Novel Functionals and the GW Approach

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1 Novel Functionals and the GW Approach Patrick Rinke FritzHaberInstitut der MaxPlanckGesellschaft, Berlin Germany IPAM 2005 Workshop III: DensityFunctional Theory Calculations for Modeling Materials and BioMolecular Properties and Functions Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 1 / 41

2 ExchangeCorrelation Exactly exchangecorrleation: energy : E xc = E tot T E ext E H potential : v xc (r) = δe xc[n] δn(r) exact v xc : dǫdr dr G 0 (r,r ;ǫ) [ Σ xc (r,r ;ǫ) v xc (r)δ(r r ) ] G(r,r;ǫ) = 0 with G 0 : noninteracting Green s function G : interacting Green s function Σ xc : selfenergy alternative: GörlingLevy perturbation theory (Phys. Rev. B 47, (1993), Phys. Rev. A 50, 196 (1994)) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 2 / 41

3 Overview Novel functionals and optimised effective potentials Exactexchange Performance of exactexchange Photoelectron spectroscopy Quasiparticles and the GW method DFT and manybody theory Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 3 / 41

4 Local, Jellium Based Functionals and SelfInteraction total energy: E tot [n] = T + E ext [n] + E H [n] + E xc [n] Hartree energy: E H = 1 occ dr dr φ i (r)φ i (r)v(r r )φ j (r )φ j (r ) 2 ij LocalDensity approximation (LDA): Exc LDA [n] = dr n(r)ǫ HEG xc (n) n=n(r)) ǫ HEG xc (n(r)): exchangecorrelation energy density of the homogeneous electron gas (HEG) Generalised Gradient Corrections (GGA): Exc GGA [n] = dr n(r)ǫ HEG xc (n, n) n=n(r)) Selfinteraction for i = j if δ i 0: δ i = 1 dr dr φ i (r) 2 v(r r ) φ i (r ) 2 + E xc [ φ i (r) 2 ] 2 Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 4 / 41

5 Orbital Functionals and the Optimised Effective Potential Method orbital dependent functionals: E xc = E xc [{φ k }] functional derivative by chain rule: v xc (r) = δe xc[{φ k }] = dr δe xc[{φ k }] δφ k (r ) δn(r) δφ k (r ) δn(r) + c.c. k chain rule applied a 2 nd time: v xc (r) = ( dr dr δexc [{φ k }] δφ k (r ) ) δφ k (r ) δv s (r ) + c.c. δvs (r ) δn(r) k with v s (r) = v ext (r) + v H (r) + v xc (r) inverse of the static density response function χ s (r,r ): χ 1 s (r,r ) = δv s(r ) δn(r) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 5 / 41

6 Optimised Effective Potential Method integral equation for v xc : dr χ s (r,r )v xc (r) = Λ xc (r) = ( dr δexc [{φ k }] δφ k (r ) ) δφ k (r ) δv s (r) + c.c. k using firstorder perturbation theory: δφ k (r ) δv s (r) = for response function: i k χ s (r,r ) = δn(r) δv s (r ) = = N k φ i (r )φ i (r) φ k (r) =: G k (r,r )φ k (r) ǫ k ǫ i i k ( δ N ) δv s (r φ k ) (r)φ k(r) k φ k (r)φ i(r)φ i (r )φ k (r ) ǫ k ǫ i + c.c. Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 6 / 41

7 Optimised Effective Potential Method optimised effective potential v xc : dr χ s (r,r )v xc (r) = Λ xc (r) can also be obtained by optimising the potential v s (r) in ] [ v s(r) φ k (r) = ǫ k φ k (r) such that the orbitals {φ k } minimise E tot [n, {φ k }], i.e. that δe tot δv s (r) = 0 vs=v OEP hence the name optimised effective potential (OEP) method (OPM) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 7 / 41

8 Overview Novel functionals and optimised effective potentials Exactexchange Performance of exactexchange Photoelectron spectroscopy Quasiparticles and the GW method DFT and manybody theory Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 8 / 41

9 ExactExchange Formalism exchange only: E xc = E x E tot = T + E ext + E H + E x Hartree energy: E H = 1 2 Exchange energy: E x = 1 2 occ ij occ selfinteraction free! ij dr dr dr φ i (r)φ i (r)v(r r )φ j (r )φ j (r ) dr φ i (r)φ j (r)v(r r )φ j (r )φ i (r ) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 9 / 41

10 ExactExchange OEP Formalism (OEPx) derivative: δe x δφ k (r ) = f k with f k = 1 if occupied and 0 otherwise exchange potential: v x (r) = dr occ k unocc j l f l φ l (r ) Fock operator or exchange selfenergy: drφ l (r)v(r r )φ k (r) [ φ k Σ x φ j φ j(r )φ k (r ] ) + c.c. χ 1 s (r,r) ǫ k ǫ j occ Σ x (r,r ) = φ i (r)v(r r )φ i (r ) i v x (r) best local potential to nonlocal exchangeself energy Σ x (r,r ) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 10 / 41

11 Comparison between OEPx and HartreeFock total energy: E tot [n, {φ k }] = T[{φ k }] + E ext [n] + E H [n] + E x [{φ k }] HartreeFock: interacting particles HartreeFock equation: ] [ v ext(r) + v H (r) φ HF k (r) + orbital decay: φ HF k (r) r e βr dr Σ x (r,r )φ HF k i N exchangeonly OEP (OEPx): noninteracting particles KohnSham equation: ] [ v ext(r) + v H (r) + v x (r) orbital decay: φ OEPx k (r) r e β kr φ OEPx k (r) = ǫ OEPx k (r ) = ǫ HF k φ OEPx k due to variational principle: E tot [n HF, {φ HF k }] E tot [n OEPx, {φ OEPx k }] k φ HF k (r) (r) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 11 / 41

12 ExactExchange Formalism with Local Correlation total energy: E tot = T + E ext + E H + E x + E c Exchange energy: E x = 1 2 occ ij dr dr φ i (r)φ j (r)v(r r )φ j (r )φ i (r ) local LDA correlation OEPx(cLDA): Ec LDA [n] = dr n(r)ǫ HEG c (n) n=n(r)) ǫ HEG c (n(r)): correlation energy density of the homogeneous electron gas (HEG) local GGA correlation OEPx(cGGA): Ec GGA [n] = dr n(r)ǫ HEG c (n, n) n=n(r)) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 12 / 41

13 Asymptotic Decay of v x and v xc exact (C. O. Almbladh and U. von Barth Phys. Rev. B 31, 3231 (1985)): with α the static polarisability OEPx: LDA: v x (r) r 1 r v xc (r) r α 2r 4 vx OEPx (r) r 1 r vxc LDA (r) r e βr/3 with β the exponential decay constant of the density Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 13 / 41

14 Overview Novel functionals and optimised effective potentials Exactexchange Performance of exactexchange Photoelectron spectroscopy Quasiparticles and the GW method DFT and manybody theory Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 14 / 41

15 OEPx versus LDA: Zn and Ga Atoms Energy [ev] p 4s 4s 3d 3d 4p I(exp)=9.39 ev LDA OEPx(cLDA) e 2 /r Zn Energy [ev] d 3d 4s 4p 4p 4s I(exp)=6.00 ev LDA OEPx(cLDA) e 2 /r Ga r [a.u.] r [a.u.] from P. Rinke et al. New J. Phys. 7, 126 (2005). The calculations were performed with the pseudopotential generator of Moukara et al. J. Phys. Cond. Mat. 12 (2000) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 15 / 41

16 Small Molecules Bond Length 8.0 deviation from experiment in % HF OEPx KLI PBE PBE0 H 2 LiH Li 2 Na 2 K 2 HF F 2 CO N 2 P 2 CH 4 H 2 O data from: S. Hamel et al., J. Chem. Phys. 116, 8276 (2002) J. Paier et al., J. Chem. Phys. 122, (2005) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 16 / 41

17 Semidconductor Lattice Constant Here EXX denotes calculations including LDA correlation and NLCC refers to nonlinear core corrections. data from: M. Städele et al., Phys. Rev. B 59, (1999) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 17 / 41

18 Overview Novel functionals and optimised effective potentials Exactexchange Performance of exactexchange Photoelectron spectroscopy Quasiparticles and the GW method DFT and manybody theory Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 18 / 41

19 PhotoElectron Spectroscopy Photoemission Inverse Photoemission Absorption hn hn hn + + GW GW BSE TDDFT Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 19 / 41

20 PhotoElectron Excitation Energies Photoemission electron removal ψ s (r) = N 1, s ˆψ(r) N removal energy ǫ s = E(N) E(N 1, s) Inverse Photoemission electron addition ψ s (r) = N ˆψ(r) N + 1, s addition energy ǫ s = E(N + 1, s) E(N) e f N : Nparticle ground state N 1, s : (N 1)particle excited state s E(N, s) : total energy in state s e s Photoemission + E kin hn E vac E F (N1)electrons Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 20 / 41

21 Ionisation Potential and Electron Affinity Ionisation potential: minimal energy to remove an electron I = E(N 1) E(N) Electron affinity: minimal energy to add an electron A = E(N) E(N + 1) I in % I=E(N1)E(N) HF KLI OEPx LDA PW91 He Be Mg Ca Sr Cu Ag Au Li Na K Rb Cs Zn E. Engel in A Primer in DFT, Springer 2003 Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 21 / 41

22 Ionisation Potential KohnSham Ionisation potential: minimal energy to remove an electron I = E(N 1) E(N) KohnSham: eigenvalue of the highest occupied KohnSham level I KS = ǫ N (N) I in % I in % I=ε N (N) I=E(N1)E(N) He Be Mg Ca Sr Cu Ag Au Li Na K Rb Cs Zn OEPx LDA OEPx KS LDA KS HF KLI OEPx LDA PW91 E. Engel in A Primer in DFT, Springer 2003 NIST: Atomic reference data Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 22 / 41

23 Ionisation Potential Small Molecules Total Energy Difference deviation from exp. Method mean absolute HF LSDA PW PBE B3LYP PBE Table: IP in ev for 86 molecules of the G3/99 test set, calculated by total energy differences, V. N. Staroverov et al., J. Chem. Phys. 119, (2003) HOMO: LUMO : highest occupied molecular orbital lowest unoccupied molecular orbital ε HOMO [ev] KohnSham Eigenvalue FH OEPx LDA B88x K 2 Na 2 Li 2 CH 4 CO H 2 N 2 F 2 LiH C 5 H 5 N P 2 C 2 H 4 CH 2 O NH 3 H 2 O I exp [ev] S. Hamel et al., J. Chem. Phys. 116, 8276 (2002) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 23 / 41

24 Band Gap of Semiconductors Band gap: E gap = I A = E(N + 1) 2E(N) + E(N 1) For solids E(N + 1) and E(N 1) cannot be reliable computed in DFT, yet. In KohnSham the highest occupied state is exact (Janak s theorem): E gap = ǫ KS N+1 (N + 1) ǫks N (N) = ǫ KS (N + 1) ǫks N+1 N+1 (N) } {{ } xc KS + ǫ N+1 (N) ǫks N }{{ (N) } Egap KS For solids: N 1 n(r) 0 for N N + 1 discontinuity in v xc upon changing the particle number ( δexc [n] xc = δn(r) δe ) ( ) xc[n] 1 N+1 δn(r) + O N N Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 24 / 41

25 Band Gap of Semiconductors Discontinuity in v xc Band gap: E gap = E KS gap + xc e KS e KS KS e N+1 (N) KS e N (N) D xc KS E gap E gap KS e N+1 (N+1) N electrons k N+1 electrons k Figure: After the addition of an electron into the conduction band (right) the xc potential and the whole bandstructure shift up by a quantity xc. Figure after R. W. Godby et al., in A Primer in DFT, Springer 2003 Figure: Exactexchange potential (KLI) shifts up almost uniformily by 1 ev upon adding 10 6 electrons. J. B. Krieger et al., Phys. Rev. A 45, 101 (1992) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 25 / 41

26 Band Gap of Semiconductors in ExactExchange OEP LDA OEPx Exp. Si SiC AlAs AlN C Table: KohnSham band gap for selected semiconductors (OEPx calculations include LDA correlation), W. G. Aulbur et al., Phys. Rev. B 62, 7121 (2000) x = CBM Σ x v x CBM VBM Σ x v x VBM x (Si)= 5.48 ev M. Städele et al., Phys. Rev. B 59, (1999) Theoretical Band Gap [ev] LDA OEPx(cLDA) CdS ZnSe GaN ZnO ZnS CdSe Experimental Band Gap [ev] P. Rinke et al. New J. Phys. 7, 126 (2005) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 26 / 41

27 ExactExchange OEP Pseudopotential versus AllElectron S. Sharma et al. Phys. Rev. Lett. 95, (2005) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 27 / 41

28 PhotoElectron Excitation Energies Photoemission electron removal ψ s (r) = N 1, s ˆψ(r) N removal energy ǫ s = E(N) E(N 1, s) Inverse Photoemission electron addition ψ s (r) = N ˆψ(r) N + 1, s addition energy ǫ s = E(N + 1, s) E(N) e f N : Nparticle ground state N 1, s : (N 1)particle excited state s E(N, s) : total energy in state s e s Photoemission + E kin hn E vac E F (N1)electrons Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 28 / 41

29 PhotoElectron Spectroscopy and Green s Functions ǫ s and ψ s (r) are solutions to the manybody Hamiltonian Ĥ Ĥ can be transformed into singleparticle form: Ĥ(r,r ;ǫ) = ĥ0(r) + Σ(r,r ;ǫ) selfenery Σ contains all electronelectron interactions Green s function G solution to Schrödinger equation: G(r,r ;ǫ) = r [Ĥ(ǫ) ǫ] 1 r ψ s (r)ψs (r ) = lim η 0 + ǫ (ǫ s s + iη sgn(e f ǫ s )) spectroscopically relevant quantity: photocurrent photocurrent weighted surface integral over diagonal part of the spectral function A: A(r,r ;ǫ) = 1 π Im G(r,r ;ǫ) = s ψ s (r)ψ s (r )δ(ǫ ǫ s ) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 29 / 41

30 Overview Novel functionals and optimised effective potentials Exactexchange Performance of exactexchange Photoelectron spectroscopy Quasiparticles and the GW method DFT and manybody theory Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 30 / 41

31 The Quasiparticle Concept Quasiparticle: singleparticle like excitation A k( ) quasiparticlepeak k qp A k (ǫ) Z k ǫ (ǫ k + iγ k ) ǫ k : excitation energy Γ k : lifetime Z k : renormalisation k Quasiparticle: electron acquires polarisation cloud new entity electron = polarisation cloud quasiparticle Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 31 / 41

32 GW Approximation Interacting Quasiparticles GW = interacting quasiparticles W(r,r ; w) G is propagator S = G W SelfEnergy: energy response of the system that the quasiparticle experiences due to its own presence GW : Σ(r,r ;ǫ) = i dǫ e iǫ δ G(r,r ;ǫ + ǫ )W (r,r ;ǫ ) 2π Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 32 / 41

33 GW Approximation Formalism GW = interacting quasiparticles W(r,r ; w) G is propagator W S = G GW selfenergy: Σ(r,r ; ǫ) = i dǫ e iǫ δ G(r,r ; ǫ + ǫ )W(r,r ; ǫ ) 2π Screened interaction: W(r,r, ǫ) = dr ε 1 (r,r ; ǫ)v(r r ) Dielectric function: ε(r,r, ǫ) = δ(r r ) dr v(r r )χ 0 (r,r ; ǫ) Polarisability: χ 0 (r,r ; ǫ) = i dǫ G(r,r ; ǫ ǫ)g(r,r; ǫ ) 2π Quasiparticle equation: ĥ 0 (r)ψ s (r) + dr Σ(r,r ; ǫ qp s )ψ s(r ) = ǫ qp s ψ s(r) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 33 / 41

34 GW Merits GW : gives accurate band gaps for many materials allows to calculate lifetimes successfully applied to bulk materials surfaces nanotubes clusters defects Theoretical Band Gap [ev] GW LDA defects on surfaces Experimental Band Gap [ev] Data taken from Aulbur et al. Solid State Phys. 54 (2000) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 34 / 41

35 Quasiparticle Band Gaps P. Rinke et al. New J. Phys. 7, 126 (2005) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 35 / 41

36 Overview Novel functionals and optimised effective potentials Exactexchange Performance of exactexchange Photoelectron spectroscopy Quasiparticles and the GW method DFT and manybody theory Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 36 / 41

37 Connection between DFT and Green s Function Theory ManyBody Green s Function Theory: noninteracting electrons: DensityFunctional Theory G 1 DFT (r,r ; ǫ) = [ǫ h 0 (r) v xc (r)] δ(r r ) interacting electrons: selfenergy (Σ xc ) Perturbation Theory: Density: G 1 (r,r ; ǫ) = [ǫ h 0 (r)] δ(r r ) Σ xc (r,r ; ǫ) adiabatic connection between noninteracting and interacting system Dyson equation: G 1 = G 1 DFT [Σ xc v xc ] n(r) = 1 µ 2π Im dǫg(r,r; ǫ) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 37 / 41

38 Connection between DFT and Green s Function Theory Density condition (recap: density in DFT is exact): 0 = n(r) n DFT (r) = 1 2π Im dǫ [G(r,r; ǫ) G DFT (r,r; ǫ)] Dyson equation: G 1 = G 1 DFT [Σ xc v xc ] G DFT = G G DFT [Σ xc v xc ] G G G DFT = G DFT [Σ xc v xc ] G ShamSchlüter equation connects exact v xc with Σ: dǫdr dr G DFT (r,r ; ǫ)[σ xc (r,r ; ǫ) v xc (r)δ(r r )] G(r,r; ǫ) = 0 Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 38 / 41

39 Further Reading A Primer in Density Functional Theory, C. Fiolhais, F. Nogueira and M. Marques, Springer 2003 (ISBN ) ManyParticle Theory, E. K. U. Gross and E. Runge and O. Heinonen, Adam Hilger 1991 Quasiparticle Calculations in Solids, W. G. Aulbur and L. Jönsson and J. W. Wilkins, Solid State Phys. : Advances in Research and Applications 54, p Electronic Excitations: DensityFunctional Versus ManyBody Green s Function Approaches, G. Onida, L. Reining and A. Rubio, Rev. Mod. Phys. 74, p Orbital functionals in densityfunctional theory: the optimized effective potential method, R. Grabo, T. Kreibich, S. Kurth and E. K. U. Gross, in Strong Coulomb Correlations in Electronic Structure Calculations: Beyond the Local Density Approximation edited by V. I. Anisimov (Gordon and Breach, New York, 2000) Combining GW calculations with exactexchange densityfunctional theory: An analysis of valenceband photoemission for compound semiconductors, P. Rinke, A. Qteish, J. Neugebauer, C. Freysoldt and M. Scheffler, New J. Phys. 7, 126 (2005) Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 39 / 41

40 KriegerLiIafrate Approximation (KLI) OPM and OEPx are computationally demanding! Is there a more efficient way? KriegerLiIafrate common energy denominator approximation: (ǫ k ǫ l ) ǫ: Gk KLI (r,r ) l k φ l (r)φ l (r ) ǫ = δ(r r ) φ k (r)φ k(r ) ǫ χ KLI s (r,r ) = N k N k l k φ k (r)φ l(r)φ l (r )φ k (r ) ǫ + c.c. φ k (r)φ k(r )[δ(r r ) φ k (r)φ k (r )] ǫ + c.c. Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 40 / 41

41 KriegerLiIafrate Approximation (KLI) inserting G KLI k (r,r ) and χ KLI (r,r ) into OPM equation: s v KLI xc (r) = 1 2n(r) N k φ k (r) 2 [ u xck (r) + v KLI xck ū xck] + c.c. with u xck (r) := 1 δe xc [{φ k }] φ k (r) δφ k (r) v xck KLI := drφ k (r)vkli xc (r)φ k (r) ū xck := drφ k (r)u xck(r)φ k (r) advantage: only occupied states are needed Patrick Rinke (FHI) Novel Functionals and the GW Approach IPAM 2005 DFT 41 / 41

Combining quasiparticle energy calculations with exact-exchange density-functional theory

Combining quasiparticle energy calculations with exact-exchange density-functional theory Patrick Rinke 1, Abdallah Qteish 1,2, Jörg Neugebauer 1,3,4, Christoph Freysoldt 1 and Matthias Scheffler 1 1 Fritz-Haber-Institut