Combining many-body methods and density-functional theory. Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France

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1 Combining many-body methods and density-funtional theory Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, Frane September 2014

2 2/1 Hybrids ombining many-body methods and DFT Goal: improve the auray of present-day DFT 1 Multionfigurational hybrids (MCSCF+DFT) and double hybrids (MP2+DFT) stati (or strong) orrelation or self-interation error 2 Range-separated hybrids (lrrpa+srdft) van der Waals dispersion interations 3 Time-dependent range-separated hybrids (lrbse+srtddft) Rydberg, harge-transfer, double exitations

3 3/1 Outline with B. Civalleri, H. J. Jensen, L. Mashio, A. Savin, K. Sharkas

4 Sharkas, Savin, Jensen, Toulouse, JCP, /1 Multionfigurational hybrids (MCSCF+DFT) Multideterminant DFT based on a linear deomposition of e-e interation { } E exat = min Ψ ˆT + ˆV ne +λŵ ee Ψ +ĒHx[n λ Ψ ] Ψ with the λ-omplement density funtional Ē λ Hx [n]

5 Sharkas, Savin, Jensen, Toulouse, JCP, /1 Multionfigurational hybrids (MCSCF+DFT) Multideterminant DFT based on a linear deomposition of e-e interation { } E exat = min Ψ ˆT + ˆV ne +λŵ ee Ψ +ĒHx[n λ Ψ ] Ψ with the λ-omplement density funtional Ē λ Hx [n] Hartree and exhange ontributions: Ē λ Hx[n] = (1 λ)e Hx [n]

6 Sharkas, Savin, Jensen, Toulouse, JCP, /1 Multionfigurational hybrids (MCSCF+DFT) Multideterminant DFT based on a linear deomposition of e-e interation { } E exat = min Ψ ˆT + ˆV ne +λŵ ee Ψ +ĒHx[n λ Ψ ] Ψ with the λ-omplement density funtional Ē λ Hx [n] Hartree and exhange ontributions: Correlation ontribution: Ē λ Hx[n] = (1 λ)e Hx [n] Ē λ [n] = E [n] λ 2 E [n 1/λ ] (1 λ 2 )E [n] n 1/λ (r) = (1/λ) 3 n(r/λ) n(r) r

7 Sharkas, Savin, Jensen, Toulouse, JCP, /1 Multionfigurational hybrids (MCSCF+DFT) Multideterminant DFT based on a linear deomposition of e-e interation { } E exat = min Ψ ˆT + ˆV ne +λŵ ee Ψ +ĒHx[n λ Ψ ] Ψ with the λ-omplement density funtional Ē λ Hx [n] Hartree and exhange ontributions: Correlation ontribution: Ē λ Hx[n] = (1 λ)e Hx [n] Ē λ [n] = E [n] λ 2 E [n 1/λ ] (1 λ 2 )E [n] n 1/λ (r) = (1/λ) 3 n(r/λ) Approximations for Ψ and E x [n] n(r) MCSCF: Ψ = n nφ n BLYP, PBE, et r

8 What value for the empirial parameter λ? O3ADD6 set: 6 energy differenes for yloaddition reations of ozone with ethylene or aetylene (aug--pvtz basis, BLYP funtional): Mean absolute error (kal/mol) DFT λ MCSCF = We take λ= 0.25 as for usual hybrid funtionals Sharkas, Savin, Jensen, Toulouse, JCP, /1

9 6/1 Hand-waving argument for multionfigurational hybrids Pure density-funtional approximations (DFA) often roughly aount for stati orrelation due to a ompensation of errors E DFA x = Ex DFA ( Ex exat +E stat +E DFA ) +E dyna

10 6/1 Hand-waving argument for multionfigurational hybrids Pure density-funtional approximations (DFA) often roughly aount for stati orrelation due to a ompensation of errors E DFA x = Ex DFA ( Ex exat +E stat +E DFA ) +E dyna Usual hybrid approximations destroy this ompensation of errors and tend to be worse for desribing stati orrelation E hybrid x = λe exat x E exat x +(1 λ)e DFA x +(1 λ)e stat +E DFA +E dyna

11 6/1 Hand-waving argument for multionfigurational hybrids Pure density-funtional approximations (DFA) often roughly aount for stati orrelation due to a ompensation of errors E DFA x = Ex DFA ( Ex exat +E stat +E DFA ) +E dyna Usual hybrid approximations destroy this ompensation of errors and tend to be worse for desribing stati orrelation E hybrid x = λe exat x E exat x +(1 λ)e DFA x +(1 λ)e stat +E DFA +E dyna Multionfigurational hybrids tend to restore the balane by adding a fration λ of stati orrelation E MCSCF+DFA x λe exat x E exat x +λe stat +E stat +(1 λ)e DFA x +(1 λ 2 )E dyna +(1 λ 2 )E DFA

12 Test of MCSCF+DFT on F 2 moleule λ = 0.25, BLYP funtional, -pvtz basis: Total energy (hartree) BLYP B3LYP MCSCF+BLYP Aurate Internulear distane R (bohr) = MCSCF+BLYP is as good as B3LYP at equilibrium and improves on it at dissoiation Sharkas, Savin, Jensen, Toulouse, JCP, /1

13 Test of MCSCF+DFT on N 2 moleule λ = 0.25, BLYP funtional, -pvtz basis: Total energy (hartree) BLYP B3LYP MCSCF+BLYP Aurate Internulear distane R (bohr) = MCSCF+BLYP has still a large error at dissoiation Sharkas, Savin, Jensen, Toulouse, JCP, /1

14 Double hybrids (MP2+DFT) Start with linear deomposition of e-e interation with single-determinant approximation { } E 0 = min Φ ˆT + ˆV ne +λŵ ee Φ +ĒHx[n λ Φ ] Φ Sharkas, Toulouse, Savin, JCP, /1

15 Double hybrids (MP2+DFT) Start with linear deomposition of e-e interation with single-determinant approximation { } E 0 = min Φ ˆT + ˆV ne +λŵ ee Φ +ĒHx[n λ Φ ] Φ Then, define the following perturbation theory: { } E α = min Ψ ˆT + ˆV ne +λˆv Hx HF +αλŵ Ψ +ĒHx[n λ Ψ ] Ψ ) where λŵ = λ(ŵee ˆV Hx HF is a Møller-Plesset-type perturbation. Sharkas, Toulouse, Savin, JCP, /1

16 Double hybrids (MP2+DFT) Start with linear deomposition of e-e interation with single-determinant approximation { } E 0 = min Φ ˆT + ˆV ne +λŵ ee Φ +ĒHx[n λ Φ ] Φ Then, define the following perturbation theory: { } E α = min Ψ ˆT + ˆV ne +λˆv Hx HF +αλŵ Ψ +ĒHx[n λ Ψ ] Ψ ) where λŵ = λ(ŵee ˆV Hx HF is a Møller-Plesset-type perturbation. At seond order, we get a double-hybrid approximation: E MP2+DFT x = λex HF +(1 λ)e x [n]+(1 λ 2 )E [n]+λ 2 E MP2 = This provides a theoretial derivation of Grimme s double hybrids with only one parameter λ Sharkas, Toulouse, Savin, JCP, /1

17 Performane of one-parameter double hybrids E MP2+DFT x = λex HF +(1 λ)e x [n]+(1 λ 2 )E [n]+λ 2 E MP2. For MP2+BLYP, optimization of λ on sets of moleular atomization energies and reation barrier heights leads to λ 0.65, orresponding to a fration λ of MP2 orrelation. Inlusion of MP2 orrelation allows one to use a larger fration of HF exhange, whih redues the self-interation error. Like the two-parameter double hybrids, the one-parameter double hybrids reah near hemial auray (1-2 kal/mol) on thermohemistry properties for systems without important stati orrelation effets. For moleular rystals, double hybrids do not improve over MP2. Sharkas, Toulouse, Savin, JCP, 2011 Sharkas, Toulouse, Mashio, Civalleri, JCP, /1

18 11/1 Outline with J. Ángyán, O. Frank, G. Jansen, E. Luppi, B. Mussard, P. Reinhardt A. Savin, W. Zhu

19 12/1 Range-separated hybrids (lrrpa+srdft) Based on a range separation of e-e interation E exat = min Ψ long-range interation { Ψ ˆT + ˆV ne +Ŵ lr ee Ψ +E sr Hx[n Ψ ] erf(µr ij ) r ij i<j Savin, in Reent developments and appliations of modern DFT, 1996 Toulouse, Colonna, Savin, PRA, 2004 } short-range density funtional

20 Range-separated hybrids (lrrpa+srdft) Based on a range separation of e-e interation E exat = min Ψ long-range interation { Ψ ˆT + ˆV ne +Ŵ lr ee Ψ +E sr Hx[n Ψ ] erf(µr ij ) r ij i<j Savin, in Reent developments and appliations of modern DFT, 1996 Toulouse, Colonna, Savin, PRA, 2004 Single-determinant approximation: { E 0 = min Φ ˆT + ˆV ne +Ŵee Φ +E lr sr Φ This is a hybrid DFT with HF exhange at long range } short-range density funtional Hx[n Φ ] } Ángyán, Gerber, Savin, Toulouse, PRA, 2005 Toulouse, Gerber, Jansen, Savin, Ángyán, PRL, /1

21 Range-separated hybrids (lrrpa+srdft) Based on a range separation of e-e interation E exat = min Ψ long-range interation { Ψ ˆT + ˆV ne +Ŵ lr ee Ψ +E sr Hx[n Ψ ] erf(µr ij ) r ij i<j Savin, in Reent developments and appliations of modern DFT, 1996 Toulouse, Colonna, Savin, PRA, 2004 Single-determinant approximation: { E 0 = min Φ ˆT + ˆV ne +Ŵee Φ +E lr sr Φ This is a hybrid DFT with HF exhange at long range } short-range density funtional Hx[n Φ ] All what is missing is the long-range orrelation energy: E exat = E 0 +E lr Ángyán, Gerber, Savin, Toulouse, PRA, 2005 Toulouse, Gerber, Jansen, Savin, Ángyán, PRL, /1 }

22 Range-separated hybrids (lrrpa+srdft) Based on a range separation of e-e interation E exat = min Ψ long-range interation { Ψ ˆT + ˆV ne +Ŵ lr ee Ψ +E sr Hx[n Ψ ] erf(µr ij ) r ij i<j Savin, in Reent developments and appliations of modern DFT, 1996 Toulouse, Colonna, Savin, PRA, 2004 Single-determinant approximation: { E 0 = min Φ ˆT + ˆV ne +Ŵee Φ +E lr sr Φ This is a hybrid DFT with HF exhange at long range } short-range density funtional Hx[n Φ ] All what is missing is the long-range orrelation energy: E exat = E 0 +E lr Ángyán, Gerber, Savin, Toulouse, PRA, 2005 } srlda srpbe, et... Toulouse, Gerber, Jansen, Savin, Ángyán, PRL, /1

23 Range-separated hybrids (lrrpa+srdft) Based on a range separation of e-e interation E exat = min Ψ long-range interation { Ψ ˆT + ˆV ne +Ŵ lr ee Ψ +E sr Hx[n Ψ ] erf(µr ij ) r ij i<j Savin, in Reent developments and appliations of modern DFT, 1996 Toulouse, Colonna, Savin, PRA, 2004 Single-determinant approximation: { E 0 = min Φ ˆT + ˆV ne +Ŵee Φ +E lr sr Φ This is a hybrid DFT with HF exhange at long range } short-range density funtional Hx[n Φ ] All what is missing is the long-range orrelation energy: } srlda srpbe, et... E exat = E 0 +E lr Ángyán, Gerber, Savin, Toulouse, PRA, 2005 lrmp2 Toulouse, Gerber, Jansen, Savin, Ángyán, PRL, 2009 lrrpa 12/1

24 Fast basis onvergene with long-range interation Behavior of the wave funtion at small intereletroni distane r 12 0 : Coulomb interation Long-range interation Ψ(r 12 ) Ψ(0) = 1+ r Ψ µ (r 12 ) Ψ µ (0) = 1+ µr π + = l P l (osθ) with l l 2 = l P l (osθ) with l e αl l=0 l= µ = 0.5 bohr angle θ angle θ Gori-Giorgi, Savin, PRA, 2006; Frank, Mussard, Luppi, Toulouse, in prep. 13/1

25 14/1 lrrpa variants from ring oupled luster doubles drpa = diret ring CCD (without exhange): E,dRPA lr = 1 2 tr[ weet lr lr ] drccd = + + Suseria, Henderson, Sorensen, JCP, 2008

26 lrrpa variants from ring oupled luster doubles drpa = diret ring CCD (without exhange): E,dRPA lr = 1 2 tr[ weet lr lr ] drccd = + + Suseria, Henderson, Sorensen, JCP, 2008 RPAx-SO2 = ring CCD with exhange (Szabo-Ostlund s variant) E,RPAx-SO2 lr = 1 2 tr[ weet lr lr ] rccdx = Heßelmann, JCP, 2011 Toulouse, Zhu, Savin, Jansen, Ángyán, JCP, /1

27 Test of lrrpa+srdft on atomization energies AE6 set: 6 atomization energies of small moleules (µ = 0.5 bohr 1, srpbe funtional, -pvqz, spin unrestrited): Mean absolute error (kal/mol) lrmp2 MP2 lrdrpa drpa lrrpax-so2 RPAx-SO2 = Range separation improves the auray for all methods = All range-separated methods give a MAE of about 5 kal/mol Mussard, Reinhardt, Ángyán, Toulouse, in prep. 15/1

28 Test of lrrpa+srdft on reation barrier heights BH6 set: 6 barrier heights of reations with small moleules (µ = 0.5 bohr 1, srpbe funtional, -pvqz, spin unrestrited): Mean absolute error (kal/mol) MP2 lrmp2 lrdrpa drpa lrrpax-so2 RPAx-SO2 = Range separation improves the auray for all methods = All range-separated methods give a MAE of about 2 kal/mol Mussard, Reinhardt, Ángyán, Toulouse, in prep. 16/1

29 Toulouse, Zhu, Savin, Jansen, Ángyán, JCP, /1 Test of lrrpa+srdft on weak intermoleular interations S22 set: 22 equilibrium interation energies of weakly-interating moleular systems from water dimer to DNA base pairs (µ = 0.5 bohr 1, srpbe funtional, aug--pvdz): % of error on interation energy H bonded dispersion dispersion +multipoles lrmp2 lrdrpa lrrpax SO system in S22 set = lrrpax-so2/avdz gives a MAE of 4% wrt CCSD(T)/CBS

30 18/1 Outline with T. Helgaker, E. Rebolini, A. Savin, A. Teale

31 19/1 Time-dependent range-separated hybrids Linear-response TDDFT equation χ 1 (ω) = χ 1 0 (ω) f Hx(ω) = exitation energies, linear-response properties

32 19/1 Time-dependent range-separated hybrids Linear-response TDDFT equation χ 1 (ω) = χ 1 0 (ω) f Hx(ω) = exitation energies, linear-response properties Range separation for exhange kernel is now standard: f x = f lr,hf x +f sr,dft x +f DFT Tawada, Tsuneda, Yanagisawa, Yanai, Hirao, JCP, 2004

33 Hedegård, Heiden, Kneht, Fromager, Jensen, JCP, /1 Time-dependent range-separated hybrids Linear-response TDDFT equation χ 1 (ω) = χ 1 0 (ω) f Hx(ω) = exitation energies, linear-response properties Range separation for exhange kernel is now standard: f x = f lr,hf x +f sr,dft x +f DFT Tawada, Tsuneda, Yanagisawa, Yanai, Hirao, JCP, 2004 Here, range separation for both exhange and orrelation kernels: Start with lrmp2+srdft sheme for ground state: E x = Ex lr,hf +Ex sr,dft +E lr,mp2 linear response f x = f lr,hf x +f sr,dft x +f lr,(2) (ω) Rebolini, Savin, Toulouse, MP, 2013; Rebolini, Savin, Toulouse, in prep. Other similar shemes: Pernal, JCP, 2012; Fromager, Kneht, Jensen, JCP, 2013;

34 20/1 MP2 orrelation energy in terms of Green funtion E MP2 = ij ab 2 ε a +ε b ε i ε j ia,jb

35 20/1 MP2 orrelation energy in terms of Green funtion E MP2 = ij ab 2 ε a +ε b ε i ε j ia,jb = i a b j + in orbital spae

36 20/1 MP2 orrelation energy in terms of Green funtion E MP2 = ij ab 2 ε a +ε b ε i ε j ia,jb = i a b j + in orbital spae in spaetime 1 (x 1,t 1 ) 4

37 20/1 MP2 orrelation energy in terms of Green funtion E MP2 = ij ab 2 ε a +ε b ε i ε j ia,jb = i a b j + in orbital spae in spaetime 1 (x 1,t 1 ) 4 d1d2d3d4g 0 (1,2)G 0 (2,1)w ee (1,4)w ee (2,3)G 0 (3,4)G 0 (4,3) + exhange term

38 21/1 Seond-order orrelation self-energy and kernel δ δg 0 (1,2) E MP2 = + Σ (2) (2,1)=

39 Seond-order orrelation self-energy and kernel δ δg 0 (1,2) δ δg 0 (3,4) E MP2 = + Σ (2) (2,1)= Ξ (2) (2,4;1,3)= 2 1 2,3 1, , , /1

40 22/1 From the Bethe-Salpeter equation to TDDFT Bethe-Salpeter equation with a kernel depending on only one frequeny: χ(ω) = χ 0 (ω)+ dω dω χ 0 (ω,ω)ξ (2) Hx (ω ±ω )χ(ω,ω)

41 From the Bethe-Salpeter equation to TDDFT Bethe-Salpeter equation with a kernel depending on only one frequeny: χ(ω) = χ 0 (ω)+ dω dω χ 0 (ω,ω)ξ (2) Hx (ω ±ω )χ(ω,ω) Define an effetive seond-order kernel: [ ] f (2) Hx (ω) = χ 1 0 (ω) dω dω χ 0 (ω,ω)ξ (2) Hx (ω ±ω )χ 0 (ω,ω) χ 1 0 (ω) whih an be used in a TDDFT-like linear-response equation χ(ω) = χ 0 (ω)+χ 0 (ω)f (2) Hx (ω)χ(ω) or, equivalently, χ 1 (ω) = χ 1 0 (ω) f(2) Hx (ω) Romaniello, Sangalli, Berger, Sottile, Molinari, Reining, Onida, JCP, 2009 Sangalli, Romaniello, Onida, Marini, JCP, /1

42 In an orbital basis Effetive long-range seond-order orrelation kernel (lrbse2): f lr,(2),ia,jb (ω) = k, + 1 2,d jk i lr a kb lr ω (ε a +ε ε k ε j ) k, d ib lr ja d lr ω (ε +ε d ε j ε i ) k,l j ik lr ka b lr ω (ε b +ε ε i ε k ) kl ib lr ja lk lr ω (ε b +ε a ε k ε l ) = The orrelation kernel an be large when ω is lose to a double exitation energy. Zhang, Steinmann, Yang, JCP, 2013 Rebolini, Savin, Toulouse, in prep. 23/1

43 In an orbital basis Effetive long-range seond-order orrelation kernel (lrbse2): f lr,(2),ia,jb (ω) = k, + 1 2,d jk i lr a kb lr ω (ε a +ε ε k ε j ) k, d ib lr ja d lr ω (ε +ε d ε j ε i ) k,l j ik lr ka b lr ω (ε b +ε ε i ε k ) kl ib lr ja lk lr ω (ε b +ε a ε k ε l ) = The orrelation kernel an be large when ω is lose to a double exitation energy. Calulation of exitation energies in two steps: 1 lrtdhf+srtdlda alulation in the TDA: A X 0 = ω 0 X 0 2 perturbative addition of lrbse2 kernel: ω = ω 0 +X 0 flr,(2) (ω 0 ) X 0 Zhang, Steinmann, Yang, JCP, 2013 Rebolini, Savin, Toulouse, in prep. 23/1

44 Test of lrbse2 on the Be atom Long-range exitation energies without the sr kernel (lrhf+srlda orbitals, TDA, d-aug--pvdz basis): Long-range exitation energies (ev) first singlet exited state Referene first triplet exited state Range-separation parameter µ (bohr -1 ) Rebolini, Toulouse, Teale, Helgaker, Savin, JCP, /1

45 Rebolini, Toulouse, Teale, Helgaker, Savin, JCP, 2014 Rebolini, Savin, Toulouse, in prep. 24/1 Test of lrbse2 on the Be atom Long-range exitation energies without the sr kernel (lrhf+srlda orbitals, TDA, d-aug--pvdz basis): Long-range exitation energies (ev) first singlet exited state first triplet exited state Referene lrtdhf Range-separation parameter µ (bohr -1 )

46 Test of lrbse2 on the Be atom Long-range exitation energies without the sr kernel (lrhf+srlda orbitals, TDA, d-aug--pvdz basis): Long-range exitation energies (ev) first singlet exited state first triplet exited state Referene lrtdhf lrbse Range-separation parameter µ (bohr -1 ) = lrbse2 better desribes the long-range exitation energies Rebolini, Toulouse, Teale, Helgaker, Savin, JCP, 2014 Rebolini, Savin, Toulouse, in prep. 24/1

47 Test of lrbse2+srtddft on exitation energies 56 singlet and triplet exitation energies of 4 small moleules N 2, CO, H 2 CO, C 2 H 4 (µ = 0.35 bohr 1, srlda funtional, TDA, Sadlej+ basis): Mean absolute error (ev) Valene Rydberg 0 TDLDA lrtdhf+srtdlda lrbse2+srtdlda = For these systems, lrbse2+srtdlda does not improve muh over lrtdhf+srtdlda Rebolini, Savin, Toulouse, in prep. 25/1

48 Summary Multionfigurational hybrids (MCSCF+DFT) extension of usual hybrid funtionals with expliit stati orrelation we still need to improve the funtional Double hybrids (MP2+DFT) we provided a theoretial derivation of one-parameter double hybrids reah near-hemial auray if no important stati or dispersion orrelation effets Range-separated hybrids (lrrpa+srdft) fast basis onvergene important to inlude exhange terms for dispersion interations Time-dependent range-separated hybrids (lrbse2+srtddft) frequeny-dependent seond-order long-range orrelation kernel more tests needed, in partiular for effet of double exitations 26/1