Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p.1/26

Transcription

1 Performance of hybrid density functional methods, screened exchange and EXX-OEP methods in the PAW approach Georg Kresse, J Paier, R Hirschl, M Marsmann Institut für Materialphysik and Centre for Computational Materials Science, Universität Wien, A-1090 Wien, Austria Funded by the Austrian FWF Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p1/26

2 Introduction PAW method exact exchange (Fock) operator in the PAW method results for small molecules exact-exchange OEP method results for insulators, semiconductors and metals b-initio ackage All calculations were performed using VASP ienna imulation periodic plane wave code Fock operator implemented by R Hirschl EXX-OEP Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p2/26

3 Basic concepts of the PAW method Wave function is written as a sum of a pseudo wave function and one center corrections Ψa = Ψa i ( φi φi ) pi Ψa Same holds for charge density nab(r) = Ψ a(r)ψb(r) = ñab(r) ñ 1 ab (r) n1 ab (r) pseudo (node less) plane waves pseudo onsite radial grids exact onsite radial grids = exact Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p3/26

4 Exact exchange in the PAW method K = K K 1 K 1 ñab = Ψ a(r) Ψb(r) K = 1 2 ab fa fb Z {ñab nab} 1 r r {ñ ab nab} pseudo PW K 1 = 1 2 ab fa fb Z {ñ 1 ab n1 ab } 1 r r {ñ1 ab n1 ab } pseudo radial K 1 = 1 2 ab fa fb Z {n 1 ab } 1 r r {n1 ab } AE radial pseudo (node less) plane waves pseudo onsite radial grids exact onsite radial grids = exact Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p4/26

5 Exact exchange in the PAW method K = K K 1 K 1 ñab = Ψ a(r) Ψb(r) K = 1 2 ab fa fb Z {ñab nab} 1 r r {ñ ab nab} pseudo PW K 1 = 1 2 ab fa fb Z {ñ 1 ab n1 ab } 1 r r {ñ1 ab n1 ab } pseudo radial K 1 = 1 2 ab fa fb Z {n 1 ab } 1 r r {n1 ab } AE radial pseudo (node less) plane waves pseudo onsite radial grids exact onsite radial grids = exact n ab!!!!!!!!!!!!! " " " " " " " " " " " " # # # # # # # # # # # # # $ $ $ $ $ $ $ $ $ $ $ $ % % % % % % % % % % % % % ' ' ' ' ' ' ' ' ' ' ' ' ' ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) * * * * * * * * * * * * *,,,,,,,,,,,, / / / / / / / / / / / / / Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p4/26

6 Basic steps in exact exchange Loop over all pairs of states a, b (includes all k-point pairs) FFT of pseudo wave functions to real space and calculate charge n ab (r) = Ψ a(r) Ψ b (r) Add augmentation charges to correct moments n ab (r) FFT of charge to reciprocal space and division by Laplace operator n ab (r) FFT n ab (G) V ab (G) = 4πe2 G 2 n ab (G) FFT back to real space to obtain potential and multiplication with wavefunction V ab (r)ψ b (r) Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p5/26

7 Systematic assessment of the accuracy: PBE Molecule Exp PBE0 VASP exp PBE0 G03 exp V G O CO NO LiF SiO CS SO ClO ClF MAE MAE for G2 set (50 molecules), values in kcal, G03 aug-cc-pv5z basis (1 kcal=43 mev) Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p6/26

8 Systematic assessment of the accuracy: PBE0 Molecule Exp PBE0 VASP exp PBE0 G03 exp V G O CO NO LiF SiO CS SO ClO ClF MAE MAE for G2 set (50 molecules), values in kcal, G03 aug-cc-pv5z basis (1 kcal=43 mev) Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p7/26

9 Systematic assessment of the accuracy: PBE0 Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p8/26

10 Hybrid and screened exchange methods Split the Coulomb operator in the exchange into two terms eg 1 r = erfc(µr) r erf(µr) r or more generally 1 4πG 2 }{{} Coulomb = 1 4πG 2 1 ε(g) 4πG }{{} 2 (1 1 ε(g) ) }{{} exact DFT Hybrid functionals: ε = 4 1/4 EXX and 3/4 DFT screened exchange: ε = q 2 T F G 2 /G 2 HSE (ωpbe): ε = (1 e G2 /4µ 2 ) 1 J Heyd, GE Scuseria, J Chem Phys 121, 1187 Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p9/26

11 EXX-OEP method M Städele, JA Majewski, and P Vogl, A Görling, Phys Rev Lett 79, 2089 (1997) S Kümmel, JP Perdew, Phys Rev Lett 90, (2003) Instead of inverting the irreducible polarizability matrix χ, the response of the system is determined by a linear response solver to obtain orbital shifts ( V EXX )Ψ (1) = ( V EXX V Fock ({Ψ (0) }))Ψ (0) where the wavefunctions Ψ (0) are eigenfunctions of Hamiltonian with the local exchange potential Update potential: ( V EXX )Ψ (0) = εψ (0) V EXX (r) = Ψ (0) (r)ψ (1) (r) cc f (1) Ψ (0) (r)ψ (0) (r) Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p10/26

12 EXX-OEP method practical considerations The residual is defined as R(r) = Ψ (0) (r)ψ (1) (r) cc = R[V EXX ] Efficient and stable Pulay mixer (RMM-DIIS) is used to minimize the norm of the residual vector P Pulay, Chem Phys Lett 73, 393 (1980) The optimizer performs an optimization in the yet visited sub space to determine V EXX (r) such that the norm of the residual vector is minimized The calculations are usually initialized from a LHF calculation (local Hartree Fock) Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p11/26

13 EXX-OEP method typical convergence log 10 V ev 400 ev Convergence is robust but fairly slow Convergence slows down, if more Fourier components of the potential are determined ie the potential cutoff is increased iteration Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p12/26

14 Convergence of the present OEP implementation Assume that the OEP method is applied to a conventional DFT functional, starting from a slightly perturbed potential V OEP = V sc δv Determine the KS eigenstates for this potential and resulting charge ( V OEP )Ψ (0) = εψ (0) ρ = Ψ (0) Ψ (0) = ρ sc χδv new potential V out = v(ρ sc χδv ) = V sc vχ δv Next determine linear response Ψ (1) ( V OEP )Ψ (1) = (V OEP V out )Ψ (0) = ε δv Ψ (0) Step along V OEP = Ψ (1) Ψ (0) cc = χε δv Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p13/26

15 Convergence of OEP method Eigenvalue spectrum of χε determines convergence response (au) 5 4 ε χ G (1/A 2 ) The error in the long range part is strongly overemphasized The error in the short range part is not visible to the mixer Preconditioner based on the inverse of the Lindhard dielectric matrix did not improve the convergence Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p14/26

16 PAW datasets All calculations are performed using LDA/ PBE pseudopotentials The pseudopotential generation code does not support exact exchange The core electrons are kept frozen in the calculations The core valence interaction is however reevaluated at the appropriate level (PBE0) In the LHF and EXX-OEP case the OEP potential is determined for the valence electrons only The core valence interaction is treated using Hartree Fock A consistent LHF treatment of valence and core electrons is feasible and presently under investigations Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p15/26

17 Results noble gas solids Γ 20 XX K Γ L Γ XX K Γ L 15 Energy (ev) Ne Energy (ev) Ar Energy (ev) Γ K-point distance Kr XX K K-point distance Γ L K-point distance line EXX o dots LDA good agreement with RJ Magyar, A Fleszar, and EKU Gross, Phys Rev B 69, (2004) Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p16/26

18 Collected results at Γ Ne Ar Kr C Si Ge ZnO ZnS GaN LDA LDA LHFc EXXc EXXc exp PBE LDA, LHFc, EXXc, PBE0 present work LDA, EXXc from: [1] RJ Magyar, A Fleszar, and EKU Gross, Phys Rev B 69, (2004) [2] KKR and LMTO: T Kotani, H Akai, Phys Rev B 54, (1996) [3] PP: M Städele, M Moukara, Majewski, P Vogl, G Görling, Phys Rev B 59, (1999) [4] PP: P Rincke, A Qteish, J Neugebauer, C Freysoldt, and M Scheffler Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p17/26

19 Some comments LDA pseudopotentials EXX only for valence core-valence HF typ PP type CdS valence gap d LDA LDA[1] 081 LDA LDA 3d 088 EXXc LDA 3d EXXc LDA 3p EXXc LDA 3s EXXc EXX[1] electrons treated as valence consistent LHF for core and valence timings: four P4 28 GHz RIM timings per iteration Ne 6x6x6 k-points, 5 sec Ne 8x8x8 k-points, 20 sec C 8x8x8 k-points, 25 sec state of the art CPU s are about 15 to 2 times faster Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p18/26

20 Nanotube (10,0): 40 atoms Energy (ev) (10,0) LDA 32 k-points Exx 8 k-points k (2π/a) EXX and LDA are practically identical for the nanotube EXX 820 mev LDA 750 mev It is known that LDA underestimates the optical gaps by a factor The experimental fundamental gaps are not known Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p19/26

21 Nanotube (7,0): 24 atoms Energy (ev) (7,0) LDA 32 k-points Exx 8 k-points increase of band gap EXX 280 mev LDA 480 mev GW 600 mev T Miyake S Saito, Phys Rev B 68, k (2π/a) Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p20/26

22 Ferromagnetic bcc Fe Desaster Hunds rule DOS (states/ev/atom) EXX LDA Energy (ev) 10x10x10 k-points: 180 seconds per iteration on four P MHz ferromagnet m= 35µ B Akai (1996): m= 34µ B 3p valence Qualitative similar results, when 3p is in the core Since the up and down potential are entirely free to vary, the solution is close to Hartree Fock Correlation, of course? Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p21/26

23 Conclusions EXX-OEP is certainly interesting, but will require improved correlation functionals Hybride functionals seem to offer significant improvements compared to usual semi-local functionals right now Reasonable energetics Reasonable one electron gaps Reasonable results for correlated systems (MnO, NiO) Fairly fast using a plane wave basis set Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p22/26

24 Acknowledgment funded by the Austrian FWF within the START program R Hirschl (did most of the work on the Fock operator) J Paier (testing) M Marsmann, I Gerber (screened exchange) J Furthmüller, F Bechstedt (model GW) Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p23/26

25 Results noble gas solids Ne Ar Kr present PP[1] present PP[1] present PP[1] LDA EXX EXXc LHFc exp optical exp PBE [1] RJ Magyar, A Fleszar, and EKU Gross, Phys Rev B 69, (2004) Computational time: 6x6x6 k-points, four P MHz: 5 seconds per iteration 8x8x8 k-points, four P MHz: 20 seconds per iteration Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p24/26

26 Group IV Elements LDA LDA EXXc EXXc exp PBE0 HSE present PP[1] present PP[1] C γ C L c C X c Si γ Si L c Si X c Ge γ Ge L c Ge X c [1] M Städele, M Moukara, Majewski, P Vogl, G Görling, Phys Rev B 59, (1999) [2] KKR and LMTO, T Kotani, H Akai, Phys Rev B 54, (1996) Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p25/26

27 Semiconductors with shallow core states type PP type ZnO ZnS CdS GaN valence gap d gap d gap d gap d LDA LDA[1] LDA LDA 3d EXXc LDA 3d EXXc LDA 3p EXXc LDA 3s EXXc EXX[1] exp PBE0 PBE HSE PBE electrons treated as valence [1] P Rincke, A Qteish, J Neugebauer, C Freysoldt, and M Scheffler Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p26/26