RPA step by step. Bastien Mussard, János G. Ángyán, Sébastien Lebègue CRM 2, FacultÃl des Sciences UniversitÃl Henry PoincarÃl, Nancy

Transcription

1 RPA step by step Bastien Mussard, János G. Ángyán, Sébastien Lebègue CRM, FacultÃl des Sciences UniversitÃl Henry PoincarÃl, Nancy Eötvös University Faculty of Science Budapest, Hungaria 0 mars 01 bastien.mussard@uhp-nancy.fr

2 Philosophy of the talk less math physical historical? more math physics gets hidden? add the aspect of equivalence to rccd Dialects Different ways to derive the RPA equation - hand waving (historical) - physics (equation of motion) - chemistry (in context of ACFDT) The main idea : - RPA treats the excitations - via FDT, knowing all the exc. gives the correlation energy /14

3 RPA : the origins The way it was thought In a gaz of electron : organized oscillations - consequences of long-range Coulomb potential - explicit (LR collective behavior) AND (SR screened interaction) Ĥ = Ĥ part + Ĥ inter + Ĥ field Ĥ = Ĥ part + Ĥ osc + Ĥ int part PhysRev. 8, 65 (1951) 3/14

4 RPA : the origins The way it was thought In a gaz of electron : organized oscillations - consequences of long-range Coulomb potential - explicit (LR collective behavior) AND (SR screened interaction) Ĥ = Ĥ part + Ĥ inter + Ĥ field Ĥ = Ĥ part + Ĥ osc + Ĥ int part Contribution from particules in phase with the oscillation - the rest, which have random phases, are zeroed out PhysRev. 8, 65 (1951) 3/14

5 In the physics department... [ [ EOM for Q 0 = q : 0 δq, H, Q ]] [ 0 = ɛ q0 0 δq, Q ] 0 RPA : an approximation for Q on a basis of operators A a, A a} (creation and destruction of p-h pairs) Q = a [ ] X a A a Y a A a Fetter and Walecka, 59, /14

6 In the physics department... [ [ EOM for Q 0 = q : 0 δq, H, Q ]] [ 0 = ɛ q0 0 δq, Q ] 0 RPA : an approximation for Q on a basis of operators A a, A a} (creation and destruction of p-h pairs) [ δxaaa δy aa a] Q = a Block Matrix Form ( A ab B a b [ ] X a A a Y a A a δq= a δx =δy δx =0 and δy =0 =0 and =0 [ [ ]] [ Aa, H,A b Xb [A a,[h,a b ]]Y b =ɛ q0 X b Aa,A b] [ A,[ a H,A b]] [ ] [ ] Xb + A a,[h,a b] Y b =ɛ q0 Y b A a,a b B ab A a b ) ( Xb Y b ) = ɛ q0 ( Nab 0 0 N a b ) ( Xb Y b ) Fetter and Walecka, 59, /14

7 Range Separated Hybrid context Constatation DFT : good at short range Wavefunction methods : suitable at long range The idea Mix the two : } E E[v int ] = E[ρ] = min φ ˆT + ˆV ne + ŴHx lr + Ŵc lr φ + E sr φ Hxc[n φ ] where φ 0 via a Euler-Lagrange equation with : Ĥ 0 = ˆT + ˆV ne + ˆV lr Hx + ˆV sr Hxc A whole spectrum of possibilities erf Μ x x 1 erf Μ x x srpbe+lrmp srtpss+lrccsd and srdft+lrrpa... PRA. 7, (005) 5/14

8 Correlation energy Remember : E RSH = min φ ˆT + ˆV } ne + ŴHx lr + Ŵ c lr φ + EHxc sr [n φ] φ Adiabatic Connection } E λ = min ψ ˆT + ˆV ne + ŴHx lr + λŵc lr ψ + E sr ψ Hxc[n ψ ] PRA. 8, 0350 (010) 6/14

9 Correlation energy Remember : E RSH = min φ ˆT + ˆV } ne + ŴHx lr + Ŵ c lr φ + EHxc sr [n φ] φ Adiabatic Connection 1 0 E λ = min ψ ˆT + ˆV ne + ŴHx lr + λŵc lr ψ + E sr ψ - Integrate the derivate (Hellmann-Feynman) E λ λ =E λ=1 E λ=0 = 1 0 ψ λ Ŵ lr c ψ λ - Closer look on E RSH E λ=1 =E RSH +E lr c E λ=0 =min ψ ˆT + ˆV ne+ŵhx lr ψ +E sr Hxc [n ψ]} ψ 1 [ ] Ec lr = dλ ψ λ Ŵc lr ψ λ φ 0 Ŵc lr φ 0 0 } Hxc[n ψ ] PRA. 8, 0350 (010) 6/14

10 Correlation energy 1 [ Remember : Ec lr = dλ ψ λ Ŵc lr 0 ψ λ φ 0 Ŵ lr c φ 0 ] or, in space-spin coordinates : Ec lr = 1 ] dλ w lr 1,,1, (1,;1, ) [n x λ,,(1,;1 lr, ) nlr λ=0,,(1,;1, ) Ec lr = 1 dλtr VP c,λ } = 1 dλ rq sp (P c,λ ) pq,rs pq,rs PRA. 8, 0350 (010) 7/14

11 Correlation energy 1 [ Remember : Ec lr = dλ ψ λ Ŵc lr 0 ψ λ φ 0 Ŵ lr c φ 0 ] or, in space-spin coordinates : Ec lr = 1 ] dλ w lr 1,,1, (1,;1, ) [n x λ,,(1,;1 lr, ) nlr λ=0,,(1,;1, ) Ec lr = 1 dλtr VP c,λ } = 1 dλ rq sp (P c,λ ) pq,rs pq,rs Fluctuation Dissipation Theorem - Manipulate n G 1 (τ) n 1 G (τ) n, δ n 1 Π(τ)=i[G (τ) G 1 (τ)g 1 (τ)] n =iπ(τ)+n 1 n 1 δ n 1 Pc,λ lr dω [ ] = Π lr πi λ(ω) Π lr 0 (ω) + PRA. 8, 0350 (010) 7/14

12 RPA Up to now from DFT to an RSH method in an ACFDT framework E lr c = 1 dλ dω π w lr c x [ Π lr λ (ω) Πlr 0 (ω)] + We haven t talked about RPA yet... This is only the framework The approximation (seen from here) A solution of the Dyson equation within RPA : G lr 1,λ = G lr 1,0 Basically : G 1,λ is constant over the adiabatic connection E lr c = 1 dλ dω π [ ] wc lr x Π lr λ(ω) Π lr 0 (ω) 8/14

13 Matrix formulation for Π 0 (ω) and Π λ (ω) Lehmann Representation [ Π 0 φ ] (1 )φa(1)φ (ω)= i a ( )φ i () + φ i ( )φa()φ a (1 )φ i (1) ω ɛ ai +iη ω ɛ ai iη i,a [Π 0 (ω)] pq,rs = 1,,1, φp(1 )φ q (1)Π 0 (ω)φ r ( )φ s () ( ) 1 0 = 0 1 ( ) ɛ 0 Λ 0 = 0 ɛ Π 0 (ω) 1 = ω Λ 0 9/14

14 Matrix formulation for Π 0 (ω) and Π λ (ω) Lehmann Representation [ Π 0 φ ] (1 )φa(1)φ (ω)= i a ( )φ i () + φ i ( )φa()φ a (1 )φ i (1) ω ɛ ai +iη ω ɛ ai iη i,a [Π 0 (ω)] pq,rs = 1,,1, φp(1 )φ q (1)Π 0 (ω)φ r ( )φ s () ( ) 1 0 = 0 1 ( ) ɛ 0 Λ 0 = 0 ɛ Π RPAx λ (ω) 1 = Π 0 (ω) 1 λw Π 0 (ω) 1 = ω Λ 0 Bethe-Salpeter Π λ (ω) 1 = Π 0 (ω) 1 f λ,hxc ( ) K K Π drpa λ (ω) 1 = Π 0 (ω) 1 V = λv K K K ia,jb = ab ij ( ) A B W = B A A ia,jb = ia jb = ib aj ib ja B ia,jb = ab ij = ab ij ab ji 9/14

15 Matrix Formulation for P c = n,λ n,0 Spectral Representation Remember : P lr c,λ = dω [ Π lr πi λ (ω) Π lr (ω)] 0 We only have Π 1 0 and Π 1 λ for now... Given the eigenvalues and eigenvectors : (Λ 0 + λf) C λ,n = ω λ,n C λ,n, with C λ,n = ( Xn,λ Construct Π λ Π λ (ω) = Cn,λ C T n,λ n ω ω n,λ + iη C n,λc T } n,λ ω + ω n,λ iη Contour integration on the upper-half plane P RPA c,λ = [ ] C n,λ C T n,λ C n,0 C T n,0 n Y n,λ ) JCTC. 7, 3116 (011) 10/14

16 Flavors of RPA } Remember : Ec lr = 1 dλtr VPc,λ Formally Ec lr = 1 dλtr VP c,λ } = 1 4 dλtr WP c,λ } Within RPA P RPA c,λ Ec drpa-i = 1 Ec drpa-ii = 1 Ec RPAx-I = 1 Ec RPAx-II = 1 4 is not fully anti-symmetric, so : dλtr dλtr dλtr dλtr VP drpa-i c,λ } } WP drpa-ii c,λ } VP RPAx-I c,λ WP RPAx-II c,λ } - Historical one - similar to SOSEX - recently introduced - Almost as old as drpa-i 11/14

17 Flavors of RPA Remember : (Λ 0 + λf) C λ,n = ω λ,n C λ,n Plasmon Formulae (for drpa-i and RPAx-II) Given the orthonormality constraint C T λ,m C λ,n = δ mn ω λ,n = C T λ,n (Λ 0 + λf) C λ,n dω λ,n dλ = CT λ,nfc λ,n 1/14

18 Flavors of RPA Remember : (Λ 0 + λf) C λ,n = ω λ,n C λ,n Plasmon Formulae (for drpa-i and RPAx-II) Given the orthonormality constraint C T λ,m C λ,n = δ mn ω λ,n = C T λ,n (Λ 0 + λf) C λ,n dω λ,n dλ = CT λ,nfc λ,n Remember : Ec lr = 1 dλtr V n [ C n,λ C T n,λ C n,0c T n,0 ] } Ec plasmon = 1 dλ n Tr } C n,λ FC T n,λ C n,0 FC T n,0 E plasmon c = 1 n ω RPA λ,n } ωλ,n TDA 1/14

19 Equivalence to (d)rccd rccd energy and amplitude E rccd c = 1 4 ij ab 0 = ij ab + tik ac ɛ ck δ bc δ jk + ɛ ck δ ac δ ik tkj cb + ic ak tkj cb + tik ac jc bk + tik ac kl cd tlj db With the same notation E rccd c = 1 4 Tr(BT) 0 = B + (A + ɛ)t + T(A + ɛ) + TBT Equivalent recently proven that this equation can be derived from RPA With the T found this way, E drccd c = E drpa c 13/14

20 Conclusion/Project A lot of different RPAs can be derived - ACFDT formulae - Plasmon derivation (analytic λ-integration) - Analytic ω-integration - in connexion with rccd All have already been implemented (Paris/Nancy) The idea now (and here) : - Search more in the direction of RPA=rCCD - Derivation and Implementation of the forces 14/14