orbitaldependent functionals XC XC [{ }] = =? () cannot be evaluated in closed form 4 Optimized effective potential (OEP) method Find XC () as the solution to the minimization problem total =0 XC () OEP = functional derivative of the functional 5 Computing the OEP Expand the Kohn‒Sham orbitals: = () =1 orbital basis functions Expand the OEP: XC = () =1 auxiliary basis functions Minimize the total energy with respect to { } and { } 6 Attempts to obtain OEP-X in finite basis sets size 7 I. First approximation to the OEP: An orbital-averaged potential (OAP) Define operator XC such that XC [{ }] XC = ∗ () The OAP is a weighted average: XC = ∗ =1 XC ∗ =1 () 8 Example: Slater potential Fock exchange operator: Xexact ≡ ∗ () Slater potential: 1 S = ∗ () () =1 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: 1 2 − ∇ + + H + XC = 2 multiply by ∗ , sum over i, divide by 1 + + H + XC = 2 =1 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: 1 2 − ∇ + + H + XC = 2 same manipulations 1 + + H + XC = 2 =1 15 Example: Slater potential through Kohn‒Sham inversion 1 2 − () + S = 4 =1 | ()|2 − − H () where 1 = 2 | =1 |2 1 2 = + 4 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: SHF = −HF + HF HF |2 =1 | HF − − HHF HF |2 | =1 HF − − HHF But if OEP ≈ HF , then XOEP ≈ Xmodel = −HF + 20 Dependence of KS inversion on orbital energies 21 II. Assumption that the OEP and HF orbitals are the same The assumption = HF leads to the eigenvalue-consistent orbitalaveraged potential (ECOAP) XECOAP = SHF 1 + HF ( − HF ) HF 2 =1 22 ECOAP ≈ KLI ≈ LHF 23 Calculated exact-exchange (EXX) energies EXX − OEP , mEh 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 An HFXC potential is the XC () which reproduces a HF density within the Kohn‒Sham scheme: 1 2 − ∇ + + H + XC () = () 2 That is, XC () is such that = =1 2 HF = 2 = HF () =1 25 Inverting the Kohn–Sham equations Kohn‒Sham equations: 1 2 − ∇ + + H + XC = 2 multiply by ∗ , sum over i, divide by 1 + + H + XC = 2 =1 local ionization potential 26 Inverting the Hartree–Fock equations Hartree‒Fock equations: 1 2 − ∇ + + H + HF = HF HF 2 same manipulations HF HF + + H + SHF 1 = HF HF HF 2 =1 Slater potential built with HF orbitals 27 Closed-form expression for the HFXC potential HF XC 1 HF = S + =1 1 2 | | − HF =1 HF HF 2 HF + HF − Here = HF , but ≠ HF , ≠ HF , and ≠ HF 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 30 31 HFXC potentials can be easily computed for molecules Numerical OEP: Makmal et al. 32 Energies from exchange potentials EXX − OEP , mEh 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, vir − EXX = 0, where vir = ∫ X 3 + ⋅ ∇() vir − EXX , mEh 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 X 1 HF = S + HF − HF HF =1 HF − 2 + HF 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