Dispersion energies from the random-phase approximation with range separation
|
|
- Samantha Nelson
- 5 years ago
- Views:
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
Théorie de la fonctionnnelle de la densité avec séparation de portée pour les forces de van der Waals
Théorie de la fonctionnnelle de la densité avec séparation de portée pour les forces de van der Waals Julien Toulouse 1 Iann Gerber 2, Georg Jansen 3, Andreas Savin 1, János Ángyán 4 1 Laboratoire de Chimie
More informationExtending Kohn-Sham density-functional theory
Extending Kohn-Sham density-functional theory Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Email: julien.toulouse@upmc.fr Web page: www.lct.jussieu.fr/pagesperso/toulouse/ January
More informationRange-separated density-functional theory with long-range random phase approximation
Range-separated density-functional theory with long-range random phase approximation Julien Toulouse 1 Wuming Zhu 1, Andreas Savin 1, János Ángyán2 1 Laboratoire de Chimie Théorique, UPMC Univ Paris 6
More informationAdiabatic-connection fluctuation-dissipation density-functional theory based on range separation
Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation Julien Toulouse 1 I. Gerber 2, G. Jansen 3, A. Savin 1, W. Zhu 1, J. Ángyán 4 1 Laboratoire de Chimie Théorique,
More informationDFT basée sur le théorème de fluctuation-dissipation avec séparation de portée pour les interactions de van der Waals
DFT basée sur le théorème de fluctuation-dissipation avec séparation de portée pour les interactions de van der Waals Julien Toulouse 1 Iann Gerber 2, Georg Jansen 3, Andreas Savin 1, János Ángyán 4 1
More informationCombining density-functional theory and many-body methods
Combining density-functional theory and many-body methods Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Vrije Universiteit Amsterdam, Netherlands November 2017 Outline 2/23 1
More informationLong-range/short-range energy decomposition in density functional theory
p. 1/2 Long-range/short-range energy decomposition in density functional theory Julien Toulouse François Colonna, Andreas Savin Laboratoire de Chimie Théorique, Université Pierre et Marie Curie, Paris
More informationCombining density-functional theory and many-body methods
Combining density-functional theory and many-body methods Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Albuquerque New Mexico, USA June 2016 Outline 2/23 1 A brief overview of
More informationMultideterminant density-functional theory for static correlation. Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France
Multideterminant density-functional theory for static correlation Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Basel, Switzerland September 2015 Kohn-Sham DFT and static correlation
More informationRPA step by step. Bastien Mussard, János G. Ángyán, Sébastien Lebègue CRM 2, FacultÃl des Sciences UniversitÃl Henry PoincarÃl, Nancy
RPA step by step Bastien Mussard, János G. Ángyán, Sébastien Lebègue CRM, FacultÃl des Sciences UniversitÃl Henry PoincarÃl, Nancy Eötvös University Faculty of Science Budapest, Hungaria 0 mars 01 bastien.mussard@uhp-nancy.fr
More informationDevelopments in Electronic Structure Theory.
Developments in Electronic Structure Theory. Bastien Mussard Laboratoire de Chimie Théorique Institut du Calcul et de la Simulation Sorbonne Universités, Université Pierre et Marie Curie bastien.mussard@upmc.fr
More informationFractional-charge and Fractional-spin errors in Range-Separated Density-Functional Theory with Random Phase Approximation.
Fractional-charge and Fractional-spin errors in Range-Separated Density-Functional Theory with Random Phase Approximation. Bastien Mussard, Julien Toulouse Laboratoire de Chimie Théorique Institut du Calcul
More informationCombining many-body methods and density-functional theory. Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France
Combining many-body methods and density-funtional theory Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, Frane September 2014 2/1 Hybrids ombining many-body methods and DFT Goal: improve
More informationAssessment of range-separated time-dependent density-functional theory for calculating C 6 dispersion coefficients
1/10 Assessment of range-separated time-dependent density-functional theory for calculating C 6 dispersion coefficients Julien Toulouse 1,2, Elisa Rebolini 1, Tim Gould 3, John F. Dobson 3, Prasenjit Seal
More informationDevelopments in Electronic Structure Theory.
Developments in Electronic Structure Theory. Bastien Mussard Laboratoire de Chimie Théorique Institut du Calcul et de la Simulation Sorbonne Universités, Université Pierre et Marie Curie bastien.mussard@upmc.fr
More informationAnalytical Gradients for the Range-Separated Random Phase Approximation Correlation Energies Using a Lagrangian Framework
Analytical Gradients for the Range-Separated Random Phase Approximation Correlation Energies Using a Lagrangian Framework Bastien Mussard a, János G. Ángyán a, Péter G. Szalay b a CRM 2, Université de
More informationA new generation of density-functional methods based on the adiabatic-connection fluctuation-dissipation theorem
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
More informationDensity functional theory in the solid state
Density functional theory in the solid state Ari P Seitsonen IMPMC, CNRS & Universités 6 et 7 Paris, IPGP Department of Applied Physics, Helsinki University of Technology Physikalisch-Chemisches Institut
More informationPseudo-Hermitian eigenvalue equations in linear-response electronic-structure theory
1/11 Pseudo-Hermitian eigenvalue equations in linear-response electronic-structure theory Julien Toulouse Université Pierre & Marie Curie and CNRS, 4 place Jussieu, Paris, France Web page: www.lct.jussieu.fr/pagesperso/toulouse/
More informationElectronic structure theory: Fundamentals to frontiers. 2. Density functional theory
Electronic structure theory: Fundamentals to frontiers. 2. Density functional theory MARTIN HEAD-GORDON, Department of Chemistry, University of California, and Chemical Sciences Division, Lawrence Berkeley
More informationCLIMBING THE LADDER OF DENSITY FUNCTIONAL APPROXIMATIONS JOHN P. PERDEW DEPARTMENT OF PHYSICS TEMPLE UNIVERSITY PHILADELPHIA, PA 19122
CLIMBING THE LADDER OF DENSITY FUNCTIONAL APPROXIMATIONS JOHN P. PERDEW DEPARTMENT OF PHYSICS TEMPLE UNIVERSITY PHILADELPHIA, PA 191 THANKS TO MANY COLLABORATORS, INCLUDING SY VOSKO DAVID LANGRETH ALEX
More informationIntroduction to density-functional theory. Emmanuel Fromager
Institut de Chimie, Strasbourg, France Page 1 Emmanuel Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS M2 lecture, Strasbourg, France. Institut
More informationRole of van der Waals Interactions in Physics, Chemistry, and Biology
Role of van der Waals Interactions in Physics, Chemistry, and Biology How can we describe vdw forces in materials accurately? Failure of DFT Approximations for (Long-Range) Van der Waals Interactions 1
More information1 Density functional theory (DFT)
1 Density functional theory (DFT) 1.1 Introduction Density functional theory is an alternative to ab initio methods for solving the nonrelativistic, time-independent Schrödinger equation H Φ = E Φ. The
More informationTime-dependent linear response theory: exact and approximate formulations. Emmanuel Fromager
June 215 ISTPC 215 summer school, Aussois, France Page 1 Time-dependent linear response theory: exact and approximate formulations Emmanuel Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie
More informationDensity-functional theory at noninteger electron numbers
Density-functional theory at noninteger electron numbers Tim Gould 1, Julien Toulouse 2 1 Griffith University, Brisbane, Australia 2 Université Pierre & Marie Curie and CRS, Paris, France July 215 Introduction
More informationIntermediate DFT. Kieron Burke and Lucas Wagner. Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA
Intermediate DFT Kieron Burke and Lucas Wagner Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA October 10-19th, 2012 Kieron (UC Irvine) Intermediate DFT Lausanne12
More informationXYZ of ground-state DFT
XYZ of ground-state DFT Kieron Burke and Lucas Wagner Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA January 5-9th, 2014 Kieron (UC Irvine) XYZ of ground-state
More informationImporting ab-initio theory into DFT: Some applications of the Lieb variation principle
Importing ab-initio theory into DFT: Some applications of the Lieb variation principle Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department
More informationModel Hamiltonians in Density Functional Theory
Centre de Recherches Mathématiques CRM Proceedings and Lecture Notes Volume 41, 2006 Model Hamiltonians in Density Functional Theory Paola Gori-Giorgi, Julien Toulouse, and Andreas Savin Abstract. The
More informationvan der Waals forces in density functional theory: Perturbational long-range electron-interaction corrections
van der Waals forces in density functional theory: Perturbational long-range electron-interaction corrections János G. Ángyán and Iann C. Gerber Laboratoire de Cristallographie et de Modélisation des Matériaux
More informationLinear response time-dependent density functional theory
Linear response time-dependent density functional theory Emmanuel Fromager Laboratoire de Chimie Quantique, Université de Strasbourg, France fromagere@unistra.fr Emmanuel Fromager (UdS) Cours RFCT, Strasbourg,
More informationThe adiabatic connection
The adiabatic connection Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway Dipartimento di Scienze
More information5/27/2012. Role of van der Waals Interactions in Physics, Chemistry, and Biology
Role of van der Waals Interactions in Physics, Chemistry, and Biology 1 Role of van der Waals Interactions in Physics, Chemistry, and Biology Karin Jacobs: Take van der Waals forces into account in theory,
More informationRandom-phase approximation and beyond for materials: concepts, practice, and future perspectives. Xinguo Ren
Random-phase approximation and beyond for materials: concepts, practice, and future perspectives Xinguo Ren University of Science and Technology of China, Hefei USTC-FHI workshop on frontiers of Advanced
More informationAdiabatic connections in DFT
Adiabatic connections in DFT Andreas Savin Torino, LCC2004, September 9, 2004 Multi reference DFT Kohn Sham: single reference Variable reference space capability to approach the exact result Overview The
More informationThe calculation of the universal density functional by Lieb maximization
The calculation of the universal density functional by Lieb maximization Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry,
More informationRange-separated hybrids
Range-separated hybrids A. Savin MSSC2009 (Torino) Hybrids Wave function (Y) + density functional (n) hybrid E pluribus unum ("Out of many one") : Seal of the USA Overview Components (Y, n) Hybrids (Y
More informationJULIEN TOULOUSE, PAOLA GORI-GIORGI, ANDREAS SAVIN
Scaling Relations, Virial Theorem, and Energy Densities for Long-Range and Short-Range Density Functionals JULIEN TOULOUSE, PAOLA GORI-GIORGI, ANDREAS SAVIN Laboratoire de Chimie Théorique, CNRS et Université
More informationOptimization of quantum Monte Carlo wave functions by energy minimization
Optimization of quantum Monte Carlo wave functions by energy minimization Julien Toulouse, Roland Assaraf, Cyrus J. Umrigar Laboratoire de Chimie Théorique, Université Pierre et Marie Curie and CNRS, Paris,
More informationTDDFT in Chemistry and Biochemistry III
TDDFT in Chemistry and Biochemistry III Dmitrij Rappoport Department of Chemistry and Chemical Biology Harvard University TDDFT Winter School Benasque, January 2010 Dmitrij Rappoport (Harvard U.) TDDFT
More informationarxiv: v2 [physics.chem-ph] 14 Jan 2013
Multi-configuration time-dependent density-functional theory based on range separation Emmanuel Fromager a, Stefan Knecht b and Hans Jørgen Aa. Jensen b a Laboratoire de Chimie Quantique, arxiv:1211.4829v2
More informationShort-range exchange and correlation energy density functionals: Beyond the local-density approximation
THE JOURNAL OF CHEMICAL PHYSICS 122, 014110 2005 Short-range exchange and correlation energy density functionals: Beyond the local-density approximation Julien Toulouse, François Colonna, and Andreas Savin
More informationRecent advances in quantum Monte Carlo for quantum chemistry: optimization of wave functions and calculation of observables
Recent advances in quantum Monte Carlo for quantum chemistry: optimization of wave functions and calculation of observables Julien Toulouse 1, Cyrus J. Umrigar 2, Roland Assaraf 1 1 Laboratoire de Chimie
More informationConvergence behavior of RPA renormalized many-body perturbation theory
Convergence behavior of RPA renormalized many-body perturbation theory Understanding why low-order, non-perturbative expansions work Jefferson E. Bates, Jonathon Sensenig, Niladri Sengupta, & Adrienn Ruzsinszky
More informationIntroduction to many-body Green-function theory
Introduction to many-body Green-function theory Julien Toulouse Laboratoire de Chimie Théorique Université Pierre et Marie Curie et CNRS, 75005 Paris, France julien.toulouse@upmc.fr July 15, 2015 Contents
More informationShort Course on Density Functional Theory and Applications VII. Hybrid, Range-Separated, and One-shot Functionals
Short Course on Density Functional Theory and Applications VII. Hybrid, Range-Separated, and One-shot Functionals Samuel B. Trickey Sept. 2008 Quantum Theory Project Dept. of Physics and Dept. of Chemistry
More informationQuantum Monte Carlo wave functions and their optimization for quantum chemistry
Quantum Monte Carlo wave functions and their optimization for quantum chemistry Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France CEA Saclay, SPhN Orme des Merisiers April 2015 Outline
More informationExchange-Correlation Functional
Exchange-Correlation Functional Aiichiro Nakano Collaboratory for Advanced Computing & Simulations Depts. of Computer Science, Physics & Astronomy, Chemical Engineering & Materials Science, and Biological
More informationDispersion Interactions from the Exchange-Hole Dipole Moment
Dispersion Interactions from the Exchange-Hole Dipole Moment Erin R. Johnson and Alberto Otero-de-la-Roza Chemistry and Chemical Biology, University of California, Merced E. R. Johnson (UC Merced) Dispersion
More informationTeoría del Funcional de la Densidad (Density Functional Theory)
Teoría del Funcional de la Densidad (Density Functional Theory) Motivation: limitations of the standard approach based on the wave function. The electronic density n(r) as the key variable: Functionals
More informationComputational Methods. Chem 561
Computational Methods Chem 561 Lecture Outline 1. Ab initio methods a) HF SCF b) Post-HF methods 2. Density Functional Theory 3. Semiempirical methods 4. Molecular Mechanics Computational Chemistry " Computational
More informationFONCTIONNELLES DE DENSITÉ AVEC SÉPARATION DE PORTÉE: AU CIEL DFT SANS ÉCHELLE DE JACOB
FONCTIONNELLES DE DENSITÉ AVEC SÉPARATION DE PORTÉE: AU CIEL DFT SANS ÉCHELLE DE JACOB, Iann Gerber, Paola Gori-Giorgi, Julien Toulouse, Andreas Savin Équipe de Modélisation Quantique et Cristallographique
More informationAdvanced Quantum Chemistry III: Part 3. Haruyuki Nakano. Kyushu University
Advanced Quantum Chemistry III: Part 3 Haruyuki Nakano Kyushu University 2013 Winter Term 1. Hartree-Fock theory Density Functional Theory 2. Hohenberg-Kohn theorem 3. Kohn-Sham method 4. Exchange-correlation
More informationOVERVIEW OF QUANTUM CHEMISTRY METHODS
OVERVIEW OF QUANTUM CHEMISTRY METHODS Outline I Generalities Correlation, basis sets Spin II Wavefunction methods Hartree-Fock Configuration interaction Coupled cluster Perturbative methods III Density
More information1 Back to the ground-state: electron gas
1 Back to the ground-state: electron gas Manfred Lein and Eberhard K. U. Gross 1.1 Introduction In this chapter, we explore how concepts of time-dependent density functional theory can be useful in the
More informationMulti-reference Density Functional Theory. COLUMBUS Workshop Argonne National Laboratory 15 August 2005
Multi-reference Density Functional Theory COLUMBUS Workshop Argonne National Laboratory 15 August 2005 Capt Eric V. Beck Air Force Institute of Technology Department of Engineering Physics 2950 Hobson
More informationIntroduction to density-functional theory
Introduction to density-functional theory Julien Toulouse Laboratoire de Chimie Théorique Université Pierre et Marie Curie et CNRS, 75005 Paris, France julien.toulouse@upmc.fr June 21, 2017 http://www.lct.jussieu.fr/pagesperso/toulouse/enseignement/introduction_dft.pdf
More informationPost Hartree-Fock: MP2 and RPA in CP2K
Post Hartree-Fock: MP2 and RPA in CP2K A tutorial Jan Wilhelm jan.wilhelm@chem.uzh.ch 4 September 2015 Further reading MP2 and RPA by Mauro Del Ben, Jürg Hutter, Joost VandeVondele Del Ben, M; Hutter,
More informationComputational Chemistry I
Computational Chemistry I Text book Cramer: Essentials of Quantum Chemistry, Wiley (2 ed.) Chapter 3. Post Hartree-Fock methods (Cramer: chapter 7) There are many ways to improve the HF method. Most of
More informationTime-dependent linear-response variational Monte Carlo.
Time-dependent linear-response variational Monte Carlo. Bastien Mussard bastien.mussard@colorado.edu https://mussard.github.io/ Julien Toulouse julien.toulouse@upmc.fr Sorbonne University, Paris (web)
More informationRandom-phase approximation correlation methods for molecules and solids
Random-phase approximation correlation methods for molecules and solids Andreas Goerling, Andreas Hesselmann To cite this version: Andreas Goerling, Andreas Hesselmann. Random-phase approximation correlation
More informationarxiv: v3 [physics.chem-ph] 16 Feb 2016
Range-separated time-dependent density-functional theory with a frequency-dependent second-order Bethe-Salpeter correlation kernel Elisa Rebolini and Julien Toulouse Sorbonne Universités, UPMC Univ Paris
More informationComparison of exchange-correlation functionals: from LDA to GGA and beyond
Comparison of ehange-correlation functionals: from LDA to GGA and beyond Martin Fuchs Fritz-Haber-Institut der MPG, Berlin, Germany Density-Functional Theory Calculations for Modeling Materials and Bio-Molecular
More informationElectric properties of molecules
Electric properties of molecules For a molecule in a uniform electric fielde the Hamiltonian has the form: Ĥ(E) = Ĥ + E ˆµ x where we assume that the field is directed along the x axis and ˆµ x is the
More informationElectronic Structure Calculations and Density Functional Theory
Electronic Structure Calculations and Density Functional Theory Rodolphe Vuilleumier Pôle de chimie théorique Département de chimie de l ENS CNRS Ecole normale supérieure UPMC Formation ModPhyChem Lyon,
More informationResolution of identity approach for the Kohn-Sham correlation energy within the exact-exchange random-phase approximation
Resolution of identity approach for the Kohn-Sham correlation energy within the exact-exchange random-phase approximation Patrick Bleiziffer, Andreas Heßelmann, and Andreas Görling Citation: J. Chem. Phys.
More informationMulti-configuration time-dependent density-functional theory based on range separation
Multi-configuration time-dependent density-functional theory based on range separation Emmanuel Fromager, Stefan Knecht, and Hans Jørgen Aa. Jensen Citation: J. Chem. Phys. 138, 8411 213); doi: 1.163/1.4792199
More informationHigh-level Quantum Chemistry Methods and Benchmark Datasets for Molecules
High-level Quantum Chemistry Methods and Benchmark Datasets for Molecules Markus Schneider Fritz Haber Institute of the MPS, Berlin, Germany École Polytechnique Fédérale de Lausanne, Switzerland دانشگاه
More informationIntermolecular Forces in Density Functional Theory
Intermolecular Forces in Density Functional Theory Problems of DFT Peter Pulay at WATOC2005: There are 3 problems with DFT 1. Accuracy does not converge 2. Spin states of open shell systems often incorrect
More informationRPA, TDDFT AND RELATED RESPONSE FUNCTIONS : APPLICATION TO CORRELATION ENERGIES
RPA, TDDFT AND RELATED RESPONSE FUNCTIONS : APPLICATION TO CORRELATION ENERGIES John Dobson, Qld Micro & Nano Tech. Centre, Griffith University, Qld, Australia FAST collaboration: Janos Angyan, Sebastien
More informationDensity Functional Theory for Electrons in Materials
Density Functional Theory for Electrons in Materials Richard M. Martin Department of Physics and Materials Research Laboratory University of Illinois at Urbana-Champaign 1 Density Functional Theory for
More informationMagnetism in transition metal oxides by post-dft methods
Magnetism in transition metal oxides by post-dft methods Cesare Franchini Faculty of Physics & Center for Computational Materials Science University of Vienna, Austria Workshop on Magnetism in Complex
More informationIntroduction to Computational Chemistry
Introduction to Computational Chemistry Vesa Hänninen Laboratory of Physical Chemistry Chemicum 4th floor vesa.hanninen@helsinki.fi September 10, 2013 Lecture 3. Electron correlation methods September
More informationLecture 8: Introduction to Density Functional Theory
Lecture 8: Introduction to Density Functional Theory Marie Curie Tutorial Series: Modeling Biomolecules December 6-11, 2004 Mark Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Science
More informationTowards a unified description of ground and excited state properties: RPA vs GW
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
More informationRabi oscillations within TDDFT: the example of the 2 site Hubbard model
Rabi oscillations within TDDFT: the example of the 2 site Hubbard model Johanna I. Fuks, H. Appel, Mehdi,I.V. Tokatly, A. Rubio Donostia- San Sebastian University of Basque Country (UPV) Outline Rabi oscillations
More informationDensity Functional Theory and Group Theoretical Analysis in the Study of Hydrogen Bonded Organic Ferroelectrics
DOMENICO DI SANTE Density Functional Theory and Group Theoretical Analysis in the Study of Hydrogen Bonded Organic Ferroelectrics TESI DI LAUREA MAGISTRALE IN FISICA Relatore: Prof. Alessandra Continenza
More informationThe potential of Potential Functional Theory
The potential of Potential Functional Theory IPAM DFT School 2016 Attila Cangi August, 22 2016 Max Planck Institute of Microstructure Physics, Germany P. Elliott, AC, S. Pittalis, E.K.U. Gross, K. Burke,
More informationDFT: Exchange-Correlation
DFT: Exchange-Correlation Local functionals, exact exchange and other post-dft methods Paul Tulip Centre for Materials Physics Department of Physics University of Durham Outline Introduction What is exchange
More informationOrbital functionals derived from variational functionals of the Green function
Orbital functionals derived from variational functionals of the Green function Nils Erik Dahlen Robert van Leeuwen Rijksuniversiteit Groningen Ulf von Barth Lund University Outline 1. Deriving orbital
More informationQuantum Mechanical Simulations
Quantum Mechanical Simulations Prof. Yan Wang Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332, U.S.A. yan.wang@me.gatech.edu Topics Quantum Monte Carlo Hartree-Fock
More informationBasic Concepts and First Principles Computations for Surface Science: Applications in Chemical Energy Conversion and Storage.
International Summer School on Basic Concepts and First Principles Computations for Surface Science: Applications in Chemical Energy Conversion and Storage Norderney, Germany, July 21 26, 2013 Let s start
More informationKey concepts in Density Functional Theory (I) Silvana Botti
From the many body problem to the Kohn-Sham scheme European Theoretical Spectroscopy Facility (ETSF) CNRS - Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France Temporary Address: Centre
More informationIntroduction to Density Functional Theory
Introduction to Density Functional Theory S. Sharma Institut für Physik Karl-Franzens-Universität Graz, Austria 19th October 2005 Synopsis Motivation 1 Motivation : where can one use DFT 2 : 1 Elementary
More informationTime-Dependent Density-Functional Theory
Summer School on First Principles Calculations for Condensed Matter and Nanoscience August 21 September 3, 2005 Santa Barbara, California Time-Dependent Density-Functional Theory X. Gonze, Université Catholique
More informationProgress & challenges with Luttinger-Ward approaches for going beyond DFT
Progress & challenges with Luttinger-Ward approaches for going beyond DFT Sohrab Ismail-Beigi Yale University Dept. of Applied Physics and Physics & CRISP (NSF MRSEC) Ismail-Beigi, Phys. Rev. B (2010)
More informationDispersion Interactions in Density-Functional Theory
Dispersion Interactions in Density-Functional Theory Erin R. Johnson and Alberto Otero-de-la-Roza Chemistry and Chemical Biology, University of California, Merced E. R. Johnson (UC Merced) Dispersion from
More informationAb-initio studies of the adiabatic connection in density-functional theory
Ab-initio studies of the adiabatic connection in density-functional theory Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry,
More informationOrbital dependent correlation potentials in ab initio density functional theory
Orbital dependent correlation potentials in ab initio density functional theory noniterative - one step - calculations Ireneusz Grabowski Institute of Physics Nicolaus Copernicus University Toruń, Poland
More informationPreface Introduction to the electron liquid
Table of Preface page xvii 1 Introduction to the electron liquid 1 1.1 A tale of many electrons 1 1.2 Where the electrons roam: physical realizations of the electron liquid 5 1.2.1 Three dimensions 5 1.2.2
More informationSelf-Consistent Implementation of Self-Interaction Corrected DFT and of the Exact Exchange Functionals in Plane-Wave DFT
Self-Consistent Implementation of Self-Interaction Corrected DFT and of the Exact Exchange Functionals in Plane-Wave DFT Kiril Tsemekhman (a), Eric Bylaska (b), Hannes Jonsson (a,c) (a) Department of Chemistry,
More informationPseudopotentials for hybrid density functionals and SCAN
Pseudopotentials for hybrid density functionals and SCAN Jing Yang, Liang Z. Tan, Julian Gebhardt, and Andrew M. Rappe Department of Chemistry University of Pennsylvania Why do we need pseudopotentials?
More informationABC of ground-state DFT
ABC of ground-state DFT Kieron Burke and Lucas Wagner Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA January 5-9th, 2014 Kieron (UC Irvine) ABC of ground-state
More informationElectron Correlation
Electron Correlation Levels of QM Theory HΨ=EΨ Born-Oppenheimer approximation Nuclear equation: H n Ψ n =E n Ψ n Electronic equation: H e Ψ e =E e Ψ e Single determinant SCF Semi-empirical methods Correlation
More informationDensity Func,onal Theory (Chapter 6, Jensen)
Chem 580: DFT Density Func,onal Theory (Chapter 6, Jensen) Hohenberg- Kohn Theorem (Phys. Rev., 136,B864 (1964)): For molecules with a non degenerate ground state, the ground state molecular energy and
More informationDensity Functional Theory - II part
Density Functional Theory - II part antonino.polimeno@unipd.it Overview From theory to practice Implementation Functionals Local functionals Gradient Others From theory to practice From now on, if not
More informationarxiv: v1 [physics.chem-ph] 24 Jan 2019
Range-separated multideterminant density-functional theory with a short-range correlation functional of the on-top pair density Anthony Ferté, Emmanuel Giner, and Julien Toulouse Laboratoire de Chimie
More informationDFT calculations of NMR indirect spin spin coupling constants
DFT calculations of NMR indirect spin spin coupling constants Dalton program system Program capabilities Density functional theory Kohn Sham theory LDA, GGA and hybrid theories Indirect NMR spin spin coupling
More information