Range-separated density-functional theory with long-range random phase approximation
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1 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 et CNRS, Paris, France 2 CRM2, Institut Jean Barriol, Université de Nancy et CNRS, Vandoeuvre-lès-Nancy, France julien.toulouse@upmc.fr Web page: June 21
2 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
3 ACFDT approach to DFT Adiabatic connection formula for correlation energy: E c = 1 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)
4 Random Phase Approximation (RPA) RPA approximation: f xc,λ = = 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
5 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]
6 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 Φ RSH, This is a hybrid DFT with exact (HF) exchange at long range.
7 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 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 ]
8 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 = = RSH+lrRPA method RPAx approximation: fc,λ lr = = RSH+lrRPAx method
9 Dependence on basis size: Ne 2 Total energy (aug-cc-pvnz basis, µ =.5, sr-pbe functional): Total energy (Hartree) Ne 2 Exact RPA RPAx RSH+lrRPAx Size of one-particle basis (n in aug-cc-pvnz) = RSH+lrRPAx has a small basis dependence
10 Interaction energy curve of Ne 2 Interaction energy (aug-cc-pv5z basis, µ =.5, sr-pbe functional): Interaction energy (mhartree) Ne 2 Accurate RPA RPAx RSH+lrRPAx Internuclear distance (Bohr) = RSH+lrRPAx is more accurate than full-range RPA and RPAx
11 Interaction energy curve of Be 2 Interaction energy (cc-pv5z basis, µ =.5, sr-pbe functional): Interaction energy (mhartree) Be 2 Accurate RPA RPAx RSH+lrRPAx Internuclear distance (Bohr) = RSH+lrRPAx is more accurate than full-range RPA and RPAx
12 Equilibrium interaction energies of a set of 22 weakly-interacting molecular systems (S22 set) from water dimer to DNA base pairs mean absolute relative error (%) small basis larger basis (avtz) RPA RPAx RSH+lrRPA RSH+lrRPAx = Both range separation and exact exchange kernel are important
13 Summary and Conclusions Summary RSH+lrRPAx method = short-range DFT + long-range RPAx Conclusions RSH+lrRPAx method overcomes many problems of standard RPA RSH+lrRPAx method seems well suited for van der Waals systems RSH+lrRPAx method has also problems (e.g., dissociation) Toulouse, Gerber, Jansen, Savin, Ángyán, Phys. Rev. Lett. 12, 9644 (29) Zhu, Toulouse, Savin, Ángyán, J. Chem. Phys., to appear Toulouse, Zhu, Ángyán, Savin, submitted to Phys. Rev. A
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