Extending Kohn-Sham density-functional theory

Transcription

1 Extending Kohn-Sham density-functional theory Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Web page: January 2011

2 Introduction: Density-functional theory (DFT) Kohn-Sham (KS) scheme } E exact = min Φ ˆT + ˆV ne Φ +E Hxc [n Φ ] Φ Φ : single-determinant wave function

3 Introduction: Density-functional theory (DFT) Kohn-Sham (KS) scheme } E exact = min Φ ˆT + ˆV ne Φ +E Hxc [n Φ ] Φ Φ : single-determinant wave function Some problems (among others) of the usual approximations for exchange-correlation functional E xc [n] (LDA, GGA,...): they do not describe well (long-range) van der Waals dispersion forces they do not describe well static correlation (e.g., bond dissociation)

4 1 Short-range DFT + long-range RPA 2 Short-range DFT + long-range DMFT 3 Double-hybrid DFT

5 1 Short-range DFT + long-range RPA with J. Ángyán, I. Gerber, G. Jansen, A. Savin, W. Zhu 2 Short-range DFT + long-range DMFT 3 Double-hybrid DFT

6 Random phase approximation (RPA) in DFT An orbital-dependent approximation: E xc = E HF x [φ i }]+E RPA c [φ i,ǫ i }] usually with non-self-consistent orbitals (post-kohn-sham).

7 Random phase approximation (RPA) in DFT An orbital-dependent approximation: E xc = E HF x [φ i }]+E RPA c [φ i,ǫ i }] usually with non-self-consistent orbitals (post-kohn-sham). = increasing interest in the DFT community

8 Random phase approximation (RPA) in DFT An orbital-dependent approximation: E xc = E HF x [φ i }]+E RPA c [φ i,ǫ i }] usually with non-self-consistent orbitals (post-kohn-sham). = increasing interest in the DFT community Encouraging results: consistent with the use of exact (HF) exchange qualitatively correct dispersion forces at (very) large separation good cohesive energies and lattice constants of solids

9 Random phase approximation (RPA) in DFT An orbital-dependent approximation: E xc = E HF x [φ i }]+E RPA c [φ i,ǫ i }] usually with non-self-consistent orbitals (post-kohn-sham). = increasing interest in the DFT community Encouraging results: consistent with the use of exact (HF) exchange qualitatively correct dispersion forces at (very) large separation good cohesive energies and lattice constants of solids But several unsatisfactory aspects: short-range correlation energies far too negative strong dependence on basis size not good for simple van der Waals dimers

10 Example: interaction energy curve of Ne 2 RPA (with PBE orbitals), aug-cc-pv5z basis: Interaction energy (mhartree) Ne 2 Accurate RPA Internuclear distance (Bohr)

11 Example: interaction energy curve of Be 2 RPA (with PBE orbitals), cc-pv5z basis: Interaction energy (mhartree) Be 2 Accurate RPA Internuclear distance (Bohr)

12 Range-separated DFT Multideterminant extension of KS scheme with range separation } E exact = min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Ψ

13 Range-separated DFT Multideterminant extension of KS scheme with range separation } E exact = min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Ψ Ŵ lr ee = i<j erf(µr ij ) r ij : long-range electron-electron interaction

14 Range-separated DFT Multideterminant extension of KS scheme with range separation } E exact = min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Ψ Ŵ lr ee = i<j erf(µr ij ) r ij : long-range electron-electron interaction EHxc sr [n] : short-range Hxc density functional

15 Range-separated DFT Multideterminant extension of KS scheme with range separation } E exact = min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Ψ Ŵ lr ee = i<j erf(µr ij ) r ij : long-range electron-electron interaction EHxc sr [n] : short-range Hxc density functional minimizing wave function Ψ lr = i c iφ i is multi-determinant

16 Range-separated DFT Multideterminant extension of KS scheme with range separation } E exact = min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Ψ Ŵ lr ee = i<j erf(µr ij ) r ij : long-range electron-electron interaction EHxc sr [n] : short-range Hxc density functional minimizing wave function Ψ lr = i c iφ i is multi-determinant parameter µ controls the range of separation.

17 Range-separated DFT Multideterminant extension of KS scheme with range separation } E exact = min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Ψ Ŵ lr ee = i<j erf(µr ij ) r ij : long-range electron-electron interaction EHxc sr [n] : short-range Hxc density functional minimizing wave function Ψ lr = i c iφ i is multi-determinant parameter µ controls the range of separation. In principle: exact

18 Range-separated DFT Multideterminant extension of KS scheme with range separation } E exact = min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Ψ Ŵ lr ee = i<j erf(µr ij ) r ij : long-range electron-electron interaction EHxc sr [n] : short-range Hxc density functional minimizing wave function Ψ lr = i c iφ i is multi-determinant parameter µ controls the range of separation. In principle: exact In practice: approximations are necessary for Ψ lr and E sr xc[n]

19 Range-separated DFT: approximations Approximations for E sr xc[n] short-range LDA short-range PBE...

20 Range-separated DFT: approximations Approximations for E sr xc[n] short-range LDA short-range PBE... Approximations for Ψ lr single-determinant = RSH method MP2 = RSH+lrMP2 method RPA = RSH+lrRPA method...

21 Range-separated hybrid (RSH) scheme Restriction to single-determinant wave functions Φ: } E RSH = min Φ ˆT + ˆV ne +Ŵee Φ +E lr Hxc[n sr Φ ] Φ

22 Range-separated hybrid (RSH) scheme Restriction to single-determinant wave functions Φ: } E RSH = min Φ ˆT + ˆV ne +Ŵee Φ +E lr Hxc[n sr Φ ] Φ So the RSH energy is E RSH = Φ RSH ˆT + ˆV ne Φ RSH +E H +E lr x,hf +E sr xc This is a hybrid DFT with exact (HF) exchange at long range.

23 Long-range correlation energy E lr c What is missing in RSH is the long-range correlation energy: E exact = E RSH +E lr c

24 Long-range correlation energy E lr c What is missing in RSH is the long-range correlation energy: E exact = E RSH +E lr c Adiabatic connection from RSH reference to exact system: E lr c = 1 0 dλ } Ψ lr λ Ŵlr Ψλ lr Φ RSH Ŵ lr Φ RSH with the long-range perturbation operator Ŵ lr = Ŵ lr ee ˆV lr Hx,HF

25 Long-range correlation energy E lr c What is missing in RSH is the long-range correlation energy: E exact = E RSH +E lr c Adiabatic connection from RSH reference to exact system: E lr c = 1 0 dλ } Ψ lr λ Ŵlr Ψλ lr Φ RSH Ŵ lr Φ RSH with the long-range perturbation operator Ŵ lr = Ŵ lr ee ˆV lr Hx,HF In an orbital basis, E lr c = dλ ps w lr qr ( P lr ) c,λ pqrs pqrs

26 Long-range correlation energy E lr c Pc,λ lr from a fluctuation-dissipation theorem P lr c,λ = dω 2π [ χ lr λ (iω) χ RSH (iω) ] + lr λ where λ lr comes from the variation of the density.

27 Long-range correlation energy E lr c Pc,λ lr from a fluctuation-dissipation theorem P lr c,λ = dω 2π [ χ lr λ (iω) χ RSH (iω) ] + lr λ where λ lr comes from the variation of the density. The long-range response function χλ lr (iω) is given by χ lr λ (iω) 1 = χ lr IP,λ (iω) 1 f lr Hxc,λ (iω)

28 Long-range correlation energy E lr c Pc,λ lr from a fluctuation-dissipation theorem P lr c,λ = dω 2π [ χ lr λ (iω) χ RSH (iω) ] + lr λ where λ lr comes from the variation of the density. The long-range response function χλ lr (iω) is given by χ lr λ (iω) 1 = χ lr IP,λ (iω) 1 f lr Hxc,λ (iω) Possible approximations: RPA approximation: fxc,λ lr = 0 = RSH+lrRPA method RPAx approximation: fc,λ lr = 0 = RSH+lrRPAx method

29 Dependence on basis size: Ar 2 Binding energy (aug-cc-pvnz basis, µ = 0.5, sr-pbe functional): Percentage of CBS binding energy Ar 2 RPA RPAx RSH+lrRPA RSH+lrRPAx avtz avqz av5z CBS One-particle basis = RSH+lrRPA(x) has a small basis dependence

30 Interaction energy curve of Ne 2 Interaction energy (aug-cc-pv5z basis, µ = 0.5, sr-pbe functional): Interaction energy (mhartree) Ne 2 Accurate RPA RPAx RSH+lrRPAx Internuclear distance (bohr) = Range separation improves RPA(x)

31 Interaction energy curve of Be 2 Interaction energy (cc-pv5z basis, µ = 0.5, sr-pbe functional): Interaction energy (mhartree) Be 2 Accurate RPA RPAx RSH+lrRPAx Internuclear distance (bohr) = Range separation improves RPA(x)

32 Equilibrium interaction energies of a set of 22 weakly-interacting molecular systems from water dimer to DNA base pairs (S22 set of Hobza and coworkers) 50 mean absolute relative error (%) RPA RPAx RSH+lrRPA RSH+lrRPAx

33 Several RPAx variants RPAx: based on ACFDT RPAx-MB (McLachlan-Ball, 1964): based on plasmon formula RPAx-SO1 (Szabo-Ostlund, 1977): based on approx. CCD RPAx-SO2 (Szabo-Ostlund, 1977): based on approx. CCD

34 Several RPAx variants RPAx: based on ACFDT RPAx-MB (McLachlan-Ball, 1964): based on plasmon formula RPAx-SO1 (Szabo-Ostlund, 1977): based on approx. CCD RPAx-SO2 (Szabo-Ostlund, 1977): based on approx. CCD Mean absolute relative errors on S22 set with aug-cc-pvdz basis: mean absolute relative error (%) RSH+ lrrpax lrrpax-mb lrrpax-so1 lrrpax-so2 = Best variants of RSH+lrRPAx gives an error 5 %

35 1 Short-range DFT + long-range RPA 2 Short-range DFT + long-range DMFT with D. Rohr and K. Pernal 3 Double-hybrid DFT

36 Density-matrix-functional theory (DMFT) Minimization of a functional of the one-particle density matrix Γ } E exact = min T[Γ]+V ne [n Γ ]+E Hxc [Γ] Γ

37 Density-matrix-functional theory (DMFT) Minimization of a functional of the one-particle density matrix Γ } E exact = min T[Γ]+V ne [n Γ ]+E Hxc [Γ] Γ e.g., Buijse-Baerends (or Müller) approximation (in a basis): E BB xc [Γ] = abcd(γ 1/2 ) ab (Γ 1/2 ) cd ac db

38 Density-matrix-functional theory (DMFT) Minimization of a functional of the one-particle density matrix Γ } E exact = min T[Γ]+V ne [n Γ ]+E Hxc [Γ] Γ e.g., Buijse-Baerends (or Müller) approximation (in a basis): E BB xc [Γ] = abcd(γ 1/2 ) ab (Γ 1/2 ) cd ac db Some advantages of DMFT over DFT: no approximation on the form of the kinetic energy more flexibility beyond single-determinant DFT since Γ(r,r ) = i n i φ i (r)φ i (r ) can have fractional orbital occupation numbers = static correlation

39 Density-matrix-functional theory (DMFT) Minimization of a functional of the one-particle density matrix Γ } E exact = min T[Γ]+V ne [n Γ ]+E Hxc [Γ] Γ e.g., Buijse-Baerends (or Müller) approximation (in a basis): E BB xc [Γ] = abcd(γ 1/2 ) ab (Γ 1/2 ) cd ac db Some advantages of DMFT over DFT: no approximation on the form of the kinetic energy more flexibility beyond single-determinant DFT since Γ(r,r ) = i n i φ i (r)φ i (r ) can have fractional orbital occupation numbers = static correlation but dynamic correlation is often better described by DFT approximations.

40 Short-range DFT + long-range DMFT Start from multideterminant range-separated DFT: } E exact = min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Ψ

41 Short-range DFT + long-range DMFT Start from multideterminant range-separated DFT: } E exact = min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Ψ Reformulation with a constrained search } E exact = min min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Γ Ψ Γ

42 Short-range DFT + long-range DMFT Start from multideterminant range-separated DFT: } E exact = min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Ψ Reformulation with a constrained search E exact = min Γ min Ψ Γ = min Γ with E lr Hxc[Γ] = min Ψ Γ Ψ Ŵlr ee Ψ Ψ ˆT + ˆV ne +Ŵ lr ee Ψ +E sr Hxc[n Ψ ] T[Γ]+Vne [n Γ ]+E lr Hxc[Γ]+E sr Hxc[n Γ ] } }

43 Short-range DFT + long-range DMFT Start from multideterminant range-separated DFT: } E exact = min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Ψ Reformulation with a constrained search E exact = min Γ min Ψ Γ = min Γ with E lr Hxc[Γ] = min Ψ Γ Ψ Ŵlr ee Ψ So, the final equation that we use is E exact = min Γ Ψ ˆT + ˆV ne +Ŵ lr ee Ψ +E sr Hxc[n Ψ ] T[Γ]+Vne [n Γ ]+E lr Hxc[Γ]+E sr Hxc[n Γ ] } T[Γ]+Vne [n Γ ]+E H [n Γ ]+E lr xc[γ]+e sr xc[n Γ ] } }

44 Short-range DFT + long-range DMFT Start from multideterminant range-separated DFT: } E exact = min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Ψ Reformulation with a constrained search E exact = min Γ min Ψ Γ = min Γ with E lr Hxc[Γ] = min Ψ Γ Ψ Ŵlr ee Ψ So, the final equation that we use is E exact = min Γ Ψ ˆT + ˆV ne +Ŵ lr ee Ψ +E sr Hxc[n Ψ ] T[Γ]+Vne [n Γ ]+E lr Hxc[Γ]+E sr Hxc[n Γ ] } T[Γ]+Vne [n Γ ]+E H [n Γ ]+E lr xc[γ]+e sr xc[n Γ ] } } approximations: lrbb srpbe

45 Dependence on basis size: H 2 Total energy (cc-pvnz basis sets, µ = 0.4) exact PBE BB srpbe+lrbb total energy in hartree cc-pvdz cc-pvtz cc-pvqz cc-pv5z = srpbe+lrbb has a small basis dependence

46 Dissociation curve of H 2 Relative energy wrt equilibrium (cc-pvtz basis, µ = 0.4) relative energy in hartree Lie&Clementi PBE srpbe+lrbb BB distance in bohr = srpbe+lrbb dissociates better than PBE but still large error due to srpbe

47 Dissociation curve of LiH Relative energy wrt equilibrium (cc-pvtz basis, µ = 0.4) relative energy in hartree Lie&Clementi PBE srpbe+lrbb BB distance in bohr = srpbe+lrbb is better than both PBE and BB

48 Dissociation curve of BH Relative energy wrt equilibrium (cc-pvtz basis, µ = 0.4) relative energy in hartree Lie&Clementi PBE srpbe+lrbb BB distance in bohr = srpbe+lrbb is better than both PBE and BB

49 Dissociation curve of FH Relative energy wrt equilibrium (cc-pvtz basis, µ = 0.4) relative energy in hartree distance in bohr Lie&Clementi PBE srpbe+lrbb BB = srpbe+lrbb is better than both PBE and BB

50 1 Short-range DFT + long-range RPA 2 Short-range DFT + long-range DMFT 3 Double-hybrid DFT with K. Sharkas and A. Savin

51 Double-hybrid approximations (Grimme, 2006) Combination of HF exchange with an exchange functional and MP2 correlation with a correlation functional: Exc DH = a x Ex HF +(1 a x )E x [n]+(1 a c )E c [n]+a c Ec MP2

52 Double-hybrid approximations (Grimme, 2006) Combination of HF exchange with an exchange functional and MP2 correlation with a correlation functional: Exc DH = a x Ex HF +(1 a x )E x [n]+(1 a c )E c [n]+a c Ec MP2 Example: B2-PLYP: a x = 0.53 and a c = 0.27, optimized for heats of formation

53 Double-hybrid approximations (Grimme, 2006) Combination of HF exchange with an exchange functional and MP2 correlation with a correlation functional: Exc DH = a x Ex HF +(1 a x )E x [n]+(1 a c )E c [n]+a c Ec MP2 Example: B2-PLYP: a x = 0.53 and a c = 0.27, optimized for heats of formation = reach near-chemical accuracy but empirical

54 A rigorous derivation of double hybrids Multideterminant extension of Kohn-Sham scheme based on linear scaling of electron-electron interaction: } E exact = min Ψ ˆT + ˆV ne +λŵ ee Ψ +ĒHxc[n λ Ψ ] Ψ where ĒHxc λ [n] is the λ-complement functional.

55 A rigorous derivation of double hybrids Multideterminant extension of Kohn-Sham scheme based on linear scaling of electron-electron interaction: } E exact = min Ψ ˆT + ˆV ne +λŵ ee Ψ +ĒHxc[n λ Ψ ] Ψ where ĒHxc λ [n] is the λ-complement functional. exchange obtained by linear scaling: Ē λ x [n] = (1 λ)e x [n]

56 A rigorous derivation of double hybrids Multideterminant extension of Kohn-Sham scheme based on linear scaling of electron-electron interaction: } E exact = min Ψ ˆT + ˆV ne +λŵ ee Ψ +ĒHxc[n λ Ψ ] Ψ where ĒHxc λ [n] is the λ-complement functional. exchange obtained by linear scaling: Ē λ x [n] = (1 λ)e x [n] correlation obtained by scaling of the density: with n 1/λ (r) = (1/λ) 3 n(r/λ) Ē λ c [n] = E c [n] λ 2 E c [n 1/λ ]

57 A rigorous derivation of double hybrids First, single-determinant approximation: E DS1H = Φ ˆT + ˆV ne Φ +E H +λe HF x +(1 λ)e x [n]+e c [n] λ 2 E c [n 1/λ ] we get a density-scaled one-parameter hybrid (DS1H).

58 A rigorous derivation of double hybrids First, single-determinant approximation: E DS1H = Φ ˆT + ˆV ne Φ +E H +λe HF x +(1 λ)e x [n]+e c [n] λ 2 E c [n 1/λ ] we get a density-scaled one-parameter hybrid (DS1H). Then, nonlinear perturbation theory starting for DS1H } E α = min Ψ ˆT + ˆV ne +λˆv Hx HF +αλŵ Ψ +ĒHxc[n λ Ψ ] Ψ ) where λŵ = λ(ŵee ˆV Hx HF is the perturbation.

59 A rigorous derivation of double hybrids First, single-determinant approximation: E DS1H = Φ ˆT + ˆV ne Φ +E H +λe HF x +(1 λ)e x [n]+e c [n] λ 2 E c [n 1/λ ] we get a density-scaled one-parameter hybrid (DS1H). Then, nonlinear perturbation theory starting for DS1H } E α = min Ψ ˆT + ˆV ne +λˆv Hx HF +αλŵ Ψ +ĒHxc[n λ Ψ ] Ψ ) where λŵ = λ(ŵee ˆV Hx HF is the perturbation. zeroth + first orders gives DS1H: E (0) +E (1) = E DS1H second order: E (2) = Φ λŵ Ψ (1) = λ 2 E MP2 c

60 In summary We have derived a density-scaled one-parameter double hybrid (DS1DH): E DS1DH xc = λex HF +(1 λ)e x [n]+e c [n] λ 2 E c [n 1/λ ]+λ 2 Ec MP2

61 In summary We have derived a density-scaled one-parameter double hybrid (DS1DH): E DS1DH xc = λex HF +(1 λ)e x [n]+e c [n] λ 2 E c [n 1/λ ]+λ 2 Ec MP2 If we neglect density scaling E c [n 1/λ ] E c [n], then we get an one-parameter double hybrid (1DH): E 1DH xc = λex HF +(1 λ)e x [n]+(1 λ 2 )E c [n]+λ 2 Ec MP2 This corresponds to Grimme s double hybrids with a x = λ and a c = λ 2.

62 Tests on atomization energies parameter λ optimized on AE6 benchmark set: mean absolute error (kcal/mol) DS1DH-PBE

63 Tests on atomization energies parameter λ optimized on AE6 benchmark set: mean absolute error (kcal/mol) 9 1DH-PBE DS1DH-PBE

64 Tests on atomization energies parameter λ optimized on AE6 benchmark set: mean absolute error (kcal/mol) 9 1DH-PBE DS1DH-BLYP 4 DS1DH-PBE

65 Tests on atomization energies parameter λ optimized on AE6 benchmark set: mean absolute error (kcal/mol) 9 1DH-PBE DS1DH-BLYP 4 DS1DH-PBE 3 2 1DH-BLYP 1 0 = neglecting density scaling may or may not be good

66 Tests on atomization energies parameter λ optimized on AE6 benchmark set: mean absolute error (kcal/mol) 9 1DH-PBE DS1DH-BLYP 4 DS1DH-PBE 3 2 1DH-BLYP B2-PLYP 1 0 = neglecting density scaling may or may not be good

67 Tests on atomization energies parameter λ optimized on AE6 benchmark set: mean absolute error (kcal/mol) DS1DH-PBE 1DH-PBE DS1DH-BLYP 1DH-BLYP B2-PLYP a x = λ = 0.55 a c = λ = neglecting density scaling may or may not be good = one parameter λ is enough a x = 0.53 a c = 0.27

68 Summary and perspectives Short-range DFT + long-range RPA it seems well suited for van der Waals systems we are studying more variants of RPAx we are working on an open-shell version Short-range DFT + long-range DMFT it is promising for bond dissociation we still need to improve the short-range functional Double-hybrid DFT we provided a rigorous derivation of double hybrids they can reach near-chemical accuracy for thermochemistry we could combine them with range separation For references: Web page: