Multideterminant density-functional theory for static correlation. Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France

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1 Multideterminant density-functional theory for static correlation Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Basel, Switzerland September 2015

2 Kohn-Sham DFT and static correlation 2/37 Kohn-Sham DFT: E exact = min Φ { } Φ ˆT + ˆV ne Φ +E Hxc [n Φ ] where Φ is single-determinant wave function Usual approximations for Exc [n]: LDA, GGA, meta-gga, hybrid. Mainly very successful but still improved accuracy is needed, particularly for systems with static correlation effects.

3 Kohn-Sham DFT and static correlation Static (or strong) correlation: systems with partially filled near-degenerate orbitals Examples: bond dissociation, transition metals,... In wave-function theory, static correlation usually requires a non-perturbative multideterminant wave function Static correlation effects are very much system-dependent, making the development of universal density-functional approximations difficult Density-functional approximations often display a partial compensation between self-interaction error and static correlation error 2/37 Kohn-Sham DFT: E exact = min Φ { } Φ ˆT + ˆV ne Φ +E Hxc [n Φ ] where Φ is single-determinant wave function Usual approximations for Exc [n]: LDA, GGA, meta-gga, hybrid. Mainly very successful but still improved accuracy is needed, particularly for systems with static correlation effects.

4 Outline 3/37 1 Overview of some DFT methods including static correlation 2 Linear-separated MCSCF+DFT method 3 Range-separated MCSCF+DFT method 4 Variants and extensions

5 Outline 4/37 1 Overview of some DFT methods including static correlation 2 Linear-separated MCSCF+DFT method 3 Range-separated MCSCF+DFT method 4 Variants and extensions

6 Some DFT methods including static correlation 5/37 1 Artificial breaking of spin/spatial symmetry (unrestricted Kohn-Sham): (see, e.g.: Cremer, MP, 2001) often useful, e.g., for potential energy surfaces but spin contamination

7 Some DFT methods including static correlation 5/37 1 Artificial breaking of spin/spatial symmetry (unrestricted Kohn-Sham): (see, e.g.: Cremer, MP, 2001) often useful, e.g., for potential energy surfaces but spin contamination 2 DFT with ensemble determinants or fractional orbital occupation numbers: Φ ˆΓ = { i w i Φ i Φ i : } E = min Tr[(ˆT + ˆV ne )ˆΓ ]+E Hxc [nˆγ ] ˆΓ better mathematical properties (Lieb, IJQC, 1983; Cancès, JCP, 2001) several approximate variants have been explored (e.g.: Slater et al., PR, 1969; Dunlap, Mei, JCP, 1983; Filatov, Shaik, CPL, 1999; Takeda et al., IJQC, 2003; Chai, JCP, 2012) improve description of static correlation, but a general-purpose approximation is still lacking

8 Some DFT methods including static correlation promosing but requires more development 5/37 1 Artificial breaking of spin/spatial symmetry (unrestricted Kohn-Sham): (see, e.g.: Cremer, MP, 2001) often useful, e.g., for potential energy surfaces but spin contamination 2 DFT with ensemble determinants or fractional orbital occupation numbers: Φ ˆΓ = { i w i Φ i Φ i : } E = min Tr[(ˆT + ˆV ne )ˆΓ ]+E Hxc [nˆγ ] ˆΓ better mathematical properties (Lieb, IJQC, 1983; Cancès, JCP, 2001) several approximate variants have been explored (e.g.: Slater et al., PR, 1969; Dunlap, Mei, JCP, 1983; Filatov, Shaik, CPL, 1999; Takeda et al., IJQC, 2003; Chai, JCP, 2012) improve description of static correlation, but a general-purpose approximation is still lacking 3 Approx from strong-interacting limit: E Hxc [n] min Ψ n Ψ Ŵ ee Ψ (e.g.: Vuckovic, Wagner, Mirtschink, Gori-Giorgi, JCTC, 2015)

9 Some DFT methods including static correlation 6/37 4 Configuration-interaction schemes with Hamiltonian matrix elements constructed from DFT: DFT/MRCI method with five empirical parameters (Grimme, Waletzke, JCP, 1999): generally good results but empirical DFT/CI method with two localized CI states from constrained DFT (Wu, Cheng, van Voorhis, JCP, 2007): needs to be made more general

10 Some DFT methods including static correlation 4 Configuration-interaction schemes with Hamiltonian matrix elements constructed from DFT: DFT/MRCI method with five empirical parameters (Grimme, Waletzke, JCP, 1999): generally good results but empirical DFT/CI method with two localized CI states from constrained DFT (Wu, Cheng, van Voorhis, JCP, 2007): needs to be made more general 5 Correlated wave-function calculation in subspace + correlation density functional: { } E = min Ψ subspace ˆT + ˆV ne +Ŵ ee Ψ subspace +Ec subspace [n Ψsubspace ] Ψ subspace but difficulty of designing approximations for Ec subspace. Some attempts: modified density depending on orbital occupation numbers (e.g.: Lie, Clementi, JCP, 1974; Pérez-Jiménez, Pérez-Jordá, PRA, 2007) approximate E subspace c based on a uniform electron gas with removed low-lying virtual orbital subspace (Savin, IJQC, 1988; Miehlich, Stoll, Savin, MP, 1997; Gutlé, Savin, PRA, 2007) approximate E subspace c using the on-top pair density of Ψ subspace (e.g: Moscardó, San-Fabián, PRA, 1991; Manni et al., JCTC, 2014) 6/37

11 Outline 7/37 1 Overview of some DFT methods including static correlation 2 Linear-separated MCSCF+DFT method 3 Range-separated MCSCF+DFT method 4 Variants and extensions

12 Linear-separated MCSCF+DFT method Multideterminant DFT based on a linear decomposition of e-e interaction: { } E exact = min Ψ ˆT + ˆV ne +λŵ ee Ψ +ĒHxc[n λ Ψ ] Ψ with the λ-complement density functional Ē λ Hxc [n] Sharkas, Savin, Jensen, Toulouse, JCP, /37

13 Linear-separated MCSCF+DFT method Multideterminant DFT based on a linear decomposition of e-e interaction: { } E exact = min Ψ ˆT + ˆV ne +λŵ ee Ψ +ĒHxc[n λ Ψ ] Ψ with the λ-complement density functional Ē λ Hxc [n] Hartree and exchange contributions: Ē λ Hx[n] = (1 λ)e Hx [n] Sharkas, Savin, Jensen, Toulouse, JCP, /37

14 Linear-separated MCSCF+DFT method Multideterminant DFT based on a linear decomposition of e-e interaction: { } E exact = min Ψ ˆT + ˆV ne +λŵ ee Ψ +ĒHxc[n λ Ψ ] Ψ with the λ-complement density functional Ē λ Hxc [n] Hartree and exchange contributions: Correlation contribution: Ē λ Hx[n] = (1 λ)e Hx [n] Ē λ c [n] = E c [n] λ 2 E c [n 1/λ ] (1 λ 2 )E c [n] with the scaled density n 1/λ (r) = (1/λ) 3 n(r/λ) Sharkas, Savin, Jensen, Toulouse, JCP, /37

15 Linear-separated MCSCF+DFT method Multideterminant DFT based on a linear decomposition of e-e interaction: { } E exact = min Ψ ˆT + ˆV ne +λŵ ee Ψ +ĒHxc[n λ Ψ ] Ψ with the λ-complement density functional Ē λ Hxc [n] Hartree and exchange contributions: Correlation contribution: Ē λ Hx[n] = (1 λ)e Hx [n] Ē λ c [n] = E c [n] λ 2 E c [n 1/λ ] (1 λ 2 )E c [n] with the scaled density n 1/λ (r) = (1/λ) 3 n(r/λ) Approximations for Ψ and E xc [n] MCSCF: Ψ = n c nφ n BLYP, PBE, etc... (MP2 = double hybrids) Implemented in DALTON Sharkas, Savin, Jensen, Toulouse, JCP, /37

16 What value for the empirical parameter λ? O3ADD6 set: 6 energy differences for cycloaddition reactions of ozone with ethylene or acetylene (aug-cc-pvtz basis, BLYP functional): Mean absolute error (kcal/mol) DFT λ MCSCF = We take λ= 0.25 as for usual hybrid functionals Sharkas, Savin, Jensen, Toulouse, JCP, /37

17 Test of MCSCF+DFT on H 2 molecule λ = 0.25, cc-pvtz basis, BLYP functional: 0.85 Wrong asymptotics of B3LYP: c R Total energy (hartree) BLYP B3LYP MCSCF+BLYP Accurate Internuclear distance R (bohr) = MCSCF+BLYP removes the wrong 1/R asymptotic term Sharkas, Savin, Jensen, Toulouse, JCP 137, (2012) 10/37

18 Dissociation of H 2 : asymptotic analysis in a minimal basis 11/37 The exact energy goes exponentially to a constant for R E exact = 1+O(e R )

19 Dissociation of H 2 : asymptotic analysis in a minimal basis 11/37 The exact energy goes exponentially to a constant for R E exact = 1+O(e R ) Cancellation of spurious 1/R terms between exact exchange and static correlation Ex HF = c 1 1 2R +O(e R ) and Ec MCSCF = c R +O(e R )

20 Dissociation of H 2 : asymptotic analysis in a minimal basis 11/37 The exact energy goes exponentially to a constant for R E exact = 1+O(e R ) Cancellation of spurious 1/R terms between exact exchange and static correlation Ex HF = c 1 1 2R +O(e R ) and Ec MCSCF = c R +O(e R ) (Semi)local density functional approximations (DFA) decay as E DFA x = c 3 +O(e R ) and E DFA c = c 4 +O(e R )

21 Dissociation of H 2 : asymptotic analysis in a minimal basis 11/37 The exact energy goes exponentially to a constant for R E exact = 1+O(e R ) Cancellation of spurious 1/R terms between exact exchange and static correlation Ex HF = c 1 1 2R +O(e R ) and Ec MCSCF = c R +O(e R ) (Semi)local density functional approximations (DFA) decay as E DFA x = c 3 +O(e R ) and E DFA c = c 4 +O(e R ) Usual hybrid functionals have a spurious 1/R term E hybrid xc = c 5 λ 2R +O(e R )

22 Dissociation of H 2 : asymptotic analysis in a minimal basis 11/37 The exact energy goes exponentially to a constant for R E exact = 1+O(e R ) Cancellation of spurious 1/R terms between exact exchange and static correlation Ex HF = c 1 1 2R +O(e R ) and Ec MCSCF = c R +O(e R ) (Semi)local density functional approximations (DFA) decay as E DFA x = c 3 +O(e R ) and E DFA c = c 4 +O(e R ) Usual hybrid functionals have a spurious 1/R term Exc hybrid = c 5 λ 2R +O(e R ) MCSCF+DFT removes the spurious 1/R term E MCSCF+DFT xc = c 6 +O(e R )

23 Test of MCSCF+DFT on Li 2 molecule 12/37 λ = 0.25, BLYP functional, cc-pvtz basis: Total energy (hartree) BLYP B3LYP MCSCF+BLYP Accurate Internuclear distance R (bohr) Sharkas, Savin, Jensen, Toulouse, JCP, 2012

24 Test of MCSCF+DFT on N 2 molecule 13/37 λ = 0.25, BLYP functional, cc-pvtz basis: Total energy (hartree) BLYP B3LYP MCSCF+BLYP Accurate Internuclear distance R (bohr) Sharkas, Savin, Jensen, Toulouse, JCP, 2012

25 Test of MCSCF+DFT on F 2 molecule λ = 0.25, BLYP functional, cc-pvtz basis: Total energy (hartree) BLYP B3LYP MCSCF+BLYP Accurate Internuclear distance R (bohr) = MCSCF+BLYP improves the dissociation but still a large error on the dissociation energy Sharkas, Savin, Jensen, Toulouse, JCP, /37

26 Outline 15/37 1 Overview of some DFT methods including static correlation 2 Linear-separated MCSCF+DFT method 3 Range-separated MCSCF+DFT method 4 Variants and extensions

27 16/37 1. DFT with static correlation 2. Linear-separated MCSCF+DFT 3. Range-separated MCSCF+DFT 4. Variants and extensions Range-separated MCSCF+DFT method Multideterminant DFT based on a range separation of e-e interaction: { } E exact = min Ψ ˆT + ˆV ne +Ŵee lr,µ Ψ +E sr,µ Ψ Hxc [n Ψ] Savin, in Recent developments and applications of modern DFT, 1996 Ŵlr,µ ee = i<j erf(µr ij ) r ij : long-range electron-electron interaction E sr,µ Hxc [n] : short-range Hxc density functional minimizing wave function Ψ lr,µ = i c iφ i is multi-determinant parameter µ controls the range of separation

28 Range-separated MCSCF+DFT method Multideterminant DFT based on a range separation of e-e interaction: { } E exact = min Ψ ˆT + ˆV ne +Ŵee lr,µ Ψ +E sr,µ Ψ Hxc [n Ψ] Savin, in Recent developments and applications of modern DFT, 1996 Ŵlr,µ ee = i<j erf(µr ij ) r ij : long-range electron-electron interaction E sr,µ Hxc [n] : short-range Hxc density functional minimizing wave function Ψ lr,µ = i c iφ i is multi-determinant parameter µ controls the range of separation Approximations for Ψ lr,µ and E sr,µ xc [n] CI or MCSCF for static correlation (MP2 or RPA for van der Waals) srlda, srggas,... See, e.g.: Toulouse, Colonna, Savin, PRA, 2004; Fromager, Toulouse, Jensen, JCP, /37

29 Short-range exchange energy: LDA E sr,µ x,lda [n] = n(r)ε sr,µ x,unif (n(r))dr For Be atom: E x sr,μ a.u LDA accurate for a short-range interaction exact LDA Asymptotic expansion for µ : Ex sr,µ = A 1 µ 2 n(r) 2 dr+ A 2 µ Μ a.u. ( ) n(r) 2 n(r) +4τ(r) dr+ 2n(r) Toulouse, Savin, Flad, IJQC, 2004; Toulouse, Colonna, Savin, PRA, /37

30 Short-range correlation energy: LDA E sr,µ c,lda [n] = n(r) [ ε c,unif (n(r)) ε lr,µ c,unif (n(r)) ]dr For Be atom: 0 E c sr,μ a.u LDA accurate for a short-range interaction exact LDA Μ a.u. Asymptotic expansion for µ : Ec sr,µ = B 1 µ 2 n 2,c (r,r)dr+ B 2 µ 3 n 2 (r,r)dr+ Toulouse, Savin, Flad, IJQC, 2004; Toulouse, Colonna, Savin, PRA, 2004; Paziani, Moroni, Gori-Giorgi, Bachelet, PRB, 2006; Gori-Giorgi, Savin, PRA, /37

31 Short-range exchange energy: GGA Short-range GGA functional of Heyd, Scuseria, and Ernzerhof (2003) based on the PBE exchange hole: ε sr,µ x,gga (n) = 1 n x,pbe (n, n,r 12 )wee sr,µ (r 12 )dr 12 2 For Be atom: E x sr,μ a.u exact LDA GGA Μ a.u. = GGA describes well a longer range of interaction Other srggas: Toulouse, Colonna, Savin, PRA, 2004; Toulouse, Colonna, Savin, JCP, 2005; Goll, Werner, Stoll, PCCP, /37

32 Short-range correlation energy: GGA Interpolation between PBE at µ = 0 and expansion of LDA for µ : ε sr,µ c,gga (n, n ) = ε c,pbe (n, n ) 1+d 1 (n)µ+d 2 (n)µ 2 For Be atom: E c sr,μ a.u exact LDA GGA Μ a.u. = GGA describes well a longer range of interaction Toulouse, Colonna, Savin, PRA, 2004; Other srggas: Toulouse, Colonna, Savin, JCP, 2005; Goll, Werner, Stoll, PCCP, /37

33 Fast basis convergence with long-range interaction Behavior of the wave function at small interelectronic distance r 12 0 : Coulomb interaction Long-range interaction Ψ(r 12 ) Ψ(0) = 1+ r Ψ lr,µ (r 12 ) Ψ lr,µ (0) = 1+ µr π + = c l P l (cosθ) with c l l 2 = c l P l (cosθ) with c l e αl l=0 l= µ = 0.5 bohr angle θ angle θ Gori-Giorgi, Savin, PRA, 2006; Franck, Mussard, Luppi, Toulouse, JCP, /37

34 Choice of the range-separation parameter µ Mean absolute error of lrhf+srlda for atomization energies of 56 molecules: µ 0.5 bohr 1 gives the minimal mean absolute error Gerber, Ángyán, CPL, /37

35 Choice of the range-separation parameter µ 23/37 Comparison of lrhf+srdft and lrmcscf+srdft for He and Be atoms: µ 0.4 bohr 1 allows one to use a single-determinant wave function for He and a multideterminant wave function for Be Fromager, Toulouse, Jensen, JCP, 2007

36 Test of lrmcscf+srdft on H 2 molecule with µ = 0.4 bohr 1 and cc-pvtz basis set: Total energy (hartree) PBE PBE0 MCSCF+PBE lrmcscf+srpbe Accurate Internuclear distance R (bohr) = MCSCF+PBE and lrmcscf+srpbe give almost identical curves Sharkas, Savin, Jensen, Toulouse, JCP, /37

37 Test of lrmcscf+srdft on Li 2 molecule 25/37 with µ = 0.4 bohr 1 and cc-pvtz basis set: Total energy (hartree) PBE PBE0 MCSCF+PBE lrmcscf+srpbe Accurate Internuclear distance R (bohr) Sharkas, Savin, Jensen, Toulouse, JCP, 2012

38 Test of lrmcscf+srdft on N 2 molecule 26/37 with µ = 0.4 bohr 1 and cc-pvtz basis set: Total energy (hartree) PBE PBE0 MCSCF+PBE lrmcscf+srpbe Accurate Internuclear distance R (bohr) Sharkas, Savin, Jensen, Toulouse, JCP, 2012

39 Conclusions on MCSCF+DFT and lrmcscf+srdft 27/37 MCSCF+DFT and lrmcscf+srdft include (a fraction of) static correlation MCSCF+DFT is a simple extension of the usual hybrid approximations lrmcscf+srdft has better basis convergence and in principle better density-functional approximations We need better density-functional approximations (possibly, including dependence on on-top pair density) The MCSCF part is not a blackbox and can be computationally expensive

40 Outline 28/37 1 Overview of some DFT methods including static correlation 2 Linear-separated MCSCF+DFT method 3 Range-separated MCSCF+DFT method 4 Variants and extensions

41 Other range-separated variants for static correlation Other methods for the long-range part: Multireference perturbation theory (NEVPT2): accounts for long-range dispersion interactions on top of MCSCF (Fromager, Cimiraglia, Jensen, PRA, 2010) Constrained-pairing mean-field theory (CPMFT): static correlation by breaking electron-number conservation (Tsuchimochi, Scuseria, Savin, JCP, 2010) Pair coupled-cluster doubles (pccd): static correlation by simplication of CCD (Garza, Bulik, Henderson, Scuseria, PCCP, 2015) Density-matrix renormalization group (DMRG): allows one to use largest active spaces (Hedegård, Knecht, Kielberg, Jensen, Reiher, JCP, 2015) Other approximation for the short-range part: Exact short-range exchange via optimized effective potential (Toulouse, Gori-Giorgi, Savin, TCA, 2005; Paziani, Moroni, Gori-Giorgi, Bachelet, PRB, 2006; Stoyanova, Teale, Toulouse, Helgaker, Fromager, JCP, 2013) 29/37

42 Range-separated DMFT+DFT method Reformulation of range-separated DFT with one-particle density matrix γ : { } E exact = min min Ψ ˆT + ˆV ne +Ŵ lr,µ γ ee Ψ +E sr,µ Ψ γ Hxc [n Ψ] { } = min T[γ]+V ne [n γ ]+E lr,µ γ Hxc [γ]+esr,µ Hxc [n γ] with E lr,µ Hxc [γ] = min Ψ γ Ψ Ŵlr,µ ee Ψ So, we finally obtain: { E exact = min T[γ]+Vne [n γ ]+E H [n γ ]+E lr,µ γ xc [γ]+exc sr,µ [n γ ] } In practice, we use the Buijse-Baerends (or Müller) approximation (in atomic-orbital basis): Exc lr,µ,bb [γ] = (γ 1/2 ) ab (γ 1/2 ) cd ac wee lr,µ db abcd Pernal, PRA, 2010; Rohr, Toulouse, Pernal, PRA, /37

43 lrdmft+srdft method: dependence on basis size Total energy of H 2 (cc-pvnz basis sets, µ = 0.4 bohr 1 ): exact BB PBE lrbb+srpbe total energy in hartree cc-pvdz cc-pvtz cc-pvqz cc-pv5z = lrbb+srpbe has a small basis dependence Rohr, Toulouse, Pernal, PRA, /37

44 Test of lrdmft+srdft method on H 2 molecule Relative energy wrt equilibrium (cc-pvtz basis, µ = 0.4 bohr 1 ): relative energy in hartree accvrate PBE lrbb+srpbe distance in bohr = lrbb+srpbe dissociates better than PBE but still large error due to srpbe BB Rohr, Toulouse, Pernal, PRA, /37

45 Test of lrdmft+srdft method on FH molecule Relative energy wrt equilibrium (cc-pvtz basis, µ = 0.4 bohr 1 ): relative energy in hartree distance in bohr accvrate PBE lrbb+srpbe = lrbb+srpbe dissociates better than PBE but still error due to srpbe BB Rohr, Toulouse, Pernal, PRA, /37

46 Extension to excited states: a time-independent approach Range-separated DFT for the ground state: E 0 = min Ψ { Ψ ˆT + ˆV ne +Ŵ lr,µ ee Ψ +E sr,µ Hxc [n Ψ] leads to the effective Schrödinger equation: ) (ˆT + ˆV ne +Ŵee lr,µ + ˆV sr,µ Hxc Ψ µ n = E n Ψ µ µ n where E µ n are approximations to the excited-state energies, intermediate between the Kohn-Sham orbital energies (µ = 0) and the exact energies (µ ). These approximate excited-state energies can be refined by extrapolation using the derivative of E µ n wrt µ: E n E µ n + µ 2 de µ n dµ } Rebolini, Toulouse, Teale, Helgaker, Savin, JCP, 2014; MP, 2015; PRA, /37

47 Extension to excited states: a time-independent approach Rebolini, Toulouse, Teale, Helgaker, Savin, JCP, 2014; MP, 2015; PRA, /37 Example for H 2 molecule near the dissociation limit (R = 3R eq ): Excitation energies (ev) nd singlet excited state 1st singlet excited state 1st triplet excited state Range-separation parameter µ (bohr -1 )

48 Extension to excited states: a time-independent approach Rebolini, Toulouse, Teale, Helgaker, Savin, JCP, 2014; MP, 2015; PRA, /37 Example for H 2 molecule near the dissociation limit (R = 3R eq ): Excitation energies (ev) nd singlet excited state 1st singlet excited state extrapolation 1st triplet excited state Range-separation parameter µ (bohr -1 ) = At µ 0.5, correct description of 1st singlet excited state (fundamental gap) and 2nd singlet excited state (double excitation)

49 Extension to excited states: time-dependent approaches 36/37 Linear-response lrmcscf+srdft method for excitation energies: description of charge-transfer and double excitations (Fromager, Knecht, Jensen, JCP, 2013; Hedegård, Heiden, Knecht, Fromager, Jensen, JCP, 2013), inclusion of environmental effects via polarizable embedding (Hedegård, Olsen, Knecht, Kongsted, Jensen, JCP, 2013) Linear-response lrdmft+srdft method for excitation energies: relatively accurate single excitations but less accurate double excitations (Pernal, JCP, 2012) Linear-response lrbse+srdft method for excitation energies: small improvement of valence and Rydberg excitations (Rebolini, Toulouse, submitted, arxiv: )

50 Conclusions and Acknowledgments 37/37 Various approaches proposed for including static correlation in DFT We propose MCSCF+DFT and lrmcscf+srdft hybrid methods There are several variants (e.g., DMFT) and extensions (e.g., excited states) We need improved density-functional approximations in these hybrid methods We need more general (e.g., open shells) and efficient implementations of these hybrid methods Acknowledgments J. Ángyán, F. Colonna, H.-J. Flad, O. Franck, E. Fromager, P. Gori-Giorgi, T. Helgaker, H. J. Aa. Jensen, E. Luppi, B. Mussard, K. Pernal, E. Rebolini, P. Reinhardt, D. Rohr, A. Savin, K. Sharkas, A. Stoyanova, A. Teale

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