A new generation of density-functional methods based on the adiabatic-connection fluctuation-dissipation theorem
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1 A new generation of density-functional methods based on the adiabatic-connection fluctuation-dissipation theorem Andreas Görling, Patrick Bleiziffer, Daniel Schmidtel, and Andreas Heßelmann Lehrstuhl für Theoretische Chemie Universität Erlangen Nürnberg J. Gebhardt, G. Gebhardt, T. Gimon, W. Hieringer, C. Neiß, I. Nikiforidis, H. Schulz, E. Trushin, K.-G. Warnick, T. Wölfle A. Görling (University Erlangen Nürnberg) Seattle / 36
2 Overview 1 Introduction Orbital-dependent functionals 2 Exact-exchange (EXX) Kohn-Sham methods 3 Direct RPA and EXXRPA correlation energy Adiabatic-connection fluctuation-dissipation theorem for KS correlation energy EXXRPA methods Direct RPA methods Selfconsistent direct RPA Correlation potentials Total energies, reaction energies Bond dissociation, static correlation Weak interactions, Van-der-Waals bonds 4 Concluding Remarks 5 Literature A. Görling (University Erlangen Nürnberg) Seattle / 36
3 Motivation Orbital-dependent functionals Orbital- and eigenvalue-dependent exchange-correlation functionals in Kohn-Sham methods in order to improve accuracy get rid of Coulomb self-interactions and self-interactions in general (qualitatively correct KS spectra, improved input data for TDDFT) enable description of static correlation (bond formation and breaking), i.e., multiconfiguration cases enable description of Van der Waals interactions and noncovalent, mean interactions A. Görling (University Erlangen Nürnberg) Seattle / 36
4 Kohn-Sham formalism Electronic Schrödinger equation [ ˆT + ˆV ee + v]ψ = E Ψ Kohn-Sham equation [ ˆT + ˆv s ]Φ = E,s Φ [ v s (r)] φ i (r) = ɛ i φ i (r) v s = v + v H + v xc Relation Kohn-Sham to real electronic system HK ˆT + v s Φ ρ Ψ ˆT + ˆVee + ˆv E = T s + U[ρ ] + E xc [ρ ] + dr v(r) ρ(r) HK v xc (r) = δe xc δρ(r) A. Görling (University Erlangen Nürnberg) Seattle / 36
5 Orbital-dependent functionals I Historic development of DFT Ground state energy of an electronic system E = T s + U + E x + E c + dr v nuc (r)ρ(r) Thomas-Fermi-Dirac E = T s [ρ]+ U[ρ] +E x [ρ] + E c [ρ]+ dr v nuc (r)ρ(r) δe/δρ(r) = µ Conventional Kohn-Sham E = T s [{φ i }]+ U[ρ] +E x [ρ] + E c [ρ]+ dr v nuc (r)ρ(r) [ ˆT + ˆv nuc + ˆv H + ˆv x + ˆv c ]φ i = ε i φ i Kohn-Sham with orbital dependent functionals E = T s [{φ i }]+ U[ρ]+E x [{φ i }] +E c [{φ i }]+ dr v nuc (r)ρ(r) [ ˆT + ˆv nuc + ˆv H + ˆv x + ˆv c ]φ i = ε i φ i A. Görling (University Erlangen Nürnberg) Seattle / 36
6 Orbital-dependent functionals II Conventional Kohn-Sham methods E = T s [{φ i }] + U[ρ ] + E x [ρ ] + E c [ρ ] + dr v nuc (r) ρ (r) [ ˆT + ˆvnuc + ˆv H [ρ ] + ˆv x [ρ ] + ˆv c [ρ ]] φ i = ε i φ i KS methods with orbital- and eigenvalue-dependent xc-functionals E = T s [{φ i }] + U[ρ ] + E x [{φ i }] + E c [{ε s }, {φ s }] + dr v nuc (r) ρ(r) [ ˆT + ˆvnuc + ˆv H [ρ ] + ˆv x [{φ i }] + ˆv c [{ε s }, {φ s }]] φ i = ε i φ i A. Görling (University Erlangen Nürnberg) Seattle / 36
7 Exact treatment of KS exchange Exchange energy occ. E x = dr dr φ i(r )φ j (r )φ j (r)φ i (r) r r i,j Exchange potential v x ( r) = δe x[{φ i }] δρ( r) dr χ (r, r ) v x (r ) = t x (r) occ. KS response function χ (r, r ) = 4 occ. t x (r) = 4 i unocc. a i unocc. a φ i (r)φ a (r) a ˆv NL x i ɛ i ɛ a φ i (r)φ a (r)φ a (r )φ i (r ) ε i ε a Plane wave methods for solid numerically stable Gaussian basis set methods for molecules numerically demanding A. Görling (University Erlangen Nürnberg) Seattle / 36
8 Exact-exchange Gaussian basis set KS method Auxiliary basis set: Electrostatic potential of Gaussian functions v x (r) = v x,k f k (r) with f k (r) = dr g k (r )/ r r k Incorporation of exact conditions to treat asymptotic of v x (r) dr ρ x (r) = 1 with ρ x (r) = v x,k g k (r) k φ HOMO v x φ HOMO = φ HOMO ˆv NL x φ HOMO Balancing of auxiliary and orbital basis sets (uncontracted orbital basis sets required) JCP 127, 5412 (27) By preprocessing of auxiliary basis set and regularization techniques, EXX calculations with standard contracted orbital basis sets possible (aug-cc-pcvxz, x = 3, 4, 5) A. Görling (University Erlangen Nürnberg) Seattle / 36
9 EXX vs. GGA (PBE) orbitals of methane 2 t2u ev 2 t2u ev 3 a1g.396 ev 1 t2u ev 2 a1g ev -5 3 a1g ev t2u ev -2 2 a1g ev contour value.32 A. Go rling (University Erlangen Nu rnberg) Seattle / 36
10 EXX vs. GGA (PBE) orbitals of methane 2 t2u ev 2 t2u ev 3 a1g.396 ev 1 t2u ev 2 a1g ev -5 3 a1g ev t2u ev -2 2 a1g ev contour value.13 A. Go rling (University Erlangen Nu rnberg) Seattle / 36
11 Band gaps of semiconductors FLAPW vs. PP EXX band gaps EXX+VWNc Exp. FLAPW a PP b Si Γ Γ Γ L Γ X SiC Γ Γ Γ L Γ X Ge Γ Γ Γ L Γ X GeAs Γ Γ Γ L Γ X C Γ Γ Γ L Γ X a PRB 83, 4515 (211) b PRB 59, 131 (1997) A. Görling (University Erlangen Nürnberg) Seattle / 36
12 Summary exact-exchange Kohn-Sham methods Problem of Coulomb self-interaction is solved Physically meaningful orbital and eigenvalue spectra (Improved input data for time-dependent DFT) Similar total energies of exact-exchange-only Kohn-Sham and HF methods but completely different orbital and eigenvalue spectra A. Görling (University Erlangen Nürnberg) Seattle / 36
13 Adiabatic-connection fluctuation-dissipation theorem for DFT correlation energy I E c = 1 2π 1 dα drdr 1 r r dω [ ] χ α (r, r, iω) χ (r, r, iω) Integration of response functions along complex frequencies 1 2π dω dr dr g(r, r ) χ α (r, r, iω) = [ = dr dr g(r, r ) ρ α 2 (r, r ) 1 ] 2 ρ(r)ρ(r ) + ρ(r)δ(r r ) dω a a 2 + ω 2 = π 2 later on g(r, r ) = 1 r r χ α (r, r, iω) = 2 n E n E (E n E ) 2 + ω 2 Ψα ˆρ(r) Ψ α n Ψ α n ˆρ(r ) Ψ α V c (α) = Ψ (α) ˆV ee Ψ (α) Φ ˆV ee Φ A. Görling (University Erlangen Nürnberg) Seattle / 36
14 Adiabatic-connection fluctuation-dissipation theorem for DFT correlation energy II E c = 1 2π 1 dα drdr 1 r r dω Integration along adiabatic connection E c = 1 [ ] χ α (r, r, iω) χ (r, r, iω) = 1 dα V c (α) with V c (α) = Ψ (α) ˆV ee Ψ (α) Φ ˆV ee Φ E c = Ψ (α) ˆT + α ˆV ee Ψ (α) Φ ˆT + α ˆV ee Φ dα V c (α) Ψ (α) = min Ψ ρ Ψ ˆT + α ˆV ee Ψ From Hellmann-Feynman theorem follows de c (α) dα = V c (α) A. Görling (University Erlangen Nürnberg) Seattle / 36
15 Response functions occ χ (r, r, iω) = 4 KS response function χ (r, r, iω) i unocc a ɛ ai ɛ 2 ai + ω2 ϕ i(r)ϕ a (r)ϕ a (r )ϕ i (r ) Basics time-dependent DFT ρ (1) = X α=1 v (1) = X v (1) s = X [v (1) + F Hxc X α=1 v (1) ] v s = v + v H + v xc v s (1) = v (1) pert + v (1) H + v(1) xc v (1) Hxc = δv Hxc δρ ρ(1) = F Hxc ρ (1) = F Hxc X α=1 v (1) Response functions χ α (r, r, iω) from EXX-TDDFT X α = X + α X F Hx X α X α = [1 αx F Hx ] 1 X X α = X [X αh Hx ] 1 X with X F Hx X = H Hx A. Görling (University Erlangen Nürnberg) Seattle / 36
16 Exact frequency-dependent exchange kernel OEP-like equation for sum f Hx (ω, r, r ) of Coulomb and EXX kernel dr dr X (r, r, ω) f Hx (ω, r, r ) X (r, r, ω) = h Hx (ω, r, r ) with h Hx (ω, r, r ) = 1 4 Y (r)λ(ω) [A + B + ] λ(ω)y(r ) + ω Y (r)λ(ω)ɛ 1 [A + B + ] ɛ 1 λ(ω)y(r) + a ˆv NL x ˆv x j Y ia(r)λ ia(ω) φ i(r)φ j(r) + ɛ a ɛ j i j a + b ˆv NL x ˆv x i Y ia(r)λ ia(ω) φ a(r)φ b (r) + ɛ b ɛ i a b i A ia,jb = 2(ai jb) (ab ji) ia,jb = δ ij ϕ a ˆv NL x B ia,jb = 2(ai bj) (aj bi) ˆv x ϕ b δ ab ϕ i ˆv NL x ˆv x ϕ j λ ia,jb = δ ia,jb 4ɛ ia ɛ 2 ia +ω2 ɛ ia,jb = δ ia,jb ɛ ia = δ ia,jb (ɛ a ɛ i) Y ia(r) = φ i(r)φ a(a) RPA: h H (r, r, ω) = Y (r)λ(ω) C λ(ω)y(r) with C ia,jb = (ia jb) A. Görling (University Erlangen Nürnberg) Seattle / 36
17 E c = 1 2π dω 1 dα RI-EXXRPA method I drdr 1 [ ] χ α(r, r, iω) χ (r, r, iω) r r Representation of χ α(iω), h x(iω) in RI basis set with respect to Coulomb norm with X α = [1 αx F Hx ] 1 X X α = X [X αh Hx ] 1 X E c = 1 2π dω 1 (using F Hx = X 1 H Hx X 1 ) ] dα T r [S 1 X [X α H Hx ] 1 X S 1 X X (iω) = D λ(iω)d with D ia,h = (ϕ iϕ a f h ) Coul and λ ia,jb = δ ia,jb 4ɛ ia ɛ 2 ia + ω2 and H(iω) = 1 4 DT λ(iω) [A+B+ ] λ(iω)d + (iω) DT λ(iω) ɛ 1 [A+B+ ] ɛ 1 λ(iω)d + W 1(iω) + W T 1 (iω) + W 2(iω) + W T 2 (iω) A. Görling (University Erlangen Nürnberg) Seattle / 36
18 RI-EXXRPA method II E c = 1 2π dω 1 ] dα T r [S 1 X [X α H Hx ] 1 X S 1 X Orthonormalize RI basis set, i.e. make S = E, and use X = ( X ) 2 1 ( X) 1 2 carry out analytic integration over coupling constant and E c = 1 2π ( ) ] dω T r [( X (iω)) 1 2 U(iω) τ 1 (iω) ln[ 1 + τ (iω) ]+1 U(iω) T ( X (iω)) 1 2 with ( X (iω)) 1 2 H(iω)( X(iω)) 1 2 = U(iω) τ (iω) U (iω) Complete exchange kernel can be treated (RI-EXXRPA+) N 5 scaling A. Görling (University Erlangen Nürnberg) Seattle / 36
19 E c = 1 2π dω 1 RI2-dRPA method RI2-dRPA ] dα T r [S 1 X [X α H H ] 1 X S 1 X For drpa, with second RI approximation and S = E, H H simplifies to H H = X X With spectral representation X (iω) = V(iω) σ(iω) V (iω) E c = 1 2π dω Tr [ln[1 + σ(iω)] σ(iω)] Only KS response function X (iω) required N 4 scaling A. Görling (University Erlangen Nürnberg) Seattle / 36
20 Self-consistent direct RPA Optimized-effective-potential equation for vc drpa dr χ (r, r ) vc drpa (r ) = t drpa c (r) with t drpa c (r) = s dr δedrpa c δφ s (r ) δφ s (r ) δv s (r) + s δe drpa c δɛ s δɛ s δv s (r) v drpa c is represented in auxiliary basis set X v drpa c = t drpa c A. Görling (University Erlangen Nürnberg) Seattle / 36
21 drpa correlation potentials I.2 v drpa c for neon atoms.15.1 v c drpa (a.u.) Exact -.15 SC-dRPA PW92 LDA -.2 PBE r(a.u.) drpa correlation potential qualitatively correct in contrast to LDA and GGA correlation potential A. Görling (University Erlangen Nürnberg) Seattle / 36
22 drpa correlation potentials II v drpa c for C 2 H 2 v[a.u.] drpa v c v x r[a.u.] A. Görling (University Erlangen Nürnberg) Seattle / 36
23 Results EXXRPA correlation methods I Deviations of total energies from CCSD(T) energies, E = E Method E CCSD(T) Orbital basis: aug-cc-pcvtz RI basis (for E c and v xc): aug-cc-pvtz/mp2-fit Energy difference to CCSD(T) [Hartree] CCSD MP2 EXXRPA@EXX EXXRPA@dRPA NH 2 CONH 2 HCOOCH 3 HCONH 2 H 2 CCO CH 3 CHO C 2 H 4 O HCOOH HNCO HCHO C 2 H 5 OH CH 3 OH C 2 H 6 C 2 H 4 C 2 H 2 CH 4 NH 3 CO 2 CO H 2 O 2 H 2 O H 2 RMS [Hartree] CCSD MP2 B3LYP EXXRPA+@dRPA EXXRPA+@EXX drpa@drpa drpa@exx A. Görling (University Erlangen Nürnberg) Seattle / 36
24 Results EXXRPA correlation methods II Deviations of total energies from CCSD(T) energies, E = E Method E CCSD(T) Orbital basis: aug-cc-pcvqz RI basis (for E c and v xc): aug-cc-pvqz/mp2-fit Energy difference to CCSD(T) [Hartree] CCSD MP2 EXXRPA@EXX EXXRPA@dRPA NH 2 CONH 2 HCOOCH 3 HCONH 2 H 2 CCO CH 3 CHO C 2 H 4 O HCOOH HNCO HCHO C 2 H 5 OH CH 3 OH C 2 H 6 C 2 H 4 C 2 H 2 CH 4 NH 3 CO 2 CO H 2 O 2 H 2 O H RMS [Hartree] CCSD MP2 B3LYP EXXRPA+@SC-RPA EXXRPA+@EXX drpa@drpa drpa@exx A. Görling (University Erlangen Nürnberg) Seattle / 36
25 Results EXXRPA correlation methods III Deviations of reaction energies from CCSD(T) reaction energies Orbital basis: aug-cc-pcvtz RI basis (for E x and v xc): aug-cc-pvtz/mp2-fit RMS [kcal/mol] drpa@drpa drpa@exx B3LYP MP2 CCSD 1 kcal/mol =.43 ev EXXRPA+@dRPA EXXRPA+@EXX A. Görling (University Erlangen Nürnberg) Seattle / 36
26 Results EXXRPA correlation methods IV Deviations of reaction energies from CCSD(T) reaction energies Orbital basis: aug-cc-pcvqz RI basis (for E x and v xc): aug-cc-pvqz/mp2-fit RMS [kcal/mol] drpa@drpa drpa@exx B3LYP MP2 CCSD 1 kcal/mol =.43 ev EXXRPA+@dRPA EXXRPA+@EXX A. Görling (University Erlangen Nürnberg) Seattle / 36
27 Dissociation of H 2 Orbital basis: aug-cc-pcvqz RI basis (for E c and v xc ): aug-cc-pvqz/mp2-fit EXXRPA SC-EXXRPA FCI HF E [a.u.] r [a.u.] Dissociation limit of other molecules (CO, N 2, etc.) is also treated correctly A. Görling (University Erlangen Nürnberg) Seattle / 36
28 Coupling constant integrand -.5 HF-RPA 1 E c = dα V c (α) -.5 HF-RPA V c (α) [a.u.] r=1.4 a r=6. a r=1. a r=2. a r=3. a EXX-RPA V c HL (α) [a.u.] r=1.4 a r=6. a r=1. a r=2. a r=3. a EXX-RPA V c (α) [a.u.] V c HL (α) [a.u.] (ia ia) coupling strength V c (α) coupling strength Vc HL (α) A. Görling (University Erlangen Nürnberg) Seattle / 36
29 Dissociation of N HF EXX CCSD RI-EXX-RPA Orbital basis: avtz RI basis: avtz Energy [hartree] r [bohr] Special treatment of singularity in ω-integrand ( τ 1 ln [ 1 + τ (iω) ] + 1 ) A. Görling (University Erlangen Nürnberg) Seattle / 36
30 Twisting of ethene Orbital basis: avqz RI basis: avqz Energy [Hartree] Torsion angle [degree] HF EXX CCSD B3LYP RI-EXXRPA+ MCSCF A. Görling (University Erlangen Nürnberg) Seattle / 36
31 RI-EXXRPA for noncovalent bonds Interaction energies in kcal/mol Complex BasisCCSD(T) drpa drpa EXXRPA+ EXXRPA+ MP2 (NH 3 ) CBS (H 2 O) CBS (HCONH 2 ) CBS (HCOOH) CBS (C 2 H 4 ) CBS C 2 H 4 C 2 H CBS (CH 4 ) CBS RMS RMS CBS a CBS obtained via F12-CCSD 1 kcal/mol =.43 ev A. Görling (University Erlangen Nürnberg) Seattle / 36
32 S22 test set of dimer binding energies E [kcal/mol] Hydrogen Standard benchmark for non-covalent interactions E = E Binding Method EBinding CCSD(T) Mixed -12 drpa@drpa EXXRPA+@dRPA Dispersion -14 MP2 PBE B3LYP NH 3 (C 2h ) 2 (H 2 O) 2 (C s) 3 (HCOOH) 2 (C 2h ) 4 (CHONH 2 ) 2 5 Uracil-Uracil (C 2h ) 6 2-Pyridoxine-2-Aminopyridine 7 AT (WC) 8 (CH 4 ) 2 (D 3d ) 9 (C 2 H 4 ) 2 (D 2d ) 1 Bz CH 4 (C 3) 11 Bz-Bz (C 2h ) 12 Pyrazine-Pyrazine (C s) 13 Uracil-Uracil (C 2) 14 Indole-Bz (stacked) 15 AT (stacked) 16 C 2 H 4 C 2 H 2 (C 2v) 17 Bz H 2 O (C s) 18 Bz NH 3 (C s) 19 Bz HCN (C s) 2 Bz-Bz (C 2v) 21 Indole-Bz (T-shaped) 22 Phenole-Phenole A. Görling (University Erlangen Nürnberg) Seattle / 36
33 Directional non-covalent interactions in Ar Br2 I CCSD(T) EXXRPA 1 kcal/mol =.43 ev A. Go rling (University Erlangen Nu rnberg) Seattle / 36
34 Directional non-covalent interactions in Ar Br2 II CCSD(T) B3LYP 1 kcal/mol =.43 ev A. Go rling (University Erlangen Nu rnberg) Seattle / 36
35 Summary EXXRPA correlation functional combines accuracy at equilibrium geometries with a correct description of dissociation (static correlation) and a highly accurate treatment of VdW interactions EXXRPA correlation functional is self-interaction free Self-consistent drpa yields qualitatively reasonable correlation potentials in contrast to conventional KS methods Promising starting point for further developments, e.g., completely self-consistent EXXRPA method or inclusion of correlation in kernel Orbital-dependent functionals open up fascinating new possibilities in DFT A. Görling (University Erlangen Nürnberg) Seattle / 36
36 Literature Phys. Rev. Lett. 79, 289 (1997) Phys. Rev. B 81, (21) EXX for solids Phys. Rev. B 83, 4515 (211) EXX for molecules Phys. Rev. Lett. 83, 5459 (1999) J. Chem. Phys. 128, 1414 (28) Phys. Rev. A 8, 1257 (29) Phys. Rev. Lett. 12, 2333 (29) Mol. Phys. 18, 359 (21) Mol. Phys. 19, 2473 (211) Review TDEXX Int. J. Quantum. Chem. 11, 222 (21) J. Chem. Phys. 134, 3412 (211) EXX-RPA Phys. Rev. Lett. 16, 931 (211) J. Chem. Phys. 136, (212) A. Görling (University Erlangen Nürnberg) Seattle / 36
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