Towards a unified description of ground and excited state properties: RPA vs GW
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- Barrie Martin
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1 Towards a unified description of ground and excited state properties: RPA vs GW Patrick Rinke Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin - Germany Towards Reality in Nanoscale Materials V Patrick Rinke (FHI) RPA/GW Levi
2 Wish list for optimum electronic structure approach d/f-electrons (e.g. Cerium) applicable across dimensionalities: molecules/clusters wires/tubes surfaces/films solids, extended systems Energy relative to E F (ev) (bio)molecules ZnO G 0 W 0@OEPx(cLDA) applicable across periodic table: from light to heavy elements including d/f electron physics/chemistry H L A G K M G Patrick Rinke (FHI) RPA/GW Levi
3 Wish list for optimum electronic structure approach Energy relative to E F (ev) d/f-electrons (e.g. Cerium) (bio)molecules ZnO G 0 W 0@OEPx(cLDA) ground/excited state: consistent description parameter free free from pathologies e.g.: self-interaction error absence of van der Waals band-gap problem computationally efficient gradients structure relaxation... H L A G K M G Patrick Rinke (FHI) RPA/GW Levi
4 Wish list for optimum electronic structure approach Energy relative to E F (ev) d/f-electrons (e.g. Cerium) (bio)molecules ZnO G 0 W 0@OEPx(cLDA) ground/excited state: consistent description parameter free free from pathologies e.g.: self-interaction error absence of van der Waals band-gap problem computationally efficient gradients structure relaxation... H L A G K M G Patrick Rinke (FHI) RPA/GW Levi
5 Treating van der Waals interactions fluctuating dipoles difficult, because they require non-local treatment fluctuating dipoles described by dielectric function ε(r,r,ω) correlation energy in random-phase approximation (RPA): E RPA c = 1 2π 0 dωtr[ln(ε(iω))+(1 ε(iω))] Patrick Rinke (FHI) RPA/GW Levi
6 Attractive features of the RPA exact exchange + random phase approximation: (EX+cRPA)@reference E EX+cRPA tot = T s +E ext +E H +E exact x +E RPA c E RPA c = + + Exact exchange (with Kohn-Sham orbitals): OEPx self-interaction error considerably reduced vdw interactions included automatically and seamlessly Screening taken into account applicable to metals/small gap systems (in contrast to MP2) Patrick Rinke (FHI) RPA/GW Levi
7 Attractive features of the RPA exact exchange + random phase approximation: (EX+cRPA)@reference E EX+cRPA tot = T s +E ext +E H +E exact x +E RPA c E RPA c = + + For an overview of RPA, see our recent review article: Exact exchange Ren, Joas, (with Rinke, Kohn-Sham Scheffler, orbitals): arxiv: v1 OEPx self-interaction error considerably reduced vdw interactions included automatically and seamlessly Screening taken into account applicable to metals/small gap systems (in contrast to MP2) Patrick Rinke (FHI) RPA/GW Levi
8 van der Waals bonding 20 Ar 15 2 Binding energy (mev) PBE 0-2 C 6 /R Reference RPA@PBE Bond length (Å) Correct asymptotic behavior crucial for large molecules Underbinding around the equilibrium distance Patrick Rinke (FHI) RPA/GW Levi
9 The SOSEX correction to RPA Digrammatic representation (motivated in coupled cluster context) E RPA+SOSEX c = nd order 3rd order = + Arising from the anti-symmetric nature of the many-body wave function RPA+SOSEX is one-electron self-correlation free D. L. Freeman, Phys. Rev. B 15, 5512 (1977). A. Grüneis et al., J. Chem. Phys. 131, (2009). J. Paier et al., J. Chem. Phys. 132, (2010); Erratum: 133, (2010). Patrick Rinke (FHI) RPA/GW Levi
10 The concept of single excitation (SE) corrections Rayleigh-Schrödinger perturbation theory: Ĥ = Ĥ0 +Ĥ Ground-state energy: E 0 = E (0) 0 +E (1) 0 +E (2) 0 + zeroth-order: E (0) 0 = Φ 0 Ĥ0 Φ 0 1st order: E (1) 0 = Φ 0 Ĥ Φ 0 2nd order: E (2) 0 = n 0 SE accounts for orbital relaxations Φ 0 Ĥ Φ n 2 E (0) 0 E n (0) = Φ 0 Ĥ Φ i,a 2 + Φ 0 Ĥ Φ ij,ab 2 i,a E (0) 0 E (0) i,a ij,ab E (0) 0 E (0) ij,ab }{{}}{{} Single excitations Double excitations = Ec SE MP2 + + Patrick Rinke (FHI) RPA/GW Levi
11 Renormalized single excitations (rse) f ia f ia f ia Ec rse i a a f i = a ij fab i j b f ai f f aj bi 2nd order 3rd order f pq = p ˆf q : Hartree-Fock operator evaluated within Kohn-Sham orbitals E SE c = occ i unocc a f ia 2 ǫ i ǫ a (ǫ i,ǫ a : Kohn-Sham orbital energies) Patrick Rinke (FHI) RPA/GW Levi
12 Renormalized single excitations (rse) f ia Ec rse i a f a i = a ij fab i j b f ai 2nd order f ia f aj 3rd order f pq = p ˆf q : Hartree-Fock operator evaluated within Kohn-Sham orbitals E SE c = occ i unocc a f ia 2 ǫ i ǫ a (ǫ i,ǫ a : Kohn-Sham orbital energies) Diagonal rse : including only terms with i = j = and a = b = occ unocc Ec rse-diag f ia 2 = Problematic for weak interactions f ii f aa i a f ia f bi Patrick Rinke (FHI) RPA/GW Levi
13 Renormalized single excitations (rse) f ia Ec rse i a f a i = a ij fab i j b f ai 2nd order f ia f aj 3rd order f pq = p ˆf q : Hartree-Fock operator evaluated within Kohn-Sham orbitals E SE c = occ i unocc a f ia 2 ǫ i ǫ a (ǫ i,ǫ a : Kohn-Sham orbital energies) Diagonal rse : including only terms with i = j = and a = b = occ unocc Ec rse-diag f ia 2 = Problematic for weak interactions f i a ii f aa 2 f ia full rse : E rse-full c = occ i unocc a f i f a Diagonalize f ij and f ab blocks separately f i, f a, and transform f ia f ia Patrick Rinke (FHI) RPA/GW Levi f ia f bi
14 Renormalized 2nd-order Perturbation Theory (r2pt) E RPA+SOSEX+rSE c = ( = RPA) ( = SOSEX) ( = rse) 2nd order 3rd order ( = 2PT) r2pt = RPA+SOSEX+rSE Patrick Rinke (FHI) RPA/GW Levi
15 van der Waals bonding 20 Ar 15 2 Binding energy (mev) PBE 0-2 C 6 /R Reference RPA@PBE Bond length (Å) Correct asymptotic behavior crucial for large molecules Underbinding around the equilibrium distance Patrick Rinke (FHI) RPA/GW Levi
16 van der Waals bonding 20 Ar 15 2 Binding energy (mev) PBE 0-2 C 6 /R RPA@PBE RPA+SOSEX Reference Bond length (Å) SOSEX has almost no effect Patrick Rinke (FHI) RPA/GW Levi
17 van der Waals bonding 20 Ar 15 2 Binding energy (mev) PBE 0-2 C 6 /R RPA@PBE RPA+SOSEX Reference RPA+SE Bond length (Å) much improved binding and asymptotics Ren, Tkatchenko, Rinke, and Scheffler, Phys. Rev. Lett. 106, (2011) Patrick Rinke (FHI) RPA/GW Levi
18 van der Waals bonding 20 Ar 15 2 Binding energy (mev) r2pt PBE 0-2 C 6 /R RPA@PBE RPA+SOSEX Reference RPA+SE Bond length (Å) best overall performance Patrick Rinke (FHI) RPA/GW Levi
19 Van der Waals interactions - the S22 benchmark set S22 set of binding energies 22 representative dimers: 7 hydrogen bonded 8 dispersion bonded 7 mixed character CCSD(T) ( gold standard ) reference values Jurečka et al., PCCP 8,1985 (2006) Patrick Rinke (FHI) RPA/GW Levi
20 Van der Waals interactions - the S22 benchmark set 58% PBE MP2 19% (EX+cRPA)@PBE 16% mean absolute percentage error Patrick Rinke (FHI) RPA/GW Levi
21 Van der Waals interactions - the S22 benchmark set 58% PBE MP2 19% (EX+cRPA)@PBE 16% (EX+cRPA+SE)@PBE 8% mean absolute percentage error X. Ren, P. Rinke, A. Tkatchenko, M. Scheffler, Phys. Rev. Lett. 106, (2011) Patrick Rinke (FHI) RPA/GW Levi
22 Van der Waals interactions - the S22 benchmark set 58% PBE MP2 19% (EX+cRPA)@PBE 16% (EX+cRPA+SE)@PBE 8% r2pt@pbe 7% mean absolute percentage error X. Ren, P. Rinke, A. Tkatchenko, M. Scheffler, Phys. Rev. Lett. 106, (2011) Patrick Rinke (FHI) RPA/GW Levi
23 Atomization energies Mean error (ev) Overbinding PBE PBE0 Underbinding G2 set PBE Solids RPA underestimates bond strengths Paier et al., J. Chem. Phys. 132, (2010); 133, (2010) Harl, Schimka, and Kresse, Phys. Rev. B 81, (2010) Patrick Rinke (FHI) RPA/GW Levi
24 Atomization energies Mean error (ev) Overbinding PBE PBE0 RPA+SOSEX Underbinding G2 set PBE Solids RPA+SOSEX SOSEX alleviates the underbinding problem Paier et al., J. Chem. Phys. 132, (2010); 133, (2010) Harl, Schimka, and Kresse, Phys. Rev. B 81, (2010) Paier et al., arxiv: Patrick Rinke (FHI) RPA/GW Levi
25 Atomization energies Mean error (ev) Overbinding PBE PBE0 RPA+SOSEX RPA+SE Underbinding G2 set PBE Solids RPA+SOSEX HF+RPA HF+RPA : EX@HF + RPA@PBE similar to RPA+SE : (EX+RPA+SE)@PBE Paier, Ren, Rinke, Scuseria, Grüneis, Kresse, and Scheffler, NJP 14, (2012) Ren, Tkatchenko, Rinke, and Scheffler, Phys. Rev. Lett. 106, (2011) Patrick Rinke (FHI) RPA/GW Levi
26 Atomization energies Mean error (ev) Overbinding PBE PBE0 RPA+SOSEX Underbinding G2 set r2pt RPA+SE PBE Solids RPA+SOSEX HF+RPA HF+RPA+SOSEX HF+RPA : EX@HF + RPA@PBE similar to RPA+SE : (EX+RPA+SE)@PBE Paier, Ren, Rinke, Scuseria, Grüneis, Kresse, and Scheffler, NJP 14, (2012) Ren, Tkatchenko, Rinke, and Scheffler, Phys. Rev. Lett. 106, (2011) Patrick Rinke (FHI) RPA/GW Levi
27 The f-electron system Cerium iso-structural (fcc-fcc) α-γ phase transition accompanied by large (15%-17%) volume collapse LDA/GGA capture only low-volume (α) phase LDA+U/SIC-LDA capture only high-volume (γ) phase LDA+DMFT so far restricted to high temperature Patrick Rinke (FHI) RPA/GW Levi
28 Cerium from first principles All electrons are treated on the same quantum mechanical level hybrid functional PBE0 25% exact exchange reduces self-interaction error physically meaningful screening compatible with GW approach cluster extrapolation approach Patrick Rinke (FHI) RPA/GW Levi
29 Cerium - cohesive energy E coh (ev) PBE α γ LDA a 0 (Å) Patrick Rinke (FHI) RPA/GW Levi
30 Cerium - cohesive energy -1.0 E coh (ev) PBE0 "α-phase" PBE α γ PBE0 "γ-phase" LDA a 0 (Å) Patrick Rinke (FHI) RPA/GW Levi
31 Cerium - cohesive energy -1.0 E coh (ev) PBE0 "α-phase" PBE α γ PBE0 "γ-phase" LDA a 0 (Å) Patrick Rinke (FHI) RPA/GW Levi
32 Cerium - electronic structure n(α) n(γ) at a=4.6å f-electrons more localized in γ-phase Patrick Rinke (FHI) RPA/GW Levi
33 Cerium - cohesive energy -1.0 wrong phase ordering E coh (ev) PBE0 "α-phase" PBE α γ PBE0 "γ-phase" LDA a 0 (Å) Patrick Rinke (FHI) RPA/GW Levi
34 Cerium - cohesive energy E coh (ev) EX+cRPA@PBE0 α-phase EX+cRPA@PBE0 γ-phase GPa correct phase ordering results for a Ce 19 cluster a 0 (Å) Patrick Rinke (FHI) RPA/GW Levi
35 TTF/TCNQ dimer at infinite separation E bind [ev] MP2 HF (EX+cRPA)@LDA Distance [Å] Patrick Rinke (FHI) RPA/GW Levi
36 Electron gas at TTF/TCNQ interface TTF and TCNQ crystals have large band gap but 2 dimensional electron gas observed at interface TTF TCNQ Nat. Mat. 7, 574 (2008) What is the origin? Patrick Rinke (FHI) RPA/GW Levi
37 Electron gas at TTF/TCNQ interface DOS DFT-PBE density of states E-E F [ev] DFT-PBE: predicts metallic interface 0.12 e/molecule charge transfer to TCNQ TTF TCNQ in seeming agreement with experiment Patrick Rinke (FHI) RPA/GW Levi
38 TTF/TCNQ dimer at infinite separation TTF 1.0 IP exp PBE TCNQ 2.8 EA PBE exp IP(TTF) > EA(TCNQ) erroneous charge transfer Patrick Rinke (FHI) RPA/GW Levi
39 TTF/TCNQ dimer at infinite separation culprit: self-interaction error in PBE add Hartree-Fock (HF) exchange to remove self-interaction PBE0-like hybrid functional: E xc (α) = α(e HF x Ex PBE )+Ec PBE Patrick Rinke (FHI) RPA/GW Levi
40 TTF/TCNQ dimer at infinite separation 4 Experiment [ev] 2 0 artificial charge transfer PBE0(α) α PBE0-like hybrid functional: E xc (α) = α(e HF x Ex PBE )+Ec PBE Patrick Rinke (FHI) RPA/GW Levi
41 TTF/TCNQ dimer at infinite separation 4 Experiment [ev] 2 0 artificial charge transfer PBE0(α) α How to choose α? Patrick Rinke (FHI) RPA/GW Levi
42 Band structures: photo-electron spectroscopy Photoemission Inverse Photoemission Absorption - hn hn - - hn GW GW BSE TDDFT Patrick Rinke (FHI) RPA/GW Levi
43 Experimental: angle-resolved photoemission spectroscopy ARPES o q= Binding Energy (ev) Masaki Kobayashi, PhD dissertation Patrick Rinke (FHI) RPA/GW Levi
44 Experimental: angle-resolved photoemission spectroscopy ARPES o q= Binding Energy (ev) Masaki Kobayashi, PhD dissertation Patrick Rinke (FHI) RPA/GW Levi
45 ARPES vs GW: the quasiparticle concept Quasiparticle: single-particle like excitation A k( ) spectral function Patrick Rinke (FHI) RPA/GW Levi
46 ARPES vs GW: the quasiparticle concept Quasiparticle: single-particle like excitation A k (ǫ) = ImG k (ǫ) ǫ k Γ k Z k : excitation energy : lifetime : renormalisation Z k ǫ (ǫ k +iγ k ) A k( ) quasiparticlepeak k qp k Patrick Rinke (FHI) RPA/GW Levi
47 ARPES vs GW: the quasiparticle concept Quasiparticle: single-particle like excitation A k (ǫ) = ImG k (ǫ) ǫ k Γ k Z k : excitation energy : lifetime : renormalisation Z k ǫ (ǫ k +iγ k ) A k( ) quasiparticlepeak k qp k Density-Functional Theory: G 0 GW: χ 0 = ig 0 G 0 W 0 = v/(1 χ 0 v) Σ 0 = ig 0 W 0 G 1 = G 1 0 Σ 0 Patrick Rinke (FHI) RPA/GW Levi
48 The screened Coulomb interaction W Work with screened coulomb interaction: W(r,r,t) = dr ε 1 (r,r,t) r r ε(r,r,t): dielectric function The challenge is to calculate W(r,r,t) from first principles! Patrick Rinke (FHI) RPA/GW Levi
49 The screened Coulomb interaction W Work with screened coulomb interaction: W(r,r,t) = dr ε 1 (r,r,t) r r ε(r,r,t): dielectric function The challenge is to calculate W(r,r,t) from first principles! we use random-phase approximation (RPA) for W: c 0 : Kohn-Sham polarizability W = v: bare Coulomb interaction formally scales with (system size) 4 Patrick Rinke (FHI) RPA/GW Levi
50 From screening to quasiparticle energies S=GW : W: screened Coulomb W G G: propagator Quasiparticle energies: G 0 W 0 scheme electron addition and removal energies correction to DFT eigenvalues ǫ DFT nk : includes image effect ǫ qp nk = ǫdft nk +Σ G 0W 0 nk (ǫ qp nk ) vxc nk Patrick Rinke (FHI) RPA/GW Levi
51 ARPES GW: wurtzite ZnO 0 ZnO G W 0 Energy relative to E F (ev) H L A G K M G Yan, Rinke, Winkelnkemper, Qteish, Bimberg, Scheffler, Van de Walle, Semicond. Sci. Technol. 26, (2011) Patrick Rinke (FHI) RPA/GW Levi
52 G 0 W 0 Band Gaps Theoretical Band Gap [ev] Ge InN LDA OEPx(cLDA) OEPx(cLDA) + G 0 W 0 CdSe CdS ZnS wz-gan zb-gan ZnO ZnSe Experimental Band Gap [ev] Rinke et al. New J. Phys. 7, 126 (2005), phys. stat. sol. (b) 245, 929 (2008) Patrick Rinke (FHI) RPA/GW Levi
53 Performance of G 0 W 0 H C S TTF -0.7 all values in ev EA TCNQ N C H IP C6H4S4 G0W0 exp exp G0W0 C12H4N4 Exp.: JACS 112, 3302 (1990), J. Chem. Phys. 60, 1177 (1974), Adv. Mat. 21,1450 (2009), Mol. Cryst. Liquid Cryst. Inc. Nonlin. Opt. 171, 255 (1989) Patrick Rinke (FHI) RPA/GW Levi
54 TTF/TCNQ dimer at infinite separation 4 Experiment [ev] 2 G 0 W PBE0(α) 0 optimal α α choose α to match G 0 W 0 Patrick Rinke (FHI) RPA/GW Levi
55 TTF/TCNQ dimer RPA binding curve E bind [ev] PBE0(α * ) (EX+cRPA)@LDA (EX+cRPA)@PBE0(α * ) Distance [Å] Patrick Rinke (FHI) RPA/GW Levi
56 TTF/TCNQ dimer implications for interface no charge transfer, but small charge rearrangement 2D electron gas not due to charge transfer electron density difference HOMO LUMO PBE optimal optimal-α calculations for interface in progress Patrick Rinke (FHI) RPA/GW Levi
57 TTF/TCNQ dimer implications for interface no charge transfer, but small charge rearrangement 2D electron gas not due to charge transfer electron density difference HOMO Can we do the same in GW? need a GW charge density self-consistent GW LUMO PBE optimal-α calculations for interface in progress optimal Patrick Rinke (FHI) RPA/GW Levi
58 G 0 W 0 for water water molecule HOMO [ev] % 50% G 0 W 10% EX 20% EX Experiment G 0 W Patrick Rinke (FHI) RPA/GW Levi
59 GW The Issue of Self-Consistency Hedin s GW equations: G(1,2) = G 0 (1,2) Γ(1,2,3) = δ(1,2)δ(1,3) P(1,2) = ig(1,2)g(2,1 + ) W(1,2) = v(1,2) + v(1,3)p(3,4)w(4,2)d(3,4) Σ(1,2) = ig(1,2)w(2,1) Dyson s equation: G 1 (1,2) = G 1 0 (1,2) Σ (1,2) selfconsistency selfconsistency Patrick Rinke (FHI) RPA/GW Levi
![scgw for water -11.0 water molecule HOMO [ev] -11.5-12.0-12.](/docs-images/75/72035792/images/60-0.jpg "5 30% 50% G 0 W 0 @PBE 10% EX 20% EX Experiment G 0 W 0 @HF scgw -13.")
60 scgw for water water molecule HOMO [ev] % 50% G 0 W 10% EX 20% EX Experiment G 0 W scgw Patrick Rinke (FHI) RPA/GW Levi
61 scgw and ionization potentials Theoretical IEs [ev] 16 LDA eigenvalue G 0 W 14 Self-Consistent GW Self-Consistent GW deviation from exp. LDA 40.0% G 0 W 5.6% scgw 2.9% GW 1.5% Experimental IEs [ev] set taken from Rostgaard, Jacobsen, and Thygesen, PRB 81, , (2010) Patrick Rinke (FHI) RPA/GW Levi
62 Dependence on the starting point: N 2 Galitskii-Migdal formula: E tot = i 2 Tr[(ω +h 0 )G(ω)] dω 2π Total Energy [Ha] N 2 scgw@pbe scgw@hf Iteration Spectrum G0W0@HF G0W0@PBE scgw@hf scgw@pbe N Energy [ev] At self-consistency no dependence on input Green s function! Patrick Rinke (FHI) RPA/GW Levi
63 The scgw density ρ(pbe) ρ(hf) ρ(scgw) ρ(hf) O C ρ(ccsd) ρ(hf) density from Green s function: ρ(r) = i σ G σσ(r,r,τ = 0 + ) Caruso, Ren, Rinke, Rubio, Scheffler: arxiv: v1 Patrick Rinke (FHI) RPA/GW Levi
64 The scgw density ρ(pbe) ρ(hf) ρ(scgw) ρ(hf) O Dipole moment (in Debye): C Exp. scgw CCSD HF PBE ρ(ccsd) ρ(hf) density from Green s function: ρ(r) = i σ G σσ(r,r,τ = 0 + ) Caruso, Ren, Rinke, Rubio, Scheffler: arxiv: v1 Patrick Rinke (FHI) RPA/GW Levi
65 CO ground state properties CO d ν vib µ E b Exp. (NIST database) sc-gw G 0 W G 0 W (EX+cRPA)@HF (EX+cRPA)@PBE PBE HF Galitskii-Migdal formula: dω E GM = i 2π Tr{[ω +h 0]G(ω)}+E ion Caruso, Ren, Rinke, Rubio, Scheffler: arxiv: v1 Patrick Rinke (FHI) RPA/GW Levi
66 TTF/TCNQ dimer revisited HOMO LUMO opt scgw Patrick Rinke (FHI) RPA/GW Levi
67 RPA and GW a consistent pair GW many-body perturbation theory RPA density-functional theory Dyson s equation: G 1 = G 1 0 Σ[G] total energy: E = E[G] e.g. Galitskii-Migdal optimized effective potential: [ ] G 0 Σ v OEP xc G0 = 0 Adiabatic connection fluctuation dissipation theorem (ACFDT): E xc = E xc [χ 0 ] = E xc [ ig 0 G 0 ] Patrick Rinke (FHI) RPA/GW Levi
68 RPA and GW a consistent pair GW many-body scgw perturbation versus scrpa: theory RPA density-functional theory RPA also iterated to self-consistency (scrpa) Dyson s equation: optimized effective potential: Do MBPT and DFT differ for the same G 1 = G 1 [ diagrams? ] 0 Σ[G] G 0 Σ v OEP xc G0 = 0 total energy: E = E[G] e.g. Galitskii-Migdal Adiabatic connection fluctuation dissipation theorem (ACFDT): E xc = E xc [χ 0 ] = E xc [ ig 0 G 0 ] Patrick Rinke (FHI) RPA/GW Levi
69 The most strongly correlated system Total Energy [Ha] H 2 scgw full CI scrpa Bond Length [Å] full CI: L. Wolniewicz, J. Chem. Phys. 99, 1851 (1993) Patrick Rinke (FHI) RPA/GW Levi
70 Kinetic energy correlation energy in G 0 W 0 : [ ] E G 0W 0 dω c = 0 2π Tr (χ 0 (iω)v) n n=2 correlation energy in RPA: [ Ec RPA dω = 0 2π Tr n=2 ] 1 n (χ 0(iω)v) n adiabatic connection recovers kinetic energy Patrick Rinke (FHI) RPA/GW Levi
71 Kinetic and correlation energy RPA kinetic energy contribution: T RPA c RPA kinetic energy: = Ec RPA E G 0W 0 c T RPA = T 0 +T RPA c RPA Coulomb correlation: U RPA c = E RPA c T RPA c Patrick Rinke (FHI) RPA/GW Levi
72 H 2 energy contributions Energy difference [Ha] E c E tot (E tot -E c ) Bond Length [Å] T E H E ext E x Bond Length [Å] Energy difference [Ha] Coulomb correlation energy makes the difference Patrick Rinke (FHI) RPA/GW Levi
73 Acknowledgements RPA+SOSEX/SE/R2PT Xinguo Ren Alex Tkatchenko Joachim Paier Andreas Grüneis Georg Kresse Gus Scuseria TTF-TCNQ Viktor Atalla Mina Yoon scrpa Daniel Rohr Maria Hellgren scgw/cerium Fabio Caruso/Marco Casadei Xinguo Ren Angel Rubio Support and Ideas Matthias Scheffler FHI-aims code Volker Blum the whole FHI-aims developer team Patrick Rinke (FHI) RPA/GW Levi
74 Cerium - spectroscopy DOS UPS+BIS PBE PBE0 α-phase DOS UPS+BIS PBE0 γ-phase E-E f (ev) description of γ-phase ok α-phase problematic UPS: Wieliczka et al. PRB 29, 3028 (1984) BIS: Wuilloud et al. PRB Patrick Rinke (FHI) 28, RPA/GW 7354 (1983) Levi
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