DFT basée sur le théorème de fluctuation-dissipation avec séparation de portée pour les interactions de van der Waals
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1 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 Laboratoire de Chimie Théorique, UPMC Univ Paris 06 et CNRS, Paris, France 2 Université de Toulouse, INSA-UPS, LPCNO, Toulouse, France 3 Fachbereich Chemie, Universität Duisburg-Essen, Essen, Allemagne 4 CRM2, Institut Jean Barriol, Université de Nancy et CNRS, Vandoeuvre-lès-Nancy, France Courriel : julien.toulouse@upmc.fr Page web : février 2009
2 1 Kohn-Sham DFT and ACFDT approach 2 Range-separated ACFDT approach 3 Some results
3 1 Kohn-Sham DFT and ACFDT approach 2 Range-separated ACFDT approach 3 Some results
4 Kohn-Sham DFT Kohn-Sham (KS) scheme { } E = min Φ ˆT + ˆV ne Φ + E Hxc [n Φ ] Φ Φ : single-determinant wave function
5 Kohn-Sham DFT Kohn-Sham (KS) scheme { } E = min Φ ˆT + ˆV ne Φ + E Hxc [n Φ ] Φ Φ : single-determinant wave function One problem (among others): Usual approximations for exchange-correlation functional E xc [n] (LDA, GGA,...) do not describe well (long-range) van der Waals dispersion forces
6 Example: interaction energy curve of Ne 2 LDA and PBE functionals, aug-cc-pv5z basis: Interaction energy (mhartree) Ne 2 Accurate LDA PBE Internuclear distance (Bohr)
7 ACFDT approach to DFT Adiabatic connection formula for correlation energy: E c = 1 0 dλ { } Ψ λ Ŵ ee Ψ λ Φ KS Ŵ ee Φ KS
8 ACFDT approach to DFT Adiabatic connection formula for correlation energy: E c = 1 0 dλ or, with a compact notation, { } Ψ λ Ŵ ee Ψ λ Φ KS Ŵ ee Φ KS E c = dλ Tr[w ee P c,λ ]
9 ACFDT approach to DFT Adiabatic connection formula for correlation energy: E c = 1 0 dλ or, with a compact notation, { } Ψ λ Ŵ ee Ψ λ Φ KS Ŵ ee Φ KS E c = dλ Tr[w ee P c,λ ] P c,λ from fluctuation-dissipation theorem dω P c,λ = 2π [χ λ(iω) χ KS (iω)]
10 ACFDT approach to DFT Adiabatic connection formula for correlation energy: E c = 1 0 dλ or, with a compact notation, { } Ψ λ Ŵ ee Ψ λ Φ KS Ŵ ee Φ KS E c = dλ Tr[w ee P c,λ ] P c,λ from fluctuation-dissipation theorem dω P c,λ = 2π [χ λ(iω) χ KS (iω)] where the response function χ λ (iω) is given by χ λ (iω) 1 = χ KS (iω) 1 f Hxc,λ (iω)
11 Random Phase Approximation (RPA) RPA approximation: f xc,λ = 0 = E c,rpa
12 Random Phase Approximation (RPA) RPA approximation: f xc,λ = 0 = E c,rpa = increasing interest in the DFT community: Perdew, Dobson, Furche, Gonze, Kresse, Scuseria,...
13 Random Phase Approximation (RPA) RPA approximation: f xc,λ = 0 = E c,rpa = increasing interest in the DFT community: Perdew, Dobson, Furche, Gonze, Kresse, Scuseria,... Encouraging results: consistent with exact exchange correct dispersion forces at (very) large separation good cohesive energies and lattice constants of solids some improvement in description of bond dissociation
14 Random Phase Approximation (RPA) RPA approximation: f xc,λ = 0 = E c,rpa = increasing interest in the DFT community: Perdew, Dobson, Furche, Gonze, Kresse, Scuseria,... Encouraging results: consistent with exact exchange correct dispersion forces at (very) large separation good cohesive energies and lattice constants of solids some improvement in description of bond dissociation But several unsatisfactory aspects: correlation energies far too negative strong dependence on basis size bump at intermediate distances in some dissociation curves embarrassing results for simple van der Waals dimers!
15 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)
16 Example: interaction energy curve of Be 2 RPA (with PBE orbitals), cc-pv5z basis: Interaction energy (mhartree) Be 2 Accurate RPA Internuclear distance (Bohr)
17 1 Kohn-Sham DFT and ACFDT approach 2 Range-separated ACFDT approach 3 Some results
18 Range-separated DFT Multideterminant extension of KS scheme with range separation { } E = min Ψ ˆT + ˆV ne + Ŵee Ψ lr + EHxc[n sr Ψ ] Ψ
19 Range-separated DFT Multideterminant extension of KS scheme with range separation { } E = min Ψ ˆT + ˆV ne + Ŵee Ψ lr + EHxc[n sr Ψ ] Ψ Ŵ lr ee = i<j erf(µr ij ) r ij : long-range electron-electron interaction
20 Range-separated DFT Multideterminant extension of KS scheme with range separation { } E = min Ψ ˆT + ˆV ne + Ŵee Ψ lr + EHxc[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
21 Range-separated DFT Multideterminant extension of KS scheme with range separation { } E = min Ψ ˆT + ˆV ne + Ŵee Ψ lr + EHxc[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
22 Range-separated DFT Multideterminant extension of KS scheme with range separation { } E = min Ψ ˆT + ˆV ne + Ŵee Ψ lr + EHxc[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. Limiting cases: µ = 0 = KS DFT µ = Standard many-body methods
23 Range-separated DFT Multideterminant extension of KS scheme with range separation { } E = min Ψ ˆT + ˆV ne + Ŵee Ψ lr + EHxc[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. Limiting cases: µ = 0 = KS DFT µ = Standard many-body methods In principle: exact
24 Range-separated DFT Multideterminant extension of KS scheme with range separation { } E = min Ψ ˆT + ˆV ne + Ŵee Ψ lr + EHxc[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. Limiting cases: µ = 0 = KS DFT µ = Standard many-body methods In principle: exact In practice: approximations are necessary for Ψ lr and E sr xc[n]
25 Range-separated DFT: approximations Approximations for E sr xc[n] short-range LDA short-range PBE...
26 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+MP2 method RPA = RSH+RPA method...
27 Range-separated hybrid (RSH) scheme Restriction to single-determinant wave functions Φ: { } E RSH = min Φ ˆT + ˆV ne + Ŵee Φ lr + EHxc[n sr Φ ] Φ
28 Range-separated hybrid (RSH) scheme Restriction to single-determinant wave functions Φ: { } E RSH = min Φ ˆT + ˆV ne + Ŵee Φ lr + EHxc[n sr Φ ] Φ The minimizing RSH determinant Φ RSH is given by ) (ˆT + ˆV ne + ˆV Hx,HF lr + ˆV Hxc sr Φ RSH = E 0 Φ RSH,
29 Range-separated hybrid (RSH) scheme Restriction to single-determinant wave functions Φ: { } E RSH = min Φ ˆT + ˆV ne + Ŵee Φ lr + EHxc[n sr Φ ] Φ The minimizing RSH determinant Φ RSH is given by ) (ˆT + ˆV ne + ˆV Hx,HF lr + ˆV Hxc sr Φ RSH = E 0 Φ RSH, So the RSH energy is E RSH = Φ RSH ˆT + ˆV ne Φ RSH + E H + E lr x,hf + E sr xc
30 Long-range correlation energy E lr c Exact energy = RSH energy + long-range correlation energy E = E RSH + E lr c
31 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 0 dλ { } Ψ lr λ Ŵ lr Ψλ lr Φ RSH Ŵ lr Φ RSH with the long-range perturbation operator Ŵ lr = Ŵ lr ee ˆV lr Hx,HF
32 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 0 dλ { } Ψ lr λ Ŵ lr 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 ]
33 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.
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 density. 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)
35 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 χ lr λ (iω) is given by χ lr λ (iω) 1 = χ lr IP,λ (iω) 1 f lr Hxc,λ (iω) Approximation: fc,λ lr = 0 = RSH+RPAx method
36 1 Kohn-Sham DFT and ACFDT approach 2 Range-separated ACFDT approach 3 Some results
37 Dependence on basis size: Ne 2 Total energy (aug-cc-pvnz basis, µ = 0.5, sr-pbe functional): Total energy (Hartree) Ne 2 Exact RPA RSH+RPAx Size of one-particle basis (n in aug-cc-pvnz) = RSH+RPAx has a small basis dependence
38 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 RSH+RPAx Internuclear distance (Bohr)
39 Interaction energy curve of Be 2 Interaction energy (cc-pv5z basis, µ = 0.5, sr-pbe functional): Interaction energy (mhartree) Be 2 Accurate RPA RSH+RPAx Internuclear distance (Bohr)
40 Conclusions and perspectives Conclusions RSH+RPAx method overcomes many problems of standard RPA RSH+RPAx method seems well suited for van der Waals systems RSH+RPAx method has also problems (e.g., dissociation) Toulouse, Gerber, Jansen, Savin, Ángyán, arxiv: Perspectives exploration of other variants of the method application to larger molecular systems (benzene dimer,...) application to solids Web page:
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