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 A brief overview of DFT/many-body hybrids 2 Range-separated hybrids for the ground state 3 Range-separated hybrids for excited states
Outline 3/23 1 A brief overview of DFT/many-body hybrids 2 Range-separated hybrids for the ground state 3 Range-separated hybrids for excited states
Overview of DFT/many-body hybrids 1 Kohn-Sham DFT (1965): { } E = min Φ ˆT + ˆV ne Φ +E Hxc [n Φ ] Φ where Φ is a single determinant Semilocal density-functional approximations (DFAs) for E xc [n]: LDA, GGAs, meta-ggas = work well in many cases but still limitations: self-interaction error, strong/static correlation, van der Waals dispersion interactions 2 Hybrid approximations (Becke 1993): E xc = a x Ex HF [Φ]+(1 a x )Ex DFA [n]+ec DFA [n] with Hartree-Fock (HF) exchange: Ex HF [Φ] = Φ Ŵ ee Φ E H [n] one empirical parameter: a x 0.25 = reduces self-interaction error 4/23
Overview of DFT/many-body hybrids 5/23 3 Double-hybrid approximations (Grimme 2006): E xc = a x Ex HF [Φ]+(1 a x )Ex DFA [n]+(1 a c )Ec DFA [n]+a c Ec MP2 with second-order Møller-Plesset (MP2) perturbative correlation: E MP2 c occ = i<j unocc a<b Φ ab ij Ŵ ee Φ 2 ε a +ε b ε i ε j two empirical parameters: a x 0.5 and a c 0.25 = further reduces self-interaction error, partially account for van der Waals dispersion, but fails for strongly correlated systems
Overview of DFT/many-body hybrids 6/23 4 General DFT/many-body hybrid scheme (Sharkas, Toulouse, Savin 2011) { } E = min Ψ ˆT + ˆV ne +λŵ ee Ψ +ĒHxc[n λ Ψ ] Ψ with one empirical parameter λ and the complement density functional: Ē λ Hx[n] = (1 λ)e Hx [n] and Ē λ c [n] = E c [n] λ 2 E c [n 1/λ ] (1 λ 2 )E c [n] Single determinant: Ψ Φ = a hybrid approximation: E xc = λex HF [Φ]+(1 λ)ex DFA [n]+(1 λ 2 )Ec DFA [n] Second-order perturbation: one-parameter double-hybrid approximation: E xc = λex HF [Φ]+(1 λ)ex DFA [n]+(1 λ 2 )Ec DFA [n]+λ 2 Ec MP2 Non-perturbative approaches: Ψ = n c nφ n = for strong correlation (Sharkas, Savin, Jensen, Toulouse, JCP, 2012)
Leininger, Stoll, Werner, Savin, CPL, 1997; Fromager, Toulouse, Jensen, JCP, 2007 7/23 1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states Overview of DFT/many-body hybrids 5 Range-separated DFT/many-body hybrid scheme (Savin 1996) { } E = min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Ψ with a long-range interaction Ŵee lr = erf(µr ij ) i<j r ij a short-range density functional EHxc sr [n] and one empirical parameter µ controlling the range of the separation Single determinant: Ψ Φ = a lrhf+srdft hybrid: E xc = E lr,hf x [Φ]+Ex sr,dfa [n]+ec sr,dfa similar to the LC scheme (Hirao et al. 2001) Many-body perturbation theory = lrmp2/lrrpa+srdft hybrids: E xc = E lr,hf x [n] [Φ]+Ex sr,dfa [n]+ec sr,dfa [n]+ec lr,mp2/rpa Ángyán, Gerber, Savin, Toulouse, PRA, 2005; Toulouse, Gerber, Jansen, Savin, Ángyán, PRL, 2009; Janesko, Henderson, Scuseria, JCP, 2009 Non-perturbative approaches: Ψ = n c nφ n
Outline 8/23 1 A brief overview of DFT/many-body hybrids 2 Range-separated hybrids for the ground state 3 Range-separated hybrids for excited states
Range-separated hybrids: lrhf+srdft 9/23 E lrhf+srdft xc = E lr,hf x Ángyán, Gerber, Savin, Toulouse, PRA, 2005 [Φ]+E sr,dfa x [n]+ec sr,dfa Short-range exchange and correlation DFAs have been developed, e.g.: short-range LDA: Ex/c srlda [n] = n(r) ǫ sr,unif x/c (n(r)) dr Toulouse, Savin, Flad, IJQC, 2004; Paziani, Moroni, Gori-Giorgi, Bachelet, PRB, 2006 [n] short-range PBE: E srpbe x/c [n] = n(r) ǫ srpbe x/c (n(r), n(r)) dr Toulouse, Colonna, Savin, JCP, 2005; Goll, Werner, Stoll, Leininger, Gori-Giorgi, Savin, CP, 2006
lrhf+srdft: reduction of the self-interaction error Dissociation of H + 2 molecule: H+ 2 H +0.5 + H +0.5 Binding energy (kcal/mol) 10 0-10 -20-30 -40-50 -60 exact lrhf+srpbe PBE0 PBE -70 0 2 4 6 8 10 Internuclear distance (Angstrom) = removal of the large self-interaction error coming from the long-range part of the PBE exchange Mussard, Toulouse, MP, 2017 10/23
Adding long-range correlation: lrmp2/lrrpa+srdft Long-range MP2: occ = E lr,mp2 c unocc i<j a<b Ángyán, Gerber, Savin, Toulouse, PRA, 2005 Φ ab ij Ŵ lr ee Φ 2 ε a +ε b ε i ε j = + Long-range direct RPA (drpa) = sum of all direct ring diagrams E lr,drpa c = + + Toulouse, Gerber, Jansen, Savin, Ángyán, PRL, 2009; Janesko, Henderson, Scuseria, JCP, 2009 Long-range RPA with exchange (RPAx-SO2) = sum of all direct + some exchange ring diagrams E lr,rpax-so2 c = + + + + + + Toulouse, Zhu, Savin, Jansen, Ángyán, JCP, 2011 11/23
Fast basis convergence of long-range perturbation theory Total energy of N 2 calculated with Gaussian basis sets (µ = 0.5 bohr 1, srpbe functional, Dunning basis cc-pvxz): Total energy (hartree) -109.25-109.30-109.35-109.40-109.45 MP2 lrmp2+srpbe VDZ VTZ VQZ V5Z V6Z Basis sets of increasing sizes = Exponential basis convergence of lrmp2+srpbe Franck, Mussard, Luppi, Toulouse, JCP, 2015 12/23
Test of lrmp2/lrrpa+srdft on atomization energies A G2 subset of 49 atomization energies of small molecules (µ = 0.5 bohr 1, srpbe functional, cc-pvqz, spin unrestricted): Standard deviation (kcal/mol) 12 8 4 0 MP2 drpa RPAx-SO2 lrmp2+srpbe lrdrpa+srpbe lrrpax-so2+srpbe -20-10 0 10 Mean error (kcal/mol) = Range separation decreases the standard deviation for all methods = All range-separated methods give a mean error of 5 kcal/mol Mussard, Reinhardt, Ángyán, Toulouse, JCP, 2015 13/23
Test of lrmp2/lrrpa+srdft on reaction barrier heights DBH24/08 set: 24 barrier heights of reactions with small molecules (µ = 0.5 bohr 1, srpbe functional, aug-cc-pvqz, spin unrestricted): Standard deviation (kcal/mol) 8 6 4 2 0-1 0 1 2 3 4 5 6 Mean error (kcal/mol) MP2 drpa RPAx-SO2 lrmp2+srpbe lrdrpa+srpbe lrrpax-so2+srpbe = Range separation improves all methods = All range-separated methods give a mean error 1 kcal/mol Mussard, Reinhardt, Ángyán, Toulouse, JCP, 2015 14/23
Toulouse, Zhu, Savin, Jansen, Ángyán, JCP, 2011 15/23 1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states Test of lrmp2/lrrpa+srdft on weak interactions S22 set: 22 equilibrium interaction energies of weakly-interacting molecular systems from water dimer to DNA base pairs (µ = 0.5 bohr 1, srpbe functional, aug-cc-pvdz): % of error on interaction energy 60 40 20 0 20 40 60 1 23 H bonded dispersion dispersion +multipoles lrmp2+srpbe lrdrpa+srpbe lrrpax SO2+srPBE 4 56 7 89 10 1112 13 1415 system in S22 set 16 1718 22 19 2021 = lrrpax-so2+srpbe/avdz gives a mean absolute error of 4%
Comparitive test for lrrpa+srdft on weak interactions Test set of 10 molecular dimers (from 6 to 32 atoms) with 8 configurations sampling the repulsive wall, minimum, and asymptotic regions (µ = 0.5 bohr 1, srpbe functional, aug-cc-pvtz): Mean absolute error (kcal/mol) LC-ωPBE-D3 0.15 B3LYP-D3 0.17 lrrpax-so2+srpbe 0.29 M06 0.31 M11 0.34 vdw-df2 0.42 = lrrpax-so2+srpbe is globally less accurate that LC-ωPBE-D3 and B3LYP-D3: the short-range xc functional still needs to be improved Taylor et al., JCP, 2016 16/23
Toward self-consistent DFT/many-body hybrids Fully self-consistent MP2+DFT hybrid using the OEP method E xc = λex HF [{φ i }]+(1 λ)ex B [n]+(1 λ 2 )Ec LYP [n]+λ 2 Ec MP2 [{φ i,ε i }] local exchange-correlation potential including the MP2 term 0 CO molecule -0.5 v xc (hartree) -1 improves accuracy of EA -1.5 BLYP (λ=0) MP2+DFT (λ=0.65) MP2 (λ=1) accurate reference -2-6 -4-2 0 2 4 6 r (bohr) Śmiga, Franck, Mussard, Buksztel, Grabowski, Luppi, Toulouse, JCP, 2016 17/23
Outline 18/23 1 A brief overview of DFT/many-body hybrids 2 Range-separated hybrids for the ground state 3 Range-separated hybrids for excited states
Time-dependent range-separated hybrids Linear-response TDDFT equation χ 1 (ω) = χ 1 0 (ω) f Hxc(ω) = excitation energies, linear-response properties Range separation for exchange kernel is now standard: f xc = f lr,hf x +f sr,dfa x +f DFA c Tawada, Tsuneda, Yanagisawa, Yanai, Hirao, JCP, 2004 Here, range separation for both exchange and correlation kernels: f xc = f lr,hf x +f sr,dfa x +f sr,dfa c +fc lr,(2) (ω) Rebolini, Savin, Toulouse, MP, 2013; Rebolini, Toulouse, JCP, 2016 Other similar schemes: Pernal, JCP, 2012; Fromager, Knecht, Jensen, JCP, 2013; Hedegård, Heiden, Knecht, Fromager, Jensen, JCP, 2013 19/23
Second-order correlation self-energy and kernel δ δg 0 (1,2) δ δg 0 (3,4) E MP2 c = + Σ (2) c (2,1)= Ξ (2) c (2,4;1,3)= 2 1 2,3 1,4 + + 2 1 2 3 1,4 2 3 2,3 2 4 2 4 + + + + 1 4 1 4 1 3 1 3 20/23
In an orbital basis 21/23 Effective long-range second-order correlation kernel (lrbse2): f lr,(2) occ c,ia,jb (ω) = k<l unocc c<d Φ a i Ŵlr ee Φ cd kl Φcd kl Ŵlr ee Φ b j ω (ε c +ε d ε k ε l ) = The correlation kernel brings the effect of the double excitations Calculation of excitation energies in two steps: 1 lrtdhf+srtdlda calculation in the TDA: A X 0 = ω 0 X 0 2 perturbative addition of lrbse2 kernel: ω = ω 0 +X 0 flr,(2) c (ω 0 ) X 0 Zhang, Steinmann, Yang, JCP, 2013 Rebolini, Toulouse, JCP, 2016
Test of lrbse2+srtddft on excitation energies 56 singlet and triplet excitation energies of 4 small molecules N 2, CO, H 2 CO, C 2 H 4 (µ = 0.35 bohr 1, srlda functional, TDA, Sadlej+ basis): 0.5 Standard deviation (ev) 0.4 0.3 0.2 Valence TDLDA Valence lrtdhf+srtdlda Valence lrbse2+srtdlda 0.1 Rydberg TDLDA Rydberg lrtdhf+srtdlda Rydberg lrbse2+srtdlda 0-1.2-1 -0.8-0.6-0.4-0.2 0 Mean error (ev) = lrbse2+srtdlda provides a slight overall improvement over lrtdhf+srtdlda Rebolini, Toulouse, JCP, 2016 22/23
Summary and Acknowledgments 23/23 DFT/many-body hybrid methods based on a decomposition of the e-e interaction into long-range and short-range parts lrhf+srdft reduces the self-interaction error lrmp2/lrrpa+srdft has a fast basis convergence lrmp2/lrrpa+srdft accounts for van der Waals dispersion interactions self-consistent MP2+DFT using the OEP method lrbse2+srtddft for excitation energies: frequency-dependent second-order long-range correlation kernel bringing the effect of the double excitations Acknowledgments J. Ángyán, A. Buksztel, F. Colonna, H.-J. Flad, O. Franck, E. Fromager, I. Gerber, I. Grabowski, G. Jansen, H. J. Aa. Jensen, E. Luppi, B. Mussard, E. Rebolini, P. Reinhardt, A. Savin, K. Sharkas, S. Śmiga, K. Szalewicz, D. Taylor, W. Zhu