Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation
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1 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, UPMC Univ Paris 06 et CNRS, Paris, France 2 Université de Toulouse, INSA-UPS, LPCNO, Toulouse, France 3 Fachbereich Chemie, Universität Duisburg-Essen, Essen, Germany 4 CRM2, Institut Jean Barriol, Université de Nancy et CNRS, Vandoeuvre-lès-Nancy, France julien.toulouse@upmc.fr Web page: January 2010
2 1 ACFDT approach to DFT 2 Range-separated ACFDT approach 3 Some results on van der Waals systems
3 1 ACFDT approach to DFT 2 Range-separated ACFDT approach 3 Some results on van der Waals systems
4 Kohn-Sham density-functional theory (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 density-functional theory (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
13 Random Phase Approximation (RPA) RPA approximation: f xc,λ = 0 = E c,rpa = increasing interest in the DFT community Encouraging results: consistent with the use of exact exchange qualitatively correct dispersion forces at (very) large separation good cohesive energies and lattice constants of solids
14 Random Phase Approximation (RPA) RPA approximation: f xc,λ = 0 = E c,rpa = increasing interest in the DFT community Encouraging results: consistent with the use of exact 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
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 ACFDT approach to DFT 2 Range-separated ACFDT approach 3 Some results on van der Waals systems
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.
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. 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. 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 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, This is a hybrid DFT with exact (HF) exchange at long range.
29 Long-range correlation energy E lr c Exact energy = RSH energy + long-range correlation energy E = E RSH + E lr c
30 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
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 With a compact notation, E lr c = dλ Tr [ w lr Pc,λ lr ]
32 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.
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. 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)
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) Possible approximations: RPA approximation: fxc,λ lr = 0 = RSH+RPA method RPAx approximation: fc,λ lr = 0 = RSH+RPAx method
35 1 ACFDT approach to DFT 2 Range-separated ACFDT approach 3 Some results on van der Waals systems
36 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 RPAx RSH+RPAx Size of one-particle basis (n in aug-cc-pvnz) = RSH+RPAx has a small basis dependence
37 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+RPAx Internuclear distance (Bohr) = Range separation improves RPA(x)
38 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+RPAx Internuclear distance (Bohr) = Range separation improves RPA(x)
39 Interaction energy curve of Ar 2 Interaction energy (aug-cc-pv5z basis, µ = 0.5, sr-pbe functional): Interaction energy (mhartree) Ar 2 Accurate RPA RPAx RSH+RPAx Internuclear distance (Bohr) = Range separation improves RPA(x)
40 Interaction energy curve of (CH 4 ) 2 Interaction energy (aug-cc-pvtz basis, µ = 0.5, sr-pbe functional): Interaction energy (mhartree) (CH 4 ) 2 CCSD(T) RSH+RPA RSH+RPAx Internuclear distance (Bohr) = Exact exchange kernel is important
41 Equilibrium interaction energies of a set of 22 weakly-interacting molecular systems (S22 set) from water dimer to DNA base pairs 50 mean absolute relative error (%) RPA RPAx RSH+RPA RSH+RPAx
42 Summary and Conclusions Summary RSH+RPAx method = short-range DFT + long-range RPAx 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, Phys. Rev. Lett. 102, (2009) Web page:
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