Dispersion energies from the random-phase approximation with range separation

Transcription

1 1/34 Dispersion energies from the random-phase approximation with range separation Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Web page: with J. Ángyán, I. Gerber, G. Jansen, A. Savin, W. Zhu August 212

2 Range-separated DFT 1/34 Extension of Kohn-Sham scheme to multideterminant wavefunction { } 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 density functional Multideterminant wave function: Ψ lr = i c iφ i Parameter µ controls separation into lr/sr

3 Range-separated DFT 1/34 Extension of Kohn-Sham scheme to multideterminant wavefunction { } 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 density functional Multideterminant wave function: Ψ lr = i c iφ i Parameter µ controls separation into lr/sr In practice: approximations for Ψ lr and E sr xc[n]

4 Range-separated DFT 1/34 Extension of Kohn-Sham scheme to multideterminant wavefunction { } 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 density functional Multideterminant wave function: Ψ lr = i c iφ i Parameter µ controls separation into lr/sr In practice: approximations for Ψ lr and E sr xc[n] srlda srpbe etc...

5 Range-separated DFT Extension of Kohn-Sham scheme to multideterminant wavefunction { } 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 density functional Multideterminant wave function: Ψ lr = i c iφ i Parameter µ controls separation into lr/sr In practice: approximations for Ψ lr and Exc[n] sr lrhf } srlda lrmp2 srpbe lrrpa vdw etc... lrcc etc... 1/34

6 2/34 1 Short-range density functionals 2 Long-range MP2 3 Long-range RPA from ACFDT 4 Long-range RPA from ring CCD

7 3/34 1 Short-range density functionals 2 Long-range MP2 3 Long-range RPA from ACFDT 4 Long-range RPA from ring CCD

8 4/34 Short-range exchange energy: LDA E sr,µ x,lda [n] = n(r)ε sr,µ x,unif (n(r))dr

9 4/34 Short-range exchange energy: LDA E sr,µ x,lda [n] = n(r)ε sr,µ x,unif (n(r))dr For Be atom: E x sr,μ a.u LDA accurate for a short-range interaction exact LDA Μ a.u.

10 4/34 Short-range exchange energy: LDA E sr,µ x,lda [n] = n(r)ε sr,µ x,unif (n(r))dr For Be atom: E x sr,μ a.u LDA accurate for a short-range interaction exact LDA Μ a.u. Asymptotic expansion for µ : Ex sr,µ = A 1 µ 2 n(r) 2 dr+ A ( 2 n(r) 2 µ 4 n(r) 2n(r) ) +4τ(r) dr+

11 5/34 Short-range correlation energy: LDA E sr,µ c,lda [n] = n(r) [ ε c,unif (n(r)) ε lr,µ c,unif (n(r)) ]dr

12 5/34 Short-range correlation energy: LDA E sr,µ c,lda [n] = n(r) [ ε c,unif (n(r)) ε lr,µ c,unif (n(r)) ]dr For Be atom: E c sr,μ a.u LDA accurate for a short-range interaction exact LDA Μ a.u.

13 5/34 Short-range correlation energy: LDA E sr,µ c,lda [n] = n(r) [ ε c,unif (n(r)) ε lr,µ c,unif (n(r)) ]dr For Be atom: E c sr,μ a.u LDA accurate for a short-range interaction exact LDA Μ a.u. Asymptotic expansion for µ : Ec sr,µ = B 1 µ 2 n 2,c (r,r)dr+ B 2 µ 3 n 2 (r,r)dr+

14 Short-range exchange energy: GGA 6/34 Short-range GGA functional of Heyd, Scuseria and Ernzerhof (23) based on the PBE exchange hole: ε sr,µ x,gga (n) = 1 n x,pbe (n, n,r 12 )wee sr,µ (r 12 )dr 12 2 For Be atom:.5 E x sr,μ a.u exact LDA GGA Μ a.u. = GGA describes well a longer range of interaction

15 Short-range correlation energy: GGA 7/34 Interpolation between PBE at µ = and expansion of LDA for µ : ε sr,µ c,gga (n, n ) = ε c,pbe (n, n ) 1+d 1 (n)µ+d 2 (n)µ 2 For Be atom:.5 E c sr,μ a.u exact LDA GGA Μ a.u. = GGA describes well a longer range of interaction

16 8/34 1 Short-range density functionals 2 Long-range MP2 3 Long-range RPA from ACFDT 4 Long-range RPA from ring CCD

17 Single-determinant reference 9/34 Restriction to single-determinant wave functions Φ = range-separated hybrid (RSH): { } E RSH = min Φ ˆT + ˆV ne +Ŵee Φ +E lr Hxc[n sr Φ ] Φ

18 Single-determinant reference 9/34 Restriction to single-determinant wave functions Φ = range-separated hybrid (RSH): { } E RSH = min Φ ˆT + ˆV ne +Ŵee Φ +E lr Hxc[n sr Φ ] Φ The minimizing determinant Φ is given by ) (ˆT + ˆV ne + ˆV Hx,HF lr + ˆV Hxc sr Φ = E Φ

19 Single-determinant reference 9/34 Restriction to single-determinant wave functions Φ = range-separated hybrid (RSH): { } E RSH = min Φ ˆT + ˆV ne +Ŵee Φ +E lr Hxc[n sr Φ ] Φ The minimizing determinant Φ is given by ) (ˆT + ˆV ne + ˆV Hx,HF lr + ˆV Hxc sr Φ = E Φ So the RSH energy is E RSH = Φ ˆT + ˆV ne Φ +E H +E lr x,hf +E sr xc This is a hybrid DFT with HF exchange at long range (similar to the LC scheme)

20 RSH reference contains no dispersion Example : Ar 2 with aug-cc-pv5z basis, µ =.5 bohr 1 :.2 Interaction energy (mhartree) Accurate PBE RSH (srpbe) Internuclear distance (bohr) 1/34

21 Long-range perturbation theory 11/34 All what is missing is the long-range correlation energy E exact = E RSH +E lr c

22 Long-range perturbation theory 11/34 All what is missing is the long-range correlation energy E exact = E RSH +E lr c Adiabatic connection from RSH reference to exact energy: { } E λ = min Ψ ˆT + ˆV ne + ˆV Hx,HF lr +λŵ lr Ψ +EHxc[n sr Ψ ] Ψ with the long-range perturbation operator Ŵ lr = Ŵ lr ee ˆV lr Hx,HF

23 Long-range perturbation theory 11/34 All what is missing is the long-range correlation energy E exact = E RSH +E lr c Adiabatic connection from RSH reference to exact energy: { } E λ = min Ψ ˆT + ˆV ne + ˆV Hx,HF lr +λŵ lr Ψ +EHxc[n sr Ψ ] Ψ with the long-range perturbation operator Ŵ lr = Ŵ lr ee ˆV lr Hx,HF Note: the density is NOT constant on this adiabatic connection

24 Long-range perturbation theory 11/34 All what is missing is the long-range correlation energy E exact = E RSH +E lr c Adiabatic connection from RSH reference to exact energy: { } E λ = min Ψ ˆT + ˆV ne + ˆV Hx,HF lr +λŵ lr Ψ +EHxc[n sr Ψ ] Ψ with the long-range perturbation operator Ŵ lr = Ŵ lr ee ˆV lr Hx,HF Note: the density is NOT constant on this adiabatic connection Zeroth + first orders: E () +E (1) = E RSH Second order: E (2) = Φ Ŵ lr Ψ lr,(1) = E lr c,mp2

25 Convergence of binding energy with basis size: Ar 2 aug-cc-pvnz basis, CP corrected, µ =.5 bohr 1, srpbe functional: 1 % of CBS binding energy MP2 lrmp2 avdz avtz avqz av5z CBS One-particle basis = lrmp2 has a fast convergence with basis size 12/34

26 Interaction energy curve of Ne 2 13/34 aug-cc-pv5z basis, CP corrected, µ =.5 bohr 1, srpbe functional:.5 Interaction energy (mhartree) Accurate MP2 lrmp Internuclear distance (bohr)

27 Interaction energy curve of Ar 2 14/34 aug-cc-pv5z basis, CP corrected, µ =.5 bohr 1, srpbe functional:.1 Interaction energy (mhartree) Accurate MP2 lrmp Internuclear distance (bohr)

28 15/34 Equilibrium interaction energies of S22 set 22 weakly-interacting molecular systems from water dimer to DNA base pairs (CP corrected, µ =.5 bohr 1, srpbe functional): % of error on interaction energy H bonded dispersion dispersion +multipoles MP2 avtz system in S22 set

29 15/34 Equilibrium interaction energies of S22 set 22 weakly-interacting molecular systems from water dimer to DNA base pairs (CP corrected, µ =.5 bohr 1, srpbe functional): % of error on interaction energy H bonded dispersion dispersion +multipoles MP2 avtz MP2 avqz system in S22 set

30 Equilibrium interaction energies of S22 set 22 weakly-interacting molecular systems from water dimer to DNA base pairs (CP corrected, µ =.5 bohr 1, srpbe functional): % of error on interaction energy H bonded dispersion dispersion +multipoles MP2 avtz MP2 avqz lrmp2 avdz system in S22 set = lrmp2 is very similar to standard MP2 15/34

31 Equilibrium interaction energies of S22 set 22 weakly-interacting molecular systems from water dimer to DNA base pairs (CP corrected, µ =.5 bohr 1, srpbe functional): % of error on interaction energy H bonded dispersion dispersion +multipoles MP2 avtz MP2 avqz lrmp2 avdz lrmp2 avtz system in S22 set = lrmp2 is very similar to standard MP2 15/34

32 16/34 1 Short-range density functionals 2 Long-range MP2 3 Long-range RPA from ACFDT 4 Long-range RPA from ring CCD

33 Long-range RPA from ACFDT 17/34 Adiabatic connection from RSH reference to exact energy: 1 { } Ec lr = dλ Ψ lr λ Ŵlr Ψλ lr Φ Ŵ lr Φ

34 Long-range RPA from ACFDT 17/34 Adiabatic connection from RSH reference to exact energy: 1 { } Ec lr = dλ Ψ lr λ Ŵlr Ψλ lr Φ Ŵ lr Φ = 1 dλ 1 2 ia,jb ab ij lr( Pc,λ lr ) ia,jb

35 Long-range RPA from ACFDT 17/34 Adiabatic connection from RSH reference to exact energy: 1 { } Ec lr = dλ Ψ lr λ Ŵlr Ψλ lr Φ Ŵ lr Φ = 1 dλ 1 2 ia,jb ab ij lr( Pc,λ lr ) ia,jb Pc,λ lr from fluctuation-dissipation theorem P lr c,λ = dω 2πi [ χ lr λ (ω) χ (ω) ] e iω+

36 Long-range RPA from ACFDT 17/34 Adiabatic connection from RSH reference to exact energy: 1 { } Ec lr = dλ Ψ lr λ Ŵlr Ψλ lr Φ Ŵ lr Φ = 1 dλ 1 2 ia,jb ab ij lr( Pc,λ lr ) ia,jb Pc,λ lr from fluctuation-dissipation theorem P lr c,λ = dω 2πi [ χ lr λ (ω) χ (ω) ] e iω+ Long-range response function χλ lr (ω) is given by χ lr λ (ω) 1 = χ (ω) 1 f lr Hxc,λ (ω)

37 Long-range RPA from ACFDT Adiabatic connection from RSH reference to exact energy: 1 { } Ec lr = dλ Ψ lr λ Ŵlr Ψλ lr Φ Ŵ lr Φ = 1 dλ 1 2 ia,jb ab ij lr( Pc,λ lr ) ia,jb Pc,λ lr from fluctuation-dissipation theorem P lr c,λ = dω 2πi [ χ lr λ (ω) χ (ω) ] e iω+ Long-range response function χλ lr (ω) is given by χ lr λ (ω) 1 = χ (ω) 1 f lr Hxc,λ (ω) Two variants: direct RPA (drpa) : f lr Hxc,λ (ω) λflr H RPA with HF exchange (RPAx) : f lr Hxc,λ (ω) λflr H +λflr x,hf 17/34

38 Convergence of binding energy with basis size: Ar 2 aug-cc-pvnz basis, CP corrected, µ =.5 bohr 1, srpbe functional: 1 % of CBS binding energy drpa (PBE) RPAx (HF) lrdrpa lrrpax avdz avtz avqz av5z CBS One-particle basis = lrrpa has a fast convergence with basis size 18/34

39 Interaction energy curve of Ne 2 19/34 aug-cc-pv5z basis, CP corrected, µ =.5 bohr 1, srpbe functional:.5 Interaction energy (mhartree) Accurate drpa (PBE) RPAx (HF) lrdrpa lrrpax Internuclear distance (bohr)

40 Interaction energy curve of Ar 2 2/34 aug-cc-pv5z basis, CP corrected, µ =.5 bohr 1, srpbe functional:.1 Interaction energy (mhartree) Accurate drpa (PBE) RPAx (HF) lrdrpa lrrpax Internuclear distance (bohr)

41 21/34 Equilibrium interaction energies of S22 set 22 weakly-interacting molecular systems from water dimer to DNA base pairs (CP corrected, µ =.5 bohr 1, srpbe functional): % of error on interaction energy H bonded dispersion dispersion +multipoles drpa (PBE) avtz system in S22 set

42 21/34 Equilibrium interaction energies of S22 set 22 weakly-interacting molecular systems from water dimer to DNA base pairs (CP corrected, µ =.5 bohr 1, srpbe functional): % of error on interaction energy H bonded dispersion dispersion +multipoles drpa (PBE) avtz drpa (PBE) avqz system in S22 set

43 21/34 Equilibrium interaction energies of S22 set 22 weakly-interacting molecular systems from water dimer to DNA base pairs (CP corrected, µ =.5 bohr 1, srpbe functional): % of error on interaction energy H bonded dispersion dispersion +multipoles drpa (PBE) avtz drpa (PBE) avqz lrdrpa avdz system in S22 set

44 Equilibrium interaction energies of S22 set 22 weakly-interacting molecular systems from water dimer to DNA base pairs (CP corrected, µ =.5 bohr 1, srpbe functional): % of error on interaction energy H bonded dispersion dispersion +multipoles drpa (PBE) avtz drpa (PBE) avqz lrdrpa avdz lrrpax avdz system in S22 set = lrrpax gives the smallest mean absolute error ( 8%) 21/34

45 22/34 1 Short-range density functionals 2 Long-range MP2 3 Long-range RPA from ACFDT 4 Long-range RPA from ring CCD

46 23/34 Long-range RPA from ring coupled cluster doubles (rccd) drpa can be formulated as a direct rccd (without exchange): E lr c,drpa = = dλ 1 2 ia,jb ia,jb ab ij lr( Pc,λ lr ) ia,jb ab ij lr( TdrCCD lr ) ia,jb Scuseria et al. 8

47 23/34 Long-range RPA from ring coupled cluster doubles (rccd) drpa can be formulated as a direct rccd (without exchange): E lr c,drpa = = dλ 1 2 ia,jb ia,jb ab ij lr( Pc,λ lr ) ia,jb ab ij lr( TdrCCD lr ) ia,jb Scuseria et al. 8 = + +

48 23/34 Long-range RPA from ring coupled cluster doubles (rccd) drpa can be formulated as a direct rccd (without exchange): E lr c,drpa = = dλ 1 2 ia,jb ia,jb ab ij lr( Pc,λ lr ) ia,jb ab ij lr( TdrCCD lr ) ia,jb Scuseria et al. 8 = + + drpa + second-order screened exchange (SOSEX) Grüneis et al. 9 Paier et al. 1 Ec,dRPA+SOSEX lr = 1 ab ij lr( T lr ) drccd 2 ia,jb ia,jb

49 23/34 Long-range RPA from ring coupled cluster doubles (rccd) drpa can be formulated as a direct rccd (without exchange): E lr c,drpa = = dλ 1 2 ia,jb ia,jb ab ij lr( Pc,λ lr ) ia,jb ab ij lr( TdrCCD lr ) ia,jb Scuseria et al. 8 = + + drpa + second-order screened exchange (SOSEX) Grüneis et al. 9 Paier et al. 1 Ec,dRPA+SOSEX lr = 1 ab ij lr( T lr ) drccd 2 ia,jb ia,jb =

50 Interaction energy curve of Ar 2 24/34 aug-cc-pv5z basis, CP corrected, µ =.5 bohr 1, srpbe functional:.1 Interaction energy (mhartree) Accurate lrdrpa lrdrpa+sosex Internuclear distance (bohr)

51 Equilibrium interaction energies of S22 set 22 weakly-interacting molecular systems from water dimer to DNA base pairs (avdz, CP corrected, µ =.5 bohr 1, srpbe functional): % of error on interaction energy H bonded dispersion dispersion +multipoles lrdrpa lrdrpa+sosex lrrpax system in S22 set = SOSEX does not improve on drpa for weak interactions 25/34

52 Long-range RPA from ring CCD with exchange (rccdx) 26/34 With antisymmetrized integrals: RPAx variant of McLachlan-Ball 1964 (equivalent to TDHF plasmon formula) E lr c,rpax-ii = 1 4 ia,jb ab ij lr( TrCCDx lr ) ia,jb

53 Long-range RPA from ring CCD with exchange (rccdx) 26/34 With antisymmetrized integrals: RPAx variant of McLachlan-Ball 1964 (equivalent to TDHF plasmon formula) E lr c,rpax-ii = 1 4 ia,jb ab ij lr( TrCCDx lr ) ia,jb With non-antisymmetrized integrals, RPAx variant of Szabo-Ostlund 1977 Ec,RPAx-SO2 lr = 1 ab ij lr( T lr ) rccdx 2 ia,jb ia,jb

54 Long-range RPA from ring CCD with exchange (rccdx) 26/34 With antisymmetrized integrals: RPAx variant of McLachlan-Ball 1964 (equivalent to TDHF plasmon formula) E lr c,rpax-ii = 1 4 ia,jb ab ij lr( TrCCDx lr ) ia,jb With non-antisymmetrized integrals, RPAx variant of Szabo-Ostlund 1977 Ec,RPAx-SO2 lr = 1 ab ij lr( T lr ) rccdx 2 ia,jb ia,jb =

55 Interaction energy curve of Ne 2 27/34 aug-cc-pv5z basis, CP corrected, µ =.5 bohr 1, srpbe functional:.5 Interaction energy (mhartree) Accurate RPAx-II RPAx-SO2 lrrpax-ii lrrpax-so Internuclear distance (bohr)

56 Interaction energy curve of Ar 2 28/34 aug-cc-pv5z basis, CP corrected, µ =.5 bohr 1, srpbe functional:.1 Interaction energy (mhartree) Accurate RPAx-II RPAx-SO2 lrrpax-ii lrrpax-so Internuclear distance (bohr)

57 Equilibrium interaction energies of S22 set 22 weakly-interacting molecular systems from water dimer to DNA base pairs (avdz, CP corrected, µ =.5 bohr 1, srpbe functional): % of error on interaction energy H bonded dispersion dispersion +multipoles lrrpax lrrpax II lrrpax SO system in S22 set = lrrpax-so2 gives a mean absolute error of 4% 29/34

58 Equilibrium interaction energies of S22 set 22 weakly-interacting molecular systems from water dimer to DNA base pairs (avdz, CP corrected, µ =.5 bohr 1, srpbe functional): % of error on interaction energy H bonded dispersion dispersion +multipoles lrrpax lrrpax II lrrpax SO2 lrccd system in S22 set = lrrpax-so2 gives a mean absolute error of 4% 29/34

59 Spin-adapted RPA for closed-shell systems 3/34 The variants based on drccd only need spin-singlet excitations: drpa with 1 K ia,jb = 2 ab ij E c,drpa = 1 2 tr[1 K 1 T drccd ]

60 Spin-adapted RPA for closed-shell systems 3/34 The variants based on drccd only need spin-singlet excitations: drpa with 1 K ia,jb = 2 ab ij E c,drpa = 1 2 tr[1 K 1 T drccd ] drpa+sosex E c,drpa+sosex = 1 2 tr[1 B 1 T drccd ] with 1 B ia,jb = 2 ab ij ab ji

61 Spin-adapted RPA for closed-shell systems 31/34 The variants based on rccdx may also need spin-triplet excitations: RPAx-II E c,rpax-ii = 1 4 tr[1 B 1 T rccdx + 3 B 3 ] T rccdx with 3 B ia,jb = ab ji

62 Spin-adapted RPA for closed-shell systems 31/34 The variants based on rccdx may also need spin-triplet excitations: RPAx-II E c,rpax-ii = 1 4 tr[1 B 1 T rccdx + 3 B 3 ] T rccdx with 3 B ia,jb = ab ji RPAx-SO2 E c,rpax-so2 = 1 2 tr[1 K 1 ] T rccdx

63 Spin-adapted RPA for closed-shell systems 31/34 The variants based on rccdx may also need spin-triplet excitations: RPAx-II E c,rpax-ii = 1 4 tr[1 B 1 T rccdx + 3 B 3 ] T rccdx with 3 B ia,jb = ab ji RPAx-SO2 E c,rpax-so2 = 1 2 tr[1 K 1 ] T rccdx RPAx-SO1 E c,rpax-so1 = 1 2 tr[1 B (1 T rccdx 3 T rccdx )]

64 Equilibrium interaction energies of S22 set 22 weakly-interacting molecular systems from water dimer to DNA base pairs (avdz, CP corrected, µ =.5 bohr 1, srpbe functional): % of error on interaction energy H bonded dispersion dispersion +multipoles lrrpax lrrpax II lrrpax SO2 lrrpax SO system in S22 set = lrrpax-so1 and lrrpax-so2 give almost identical results 32/34

65 Computational cost 33/34 drpa straightforward algorithm: N 6 algorithm with RI: N 4 drpa+sosex algorithm with RI: N 5 RPAx variants straightforward algorithm: N 6 (N 3 on 3 v) CCD N 6 (N 2 on 4 v)

66 Summary long-range RPA + short-range DFT seems well suited for van der Waals interactions has a fast basis convergence several variants with exchange: RPAx-SO2 seems the best warning: RPAx can have instabilities can RPAx be made fast? Ángyán, Gerber, Savin, Toulouse, PRA 72, 1251 (25) Toulouse, Gerber, Jansen, Savin, Ángyán, PRL 12, 9644 (29) Toulouse, Zhu, Ángyán, Savin, PRA 82, 3252 (21) Zhu, Toulouse, Savin, Ángyán, JCP 132, (21) Toulouse, Zhu, Savin, Jansen, Ángyán, JCP 135, (211) Ángyán, Liu, Toulouse, Jansen, JCTC 7, 3116 (211) Web page: 34/34

67 35/34 Short-range exchange energy: Gradient expansion (GEA) ε sr,µ x,gea (n) = Esr,µ x,lda [n]+ drn(r)c µ x (n) n 2 For Be atom:.5 E x sr,μ a.u exact LDA GEA Μ a.u. = GEA describes well a longer range of interaction

68 36/34 Short-range correlation energy: Gradient expansion (GEA) E sr,µ c,gea [n] = Esr,µ c,lda [n]+ with C µ c (n) C µ x (n). drn(r)c µ c (n) n 2 For Be atom:.5 E c sr,μ a.u exact LDA GEA Μ a.u. = GEA describes well a longer range of interaction

69 37/34 Short-range PBE exchange functional Ex sr,µ [n] = drnε sr,µ x,unif (n)f x(s, µ) with the scaled gradient s = n /(2k F n), the scaled parameter µ = µ/(2k F ) and the enhancement factor F x (s, µ) = 1+κ( µ) κ( µ) 1+b( µ)s 2 /κ( µ) In the slowly varying limit s F x (s, µ) 1+b( µ)s 2 b( µ) is the second-order coefficient of the gradient expansion In the rapidly varying limit s F x (s, µ) 1+κ( µ) κ( µ) is chosen as the largest value satisfying the Lieb-Oxford bound E sr,µ x E x C n(r) 4/3 dr

70 Short-range PBE correlation functional Ec sr,µ [n] = [ ] drn ε sr,µ c,unif (r s)+h(r s,t,µ) with the scaled gradient t = n /(2k s n). For H(r s,t,µ), same ansatz than for the original PBE satisfying: the slowly varying limit t H(r s,t,µ) β(r s,µ)t 2 β(r s,µ) is chosen so as to cancel the exchange gradient expansion. the rapidly varying limit t H(r s,t,µ) ε sr,µ c,unif (r s) the high-density limit under uniform scaling, i.e. n(r) λ 3 n(λr) and λ H(λ 1 r s,λ 1/2 t,µ) γlnλ 38/34

71 39/34 Long-range correlation energy E lr c Exact energy = RSH energy + long-range correlation energy E = E RSH +E lr c

72 39/34 Long-range correlation energy E lr c Exact energy = RSH energy + long-range correlation energy E = E RSH +E lr c Adiabatic connection from RSH reference to exact system: E lr c = 1 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

73 39/34 Long-range correlation energy E lr c Exact energy = RSH energy + long-range correlation energy E = E RSH +E lr c Adiabatic connection from RSH reference to exact system: E lr c = 1 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 With a compact notation, E lr c = dλ Tr [ w lr Pc,λ lr ]

74 4/34 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 one-particle density matrix.

75 4/34 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 one-particle density matrix. The long-range response function χ lr λ (iω) is given by χ lr λ (iω) 1 = χ lr IP,λ (iω) 1 f lr Hxc,λ (iω)

76 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 one-particle density matrix. The long-range response function χ lr λ (iω) is given by χ lr λ (iω) 1 = χ lr IP,λ (iω) 1 f lr Hxc,λ (iω) Several possible approximations: MP2 approximation (2 nd order in w lr RPA approximation: f lr xc,λ ee) = RSH+MP2 method = = RSH+RPA method RPAx approximation: f lr c,λ = = RSH+RPAx method 4/34

77 41/34 Details: adiabatic connection formula for E lr c Adiabatic connection starting from the RSH reference: { E λ = min Ψ ˆT + ˆV ne + ˆV Hx,HF lr +λŵ lr Ψ +E sr Ψ Hxc[n Ψ ] with the long-range perturbation operator Ŵ lr = Ŵ lr ee ˆV lr Hx,HF }

78 41/34 Details: adiabatic connection formula for E lr c Adiabatic connection starting from the RSH reference: { E λ = min Ψ ˆT + ˆV ne + ˆV Hx,HF lr +λŵ lr Ψ +E sr Ψ Hxc[n Ψ ] with the long-range perturbation operator Ŵ lr = Ŵ lr ee ˆV lr Hx,HF Taking the derivative wrt λ and reintegrating E = E λ= + 1 dλ Ψ lr λ Ŵlr Ψ lr λ }

79 41/34 Details: adiabatic connection formula for E lr c Adiabatic connection starting from the RSH reference: { E λ = min Ψ ˆT + ˆV ne + ˆV Hx,HF lr +λŵ lr Ψ +E sr Ψ Hxc[n Ψ ] with the long-range perturbation operator Ŵ lr = Ŵ lr ee ˆV lr Hx,HF Taking the derivative wrt λ and reintegrating E = E λ= + 1 dλ Ψ lr λ Ŵlr Ψ lr λ where E λ= = Φ RSH ˆT + ˆV ne + ˆV lr Hx,HF Φ RSH +E sr Hxc [n Φ RSH ] = E RSH Φ RSH Ŵ lr Φ RSH }

80 41/34 Details: adiabatic connection formula for E lr c Adiabatic connection starting from the RSH reference: { E λ = min Ψ ˆT + ˆV ne + ˆV Hx,HF lr +λŵ lr Ψ +E sr Ψ Hxc[n Ψ ] with the long-range perturbation operator Ŵ lr = Ŵ lr ee ˆV lr Hx,HF Taking the derivative wrt λ and reintegrating E = E λ= + 1 dλ Ψ lr λ Ŵlr Ψ lr λ where E λ= = Φ RSH ˆT + ˆV ne + ˆV lr Hx,HF Φ RSH +E sr Hxc [n Φ RSH ] = E RSH Φ RSH Ŵ lr Φ RSH so the long-range correlation energy is E lr c = 1 dλ { } Ψ lr λ Ŵlr Ψλ lr Φ RSH Ŵ lr Φ RSH }

81 42/34 Details: Green function and Dyson equation The term λ lr coming from the variation of the one-particle density has the following form lr λ = F[Glr λ ] F[G RSH] where Gλ lr is the long-range two-point Green function and G RSH is the RSH reference Green function.

82 42/34 Details: Green function and Dyson equation The term λ lr coming from the variation of the one-particle density has the following form lr λ = F[Glr λ ] F[G RSH] where Gλ lr is the long-range two-point Green function and G RSH is the RSH reference Green function. The long-range Green function is given by the following self-consistent Dyson equation ( ) G lr 1 λ = G 1 RSH λ( Σ lr Hx[Gλ lr ] Σlr Hx[G RSH ] ) Σ lr c,λ [Glr λ ] where Σ lr are long-range self-energies.

83 43/34 Details: four-point long-range response function Time-ordered four-point response function, using 1 = (x 1 t 1 ) etc..., [ ] iχ lr λ (1,2;1,2 ) = Ψ lr λ ˆT δĝ(1,1 ) δĝ(2,2 ) Ψλ lr

84 43/34 Details: four-point long-range response function Time-ordered four-point response function, using 1 = (x 1 t 1 ) etc..., [ ] iχ lr λ (1,2;1,2 ) = Ψ lr λ ˆT δĝ(1,1 ) δĝ(2,2 ) Ψλ lr It is given by a Bethe-Salpeter-type equation (χ lr λ ) 1 = (χ lr IP,λ ) 1 λf lr Hx f lr c,λ with the independent-particle response function and the four-point kernels χ lr IP,λ = iglr λ Glr λ λf lr Hx = iλ δσlr Hx δg lr λ f lr c,λ δgλ lr c,λ = iδσlr

85 44/34 Details: RPA and RPAx approximations The RPA approximation: Σ lr xc,λ = implies Gλ lr = G RSH lr λ = χ lr IP,λ = ig RSHG RSH = χ RSH fxc,λ lr =

86 Details: RPA and RPAx approximations The RPA approximation: Σ lr xc,λ = implies Gλ lr = G RSH lr λ = χ lr IP,λ = ig RSHG RSH = χ RSH fxc,λ lr = Similarly, the RPAx approximation: Σ lr c,λ = implies Gλ lr = G RSH lr λ = χ lr IP,λ = ig RSHG RSH = χ RSH fc,λ lr = 44/34

87 45/34 Details: implementation of RPAx correlation energy In a basis, the RPAx long-range correlation energy is E lr c,rpax = dλ iajb ij ŵee ab lr ( Pc,RPAx,λ lr ) iajb

88 45/34 Details: implementation of RPAx correlation energy In a basis, the RPAx long-range correlation energy is E lr c,rpax = dλ iajb ij ŵee ab lr ( Pc,RPAx,λ lr ) iajb The long-range RPAx second-order density matrix is given by P lr c,rpax,λ = (A λ B λ ) 1/2 M λ 1/2 (A λ B λ ) 1/2 1 with M λ = (A λ B λ ) 1/2 (A λ +B λ )(A λ B λ ) 1/2 and the singlet orbital rotation Hessians and (A λ ) iajb = (ǫ a ǫ i )δ ij δ ab +2λ aj ŵ lr ee ib λ aj ŵ lr ee bi (B λ ) iajb = 2λ ab ŵ lr ee ij λ ab ŵ lr ee ji

89 46/34 Details: ring CCD amplitude equations direct ring CCD 1 K+ 1 L 1 T drccd + 1 T drccd 1 L+ 1 T drccd 1 K 1 T drccd = with 1 K ia,jb = 2 ab ij and 1 L ia,jb = (ǫ a ǫ i )δ ij δ ab +2 ab ij ring CCD with exchange 1 B+ 1 A 1 T rccdx + 1 T rccdx 1 A+ 1 T rccdx 1 B 1 T rccdx = 3 B+ 3 A 3 T rccdx + 3 T rccdx 3 A+ 3 T rccdx 3 B 3 T rccdx = with 1 B ia,jb = 2 ab ij ab ji and 1 A ia,jb = (ǫ a ǫ i )δ ij δ ab +2 ib aj ib ja and 3 B ia,jb = ab ji and 3 A ia,jb = (ǫ a ǫ i )δ ij δ ab ib ja