Introduction to multiconfigurational quantum chemistry. Emmanuel Fromager

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1 Introduction to muticonfigurationa quantum chemistry Emmanue Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS Second schoo in computationa physics, Les Houches, France, Second schoo in computationa physics, Les Houches, France Page 1

2 FCI HF MP2 H 2 (1 1 Σ g +, aug cc pvqz) Energy (a.u.) Interatomic distance (a.u.) Second schoo in computationa physics, Les Houches, France Page 2

3 H 2 in the equiibrium geometry: Static correation Ψ 0 = C 0 1σ α g 1σβ g +... where C 0 2 = 98% no static correation In the dissociation imit: H A... H B and NOT H A... H+ B or H+ A... H B φ 1σg (r) = 1 ) (φ 1sA (r) + φ 1sB (r) 2 and φ 1σu (r) = 1 ) (φ 1sA (r) φ 1sB (r) 2 ( 1s α A 1sβ B + 1sα B 1sβ A + 1sα A 1sβ A + 1sα B 1sβ B ) 1σ α g 1σβ g = 1 2 ( 1s α A 1sβ B + 1sα B 1sβ A 1sα A 1sβ A 1sα B 1sβ B ) 1σ α u 1σβ u = 1 2 Ψ 0 = 1 ) ( 1σg α 1σg β 1σu α 1σu β 2 strong static correation Second schoo in computationa physics, Les Houches, France Page 3

4 Exact ground-state wave function in the dissociation imit: Ψ 0 (r 1, σ 1, r 2, σ 2 ) = Ψ 0 (r 1, r 2 ) 1 ) (α(σ 1 )β(σ 2 ) α(σ 2 )β(σ 1 ) 2 where Ψ 0 (r 1, r 2 ) = φ 1σg (r 1 )φ 1σg (r 2 ) φ 1σu (r 1 )φ 1σu (r 2 ) If eectron 1 is bound to H A, around r 2 = r 1 we have r 1 r 2 << r 1 r A << r B r A φ 1sB (r 2 ) = φ 1s ( r 2 r B ) = φ 1s ( r 2 r 1 + r 1 r A + r A r B ) φ 1s ( r A r B ) φ 1sB (r 1 ) = φ 1s ( r 1 r B ) = φ 1s ( r 1 r A + r A r B ) φ 1s ( r A r B ) ( ) Ψ 0 (r 1, r 2 ) φ 1s ( r A r B ) φ 1s ( r 1 r A ) + φ 1s ( r 2 r A ) φ 1s (r) = e ζr Ψ 0 (r 1, r 2 ) e ζ r A r B Concusion: static correation has nothing to do with the cusp condition... It is here a ong-range type of correation (eft/right correation) Second schoo in computationa physics, Les Houches, France Page 4

5 FCI HF CISD 2/2 H 2 (1 1 Σ g +, aug cc pvqz) Energy (a.u.) Interatomic distance (a.u.) Second schoo in computationa physics, Les Houches, France Page 5

6 Muti-Configurationa Sef-Consistent Fied mode (MCSCF) The MCSCF mode consists in performing a CI cacuation with a reoptimization of the orbitas Ψ(κ, C) = e ˆκ ( i ) C i i The MCSCF mode is a muticonfigurationa extension of HF which aims at describing static correation: a imited number of determinants shoud be sufficient. Short-range dynamica correation is treated afterwards (post-mcscf modes) Choice of the determinants: active space H...H 2 eectrons in 2 orbitas (1σ g, 1σ u ) 2/2 Be 2 eectrons in 4 orbitas (2s, 2p x, 2p y, 2p z ) 2/ Second schoo in computationa physics, Les Houches, France Page 6

7 Muti-Configurationa Sef-Consistent Fied mode (MCSCF) Compete Active Space (CAS) for Be: 1s 2 2s 2, 1s 2 2p 2 x, 1s 2 2p 2 y, 1s 2 2p 2 z, if a the determinants are incuded in the MCSCF cacuation CASSCF if a Restricted Active Space (RAS) is used RASSCF The orbita space is now divided in three: douby occupied moecuar orbitas (inactive) φ i, φ j,... 1s active moecuar orbitas φ u, φ v,... 2s, 2p x, 2p y, 2p z unoccupied moecuar orbitas φ a, φ b,... 3s, 3p, 3d, Second schoo in computationa physics, Les Houches, France Page 7

8 Muti-Configurationa Sef-Consistent Fied mode (MCSCF) EX6: In order to iustrate with H 2 the fact that active orbitas can be partiay occupied, show that the active part of the density matrix A D, defined as A D vw = Ψ Êvw Ψ, where Ψ = 1 ) ( 1σ α 1 + c 2 g 1σβ g c 1σα u 1σβ u, equas A D = c 2 0 2c c 2 Note: In the particuar case of a singe determinanta wave function (c = 0) the active density matrix reduces to Second schoo in computationa physics, Les Houches, France Page 8

9 Iterative optimization of the orbita rotation vector κ and the CI coefficients C i : Ψ (0) = C (0) i i normaized starting wave function i Ψ(λ) = e ˆκ Ψ(0) + ˆQ δ 1 + δ ˆQ δ convenient parametrization λ = ˆQ = 1 Ψ (0) Ψ (0), δ = i δ i i, Ψ (0) ˆQ δ = 0, Ψ(λ) Ψ(λ) = 1. κ pq. δ i. p > q MCSCF energy expression: E(λ) = Ψ(λ) Ĥ Ψ(λ) Variationa optimization: E [1] λ + = and E c[1] λ + = E(λ) δ λ+ Eo[1] λ + E c[1] λ + = 0 where E o[1] λ + = E(λ) κ λ Second schoo in computationa physics, Les Houches, France Page 9

10 Newton method: E(λ) E(0) + λ T E [1] λt E [2] 0 λ E[1] λ E [1] E[2] 0 λ + = 0 E [2] 0 λ + }{{} = E[1] 0 Newton step Convergence reached when E [1] 0 = 0 ( ) EX7: Show that E o[1] 0,pq = Ψ(0) [Êpq Êqp, Ĥ] Ψ(0) and E c[1] 0 = 2 H CAS E(0) C (0) where H CAS ij = i Ĥ j and C(0) =. C (0) i. Note: E o[1] 0 = 0 is known as generaized Briouin theorem Second schoo in computationa physics, Les Houches, France Page 10

11 Muti-Configurationa Sef-Consistent Fied mode (MCSCF) EX8: We consider in this exercise a different parametrization of the MCSCF wave function: Ψ(κ, S) = e ˆκ e Ŝ Ψ (0) where Ŝ = K ) S K ( K Ψ (0) Ψ (0) K, Ψ (0) K = 0, K K = δ KK and i i = Ψ (0) Ψ (0) + i K K K Derive the corresponding MCSCF gradient and show that the optimized MCSCF wave functions obtained with this parametrization and the previous one are the same. Note: This doube exponentia form is convenient for computing response properties at the MCSCF eve and performing state-averaged MCSCF cacuations Second schoo in computationa physics, Les Houches, France Page 11

12 FCI HF CISD 2/2 MCSCF 2/2 H 2 (1 1 Σ g +, aug cc pvqz) Energy (a.u.) Interatomic distance (a.u.) Second schoo in computationa physics, Les Houches, France Page 12

13 Muti-Reference Perturbation Theory (MRPT) Muticonfigurationa extension of MP2: unperturbed wave function Φ 0 Ψ (0) unperturbed energy E (0) = 2 i ε i??? perturbers D??? zeroth-order excited energies E (0) + ε a + ε b ε i ε j??? unperturbed Hamitonian ˆF E (0) Φ 0 Φ 0 + D D ˆF D D D??? Second schoo in computationa physics, Les Houches, France Page 13

14 MCSCF Fock operator: f pq = h pq + rs An introduction to post-hartree-fock quantum chemistry (II) Muti-Reference Perturbation Theory (MRPT) ˆF = f pq Ê pq p,q ( pr qs 1 2 pr sq ) D rs and D rs = Ψ (0) Êrs Ψ (0) The density matrix is spit into inactive and active parts: D = I D + A D I D ij = 2δ ij, A D uv = Ψ (0) Êuv Ψ (0) In genera, the MCSCF wave function Ψ (0) is not eigenfunction of the MCSCF Fock operator (which is a one-eectron operator): Ĥ CAS Ψ (0) = E 0 Ψ (0), E 0 : converged MCSCF energy Ĥ CAS = ˆP CAS Ĥ ˆP CAS, ˆP CAS = i i i Second schoo in computationa physics, Les Houches, France Page 14

15 Muti-Reference Perturbation Theory (MRPT) EX9: Show that the Hamitonian and the Fock operator reduce, within the CAS space, to ( Ĥ CAS = 2 h ii + ( ) ) CAS 2 ij ij ij ji ˆP i i,j + uv ( h uv + i ( ) ) 2 ui vi ui iv Ê uv ˆP CAS uvwx ) uw vx (Êuv Ê wx δ vw Ê ux ˆP CAS ˆF CAS = ( 2 i h ii + 2 i,j ( ) ) CAS 2 ij ij ij ji ˆP + uv ( h uv + i ( ) ) 2 ui vi ui iv Ê uv ˆP CAS + iuv + uvwx ( ) 2 ui vi ui iv A CAS D uv ˆP ( uw vx 1 2 uw xv ) A D wx Ê uv ˆP CAS Second schoo in computationa physics, Les Houches, France Page 15

16 Muti-Reference Perturbation Theory (MRPT) MRPT2 based on ˆF is known as CASPT2: Ĥ 0 = Ψ (0) ˆF Ψ (0) }{{} Ψ(0) Ψ (0) + ˆQĤ0 ˆQ E (0) At the CASPT2 eve the zeroth-order energy does not equa the MCSCF one (as for MP2) MRPT2 based on ĤCAS is known as N-Eectron Vaence PT2 (NEVPT2): Ĥ 0 = Ψ (0) ĤCAS Ψ (0) }{{} Ψ(0) Ψ (0) + ˆQĤ0 ˆQ E 0 At the NEVPT2 eve the zeroth-order energy equas the MCSCF one Second schoo in computationa physics, Les Houches, France Page 16

17 Muti-Reference Perturbation Theory (MRPT) Choice of the perturbers: eight types of excitation subspaces for the first-order correction to the wave function S (0) ij,ab (j i, b a) S( 1) i,ab (b a) S (+1) ij,a (j i) S ( 2) ab (b a) S (+2) ij (j i) S (0) i,a S ( 1) a S (+1) i Contracted perturbation theory: for exampe, states in S ( 2) ab are expressed as Ê au Ê bv Ψ (0) = i C (0) i Ê au Ê bv i optimized at the MCSCF eve (kept frozen in the MRPT2 cacuation) Second schoo in computationa physics, Les Houches, France Page 17

18 Unperturbed Hamitonian in CASPT2: Ĥ 0 = Ψ (0) ˆF ( ( ) Ψ (0) Ψ (0) Ψ (0) + ˆP CAS Ψ (0) Ψ ) (0) ˆF ˆP CAS Ψ (0) Ψ (0) ( + k, ˆP S (k) ) ˆF ( k, ˆP S (k) ) where ˆP S (k) projects onto the excitation subspace S (k). Intruder state probem: Ψ (1) = P Ψ (0) P Ψ(0) P Ĥ Ψ(0) E (0) E (0) P where Ψ (0) P Ĥ Ψ(0) is sma and E (0) E (0) P Introduction of a eve shift parameter ε: E (0) E (0) P E (0) E (0) P + ε Intruder state probems can be avoided when using a different zeroth-order Hamitonian Second schoo in computationa physics, Les Houches, France Page 18

19 ELSEVIER 2 December 1994 Chemica Physics Letters 230 (1994) C H E MI C A L P H Y SI C S L E T T E R S The Cr2 potentia energy curve studied with muticonfigurationa second-order perturbation theory K. Andersson, B.O. Roos, P.-A. Mamqvist, P.-O. Widmark Department of Theoretica Chemistry, Chemica Centre, University ofund, P.O. Box 124, S Lund, Sweden Received 30 August Second schoo in computationa physics, Les Houches, France Page 19

20 s B O N D DIS W U C E (.4 U Fig. 4. The potentia curve of Cr, obtained with CASPT2 above and the weight of the CASSCF wavefunction in the first-order wavefunction beow. The basis set is of AN0 type of size 8s7p6d4f and the active space is formed by 3d and 4s orbitas. The 3s, 3p correation effects have been taken into account but no reativis- tic effects are incuded Second schoo in computationa physics, Les Houches, France Page 20

21 ELSEVIER 27 October 1995 Chemica Physics Letters 245 (1995) CHEMICAL PHYSICS LETTERS Muticonfigurationa perturbation theory with eve shift - the Cr 2 potentia revisited Bj~rn O. Roos, Kerstin Andersson Department of Theoretica Chemistry. Chemica Centre, P.O. Box 124, S-221 O0 Lund, Sweden Received 26 June 1995; in fina form 25 August Second schoo in computationa physics, Les Houches, France Page 21

22 II , I I I I BOND DISTANCE (au) Fig. 2. The potentia curve for the ground state of Cr 2 for three vaues of the eve shift, 0.05 (soid ine), O.10, (dashed ine), and 0.20 H (dotted ine). The LS correction has been appied. The ower diagram gives the corresponding weight of the CASSCF reference function Second schoo in computationa physics, Les Houches, France Page 22

23 Defining a two-eectron Fock operator: Dya s Hamitonian ε i = â i,σ Ψ (0) Ĥ â i,σψ (0) + E 0 ε a = â a,σψ (0) Ĥ â a,σψ (0) E 0 Ĥ D = i ε i Ê ii + a ε a Ê aa HF-type Fock operator contribution ( + 2 h ii + i i,j + ( h uv + uv i ( ) ) 2 ij ij ij ji 2 i ε i ( ) ) 2 ui vi ui iv Ê uv uvwx ) uw vx (Êuv Ê wx δ vw Ê ux ĤD Ψ (0) = ĤCAS Ψ (0) = E 0 Ψ (0) When Ψ (0) reduces to the HF determinant Φ 0, ε i and ε a equa the usua canonica HF orbita energies (Koopmans theorem) and Dya s Hamitonian reduces to the (shifted) Fock operator Ĥ D ˆF + E HF 2 i ε i = ˆF D Second schoo in computationa physics, Les Houches, France Page 23

24 Doube contraction: keep one perturber per excitation subspace. For exampe in S ( 2) ab {ÊauÊbv Ψ (0) } uv Ψ ( 2) ab = uv C ( 2) ab,uvêauêbv Ψ (0) contracted strongy contracted Advantage: the number of perturbers is reduced and they are now orthonorma. Choice of the C ( 2) ab,uv coefficients: anaogy with a contracted formuation of MP2. EX10: Show that Ĥ Φ 0 projected onto the douby-excited determinants can be rewritten as D D Ĥ Φ 0 = 1 2 D a,b,i,j ab ij ÊaiÊbj Φ 0 = a b i j Ψ (0) ij,ab where Ψ (0) ij,ab = ˆP (0) Ĥ Φ 0 S ij,ab = (1 1 2 δ ij)(1 1 ( ) 2 δ ab) ab ij ÊaiÊbj + ab ji ÊbiÊaj Φ Second schoo in computationa physics, Les Houches, France Page 24

25 EX11: The purpose of this exercise is to show that MP2 can be rewritten in a contracted form, using the perturber wave functions Ψ (0) ij,ab and the shifted Fock operator ˆF D. (i) Prove that the square norm of Ψ (0) ij,ab equas N (0) ij,ab = Ψ(0) ij,ab Ψ(0) ij,ab = 4(1 1 2 δ ij)(1 1 ( ) 2 δ ab) ab ij 2 + ab ji 2 ab ij ab ji (ii) The unperturbed Hamitonian in the contracted MP2 is defined as Ĥ 0 = Φ 0 ˆF D Φ 0 Φ 0 Φ 0 + a b i j Ψ (0) ij,ab ˆF D Ψ (0) ij,ab N (0) ij,ab Ψ (0) ij,ab Ψ(0) ij,ab N (0) ij,ab Show that the first- and second-order corrections to the energy equa E (1) = 0 and E (2) = a b i j N (0) ij,ab ε i + ε j ε a ε b (iii) Concude that the standard MP2 energy expression is thus recovered through second order Second schoo in computationa physics, Les Houches, France Page 25

26 Perturber wave functions: An introduction to post-hartree-fock quantum chemistry (II) Unperturbed Hamitonian: Ĥ 0 = E 0 Ψ (0) Ψ (0) + k, Strongy Contracted NEVPT2 Ψ (k) E (k) N (k) First and second-order energy corrections: = ˆP (k) Ĥ Ψ (0), S Ψ (k) Ψ (k), E (k) N (k) = Ψ (k) Ψ (k) = Ψ(k) ĤD Ψ (k) N (k) E (1) = 0 and E (2) = k, 1 N (k) Ψ (k) Ĥ Ψ(0) 2 E 0 E (k) = k, N (k) E 0 E (k) Expicit expressions for the zeroth-order energies: Ĥ D = ˆF D + A Ĥ D, A Ĥ D = uv ( h uv + i ( 2 ui vi ui iv ) ) Ê uv uvwx ) uw vx (Êuv Ê wx δ vw Ê ux Second schoo in computationa physics, Les Houches, France Page 26

27 N (k) E (k) = Ψ (k) An introduction to post-hartree-fock quantum chemistry (II) ĤD ˆPS (k) Ĥ Ψ (0) = Ψ (k) [ĤD, ˆP S (k) Ĥ] Ψ (0) + Ψ (k) ˆP (k) Ĥ ĤD Ψ (0) S }{{} = Ψ (k) [ĤD, ˆP (k) Ĥ] Ψ (0) + E 0 N (k) S E 0 Ψ (0) the denominators in the NEVPT2 energy expansion thus become E 0 E (k) = 1 N (k) = 1 N (k) S (0) ij,ab subspace: Ψ (k) Ψ (k) [ĤD, ˆP (k) Ĥ] Ψ (0) S [ ˆF D, ˆP S (k) Ĥ] Ψ (0) 1 N (k) Ψ (k) [ A Ĥ D, ˆP (k) Ĥ] Ψ (0) S [ ˆF D, ˆP (0) Ĥ] Ψ (0) = (ε a + ε S b ε i ε j ) Ψ (0) ij,ab, [A Ĥ D, ˆP (0) Ĥ] Ψ (0) = 0 ij,ab S ij,ab E 0 E (0) ij,ab = ε i + ε j ε a ε b MP2-type denominator Second schoo in computationa physics, Les Houches, France Page 27

28 S (+1) ij,a subspace: [ ˆF D, ˆP (+1) Ĥ] Ψ (0) = (ε a ε i ε j ) Ψ (+1) S ij,a, [A Ĥ D, ˆP (+1) ij,a S ij,a Ĥ] Ψ (0) = 0 ε (+1) ij,a = 1 N (+1) Ψ (+1) ij,a [A Ĥ D, ˆP (+1) S ij,a ij,a Ĥ] Ψ (0) effective one-eectron energy E 0 E (+1) ij,a = ε i + ε j ε a ε (+1) ij,a Second schoo in computationa physics, Les Houches, France Page 28

29 JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER NOVEMBER 2002 n-eectron vaence state perturbation theory: A spiness formuation and an efficient impementation of the strongy contracted and of the partiay contracted variants Ceestino Angei and Renzo Cimiragia a) Dipartimento di Chimica, Università di Ferrara, Via Borsari 46, I Ferrara, Itay Jean-Pau Marieu IRSAMC, Laboratoire de Physique Quantique, Université Pau Sabatier, 118 route de Narbonne, Tououse Cedex, France Second schoo in computationa physics, Les Houches, France Page 29

30 Energy (a.u.) FCI HF CISD 2/2 MCSCF 2/2 sc NEVPT2 H 2 (1 1 Σ g +, aug cc pvqz) Interatomic distance (a.u.) Second schoo in computationa physics, Les Houches, France Page 30

31 Non-perturbative post-mcscf modes Muti-Reference CI (MRCI): Ψ(C 1,..., C i,...) = ( C 0,i i + i S C S,i Ŝ i + D C D,i ˆD i +... ) where the CI coefficients C 0,i, C S,i, C D,i,... are optimized variationay. In the so-caed "internay contracted formuation", the number of CI coefficients to be optimized is reduced as foows C i = CC (0) i, C (0) i : CI coefficients optimized at the MCSCF eve Ψ(C) = C 0 Ψ (0) + S C S Ŝ Ψ (0) + D C D ˆD Ψ (0) +... MRCI is accurate for sma moecues but it has size-consistency probems and it is ess and ess practica for arge moecuar systems Second schoo in computationa physics, Les Houches, France Page 31

32 Non-perturbative post-mcscf modes Muti-Reference CC (MRCC): Ψ(t 1,..., t i,...) = i C i e ˆT i i state universa formuation In the internay contracted formuation Ψ(t) = e ˆT Ψ (0) ˆT i = ˆT and C i = C (0) i Probem: the BCH expansion does not truncate! Second schoo in computationa physics, Les Houches, France Page 32

33 Post-HF quantum chemistry for excited states Time-dependent response theory: Ĥ(t) = Ĥ + ε ˆV cos(ωt) Ĥ(t) Ψ(t) = i d dt Ψ(t) Ĥ(t) Ψ(t) i d dt Ψ(t) = Q(t) Ψ(t) where Ψ(t) = e +i t t0 Q(t)dt Ψ(t), d dt Ψ(t) = 0 when ε = 0 Q(t) = Ψ(t) Ĥ(t) i d dt Ψ(t) is referred to as time-dependent quasienergy Q = t1 t 0 Q(t)dt is simpy referred to as quasienergy The exact time-dependent wave function Ψ(t) can be obtained in two ways: ε δq = 0 Ĥ(t) Ψ(t) i d dt Ψ(t) = Q(t) Ψ(t) variationa formuation non-variationa formuation Second schoo in computationa physics, Les Houches, France Page 33

34 Post-HF quantum chemistry for excited states Approximate parametrized time-dependent wave function: Ψ(λ(t)) Linear response: λ(t) = λ(t) ε ε }{{} poes excitation energies Linear response equations for variationa methods (TD-HF, TD-MCSCF) d dε δq = 0 0 Linear response equations for non-variationa methods (ike CC) can be obtained from the (CC) quasienergy expression to which conditions (CC equations) scaed by Lagrange mutipiers are added Second schoo in computationa physics, Les Houches, France Page 34

35 Post-HF quantum chemistry for excited states State-averaged MCSCF mode: simutaneous optimization of the ground and the owest n 1 excited states at the MCSCF eve. Iterative procedure: n initia orthonorma states are buit from the same set of orbitas. Ψ (0) I = i C (0) I,i i, I = 1,..., n Doube exponentia parametrization: Ψ I (κ, S) = e ˆκ e Ŝ Ψ (0) I where Ŝ = n J=1 K>J S KJ ( Ψ (0) K Ψ(0) J ) Ψ(0) J Ψ(0) K and i i i = K Ψ (0) K Ψ(0) K Second schoo in computationa physics, Les Houches, France Page 35

36 Post-HF quantum chemistry for excited states State-averaged energy: E(κ, S) = n w I Ψ I (κ, S) Ĥ Ψ I(κ, S) I=1 where w I are arbitrary weights. In the so-caed "equa weight" state-averaged MCSCF cacuation w I = 1 n. Variationa optimization: E(κ, S) κ = E(κ, S) S = 0 Short-range dynamica correation can then be recovered, for exampe, at the Muti-State CASPT2 or NEVPT2 eves Second schoo in computationa physics, Les Houches, France Page 36

37 Summary and outook Post-HF methods are as various as eectron correation effects They are usuay spit into two types: singe- and muti-reference methods Singe-reference methods are used for modeing short-range and ong-range dynamica correation: the state-of-the-art method is Couped Custer The description of the Couomb hoe is significanty improved when using expicity correated methods (faster convergence with respect to the basis set). Muti-reference methods describe both static and dynamica correations (muticonfigurationa systems). The muti-reference extension of singe-reference methods is neither obvious nor unique. It is sti a chaenging area Second schoo in computationa physics, Les Houches, France Page 37

38 Summary and outook High accuracy in wave function theory usuay means high computationa cost. Current deveopments: paraeization, atomic-orbita-based formuations (inear scaing), density-fitting (two-eectron integras),... In this respect DFT is much more appeaing to chemists and physicists Kohn-Sham DFT is a singe-reference method which shoud be abe, in its exact formuation, to describe muticonfigurationa systems Standard approximate exchange-correation functionas cannot treat adequatey static correation The deveopment of rigorous hybrid MCSCF-DFT modes is promising but sti quite chaenging. Indeed, in wave function theory, the muticonfigurationa extension of the HF method is ceary formuated. What the muticonfigurationa extension of Kohn-Sham DFT shoud be is uncear. Current deveopments: DFT based on fictitious partiay-interacting systems, ensembe DFT, Second schoo in computationa physics, Les Houches, France Page 38