Lucia Reining - TDDFT.org

How to describe…………………1) Linear response in TDDFTa) TDDFT and spectroscopyb) TDDFT in practice2) Linear response in MBPTa) From Kohn-Sham to Quasi-Particlesb) Hedin’s equations and the BSEc) BSE in practice3) Comparisons…..4) …..and Combinations5) Outlook

TDDFT and spectroscopyV ext (r,t) ρ(r,t)responseLinear response: ρ ind = χ V extContent of χ ?

What is a spectrum?

hνWhat is a spectrum?

hνWhat is a spectrum?

hνWhat is a spectrum?

hνWhat is a spectrum?

Independent electrons and transitionshνcvIm [χ 0 ] ~ Σ vc || 2 δ (E c -E v -ω)

…but also:Perturbation -- ResponseResponse => Spectrum

Perturbation -- ResponseResponse => SpectrumPerturbation = external macroscopic

Perturbation -- ResponseResponse => SpectrumPerturbation = external macroscopic

Perturbation -- ResponseResponse => SpectrumPerturbation = external macroscopic

Perturbation -- ResponseResponse => SpectrumPerturbation = external macroscopic + inducedinduced = macroscopic

Perturbation -- ResponseResponse => SpectrumPerturbation = external macroscopic + inducedinduced = macroscopic + microscopic

V ind = (δV H /δρ + δV xc /δρ) δρPerturbation -- ResponseResponse => SpectrumPerturbation = external macroscopic + inducedinduced = macroscopic + microscopic

V ind = (δV H /δρ + δV xc /δρ) δρSelf-consistentPerturbation -- ResponseResponse => SpectrumPerturbation = external macroscopic + inducedinduced = macroscopic + microscopic

ρ ind = χ V extχ = χ 0 + χ 0 (v+f xc ) χ

ρ ind = χ V extχ = χ 0 + χ 0 (v+f xc ) χ v=δV H /δρ f xc =δV xc /δρVariation of “internal” potentialsEELS - response to V ext : -Im(χ)Absorption - response to V ext +V Hmacro: Im(χ)χ = χ 0 + χ 0 (v+f xc ) χ ρ ind = χ (V ext +V H macro )v = v – v(G=0)

1b) TDDFT in practicei) Finite system: 1 electron V xc = -V H(T+ V ext + V H –V H ) ϕ = ε ϕf xc = -vχ = χ 0χ = χ 0 + χ 0 (v+f xc ) χ“χ 0 is fine….” (for χand χ)more electrons: screening of f xcv becomes dominant“RPA is fine….” (for χand χ)“TDLDA is fine….” (for χand χ)

) Infinite system: long range important!RPA: α=1RPAABS: χ = χ 0 + χ 0 (v- v 0 ) χEELS: χ = χ 0 + χ 0 (v) χGeneral: v-αv 0Sottile et al (2002)EELS: α=0

EELS : v is dominantOlevano and Reining 2000

EELS : v is dominant“RPA is fine…”“TDLDA is fine…”Olevano and Reining 2000

Absorption: no long range in RPA.Any long range in f xc is therefore dramatic…..

How to get f xc ?hνcvWhat are these?Which interactions ?It is difficult to think in the Kohn-Sham world

So which potential in the “real world”?cv

So which potential ?cv

So which potential ?cv

So which potential ?cvHole - (N-1) electrons

Koopman’s theorem(e.g. Hartee Fock)(-∇ 2 /2 + V ext + V H + V xc ) |n> = E n |n>(-∇ 2 /2 + V ext + V H + Σ x |n> = E n |n>V xc (r) versus Σ x (r,r’)

ExchangeRelaxation - dynamical correlations(-∇ 2 /2 + V ext + V H + Σ x ) |n> = E n |n>(-∇ 2 /2 + V ext + V H + Σ(E n ) ) |n> = E n |n>Σ x (r,r’) versus Σ(r,r’,E n )

Hartree :Hartree - Fock :Beyond :ρ(r)ρ(r,r’)ρ(r,r’,t,t’)

One-particle Green’s function:G(x,x’,t,t’) = -iIf |N> = 1 Slater determinant:for t

From ∂ρ/∂tto:∂/∂t G =……….……leads to G (2) (4 field operators)Introduce potential Σ..... (G -1 = G H-1– Σ)…….to be defined implicitlyPerturbation δV ext Σ = - iGv(δG -1 /δV ext )

From ∂ρ/∂tto:∂/∂t G =……….……leads to G (2) (4 field operators)Introduce potential Σ..... (G -1 = G H-1– Σ)…….to be defined implicitlyPerturbation δV ext Σ = - iGv(δG -1 /δV ext )

Σ = i ∫GWΓW = v + ∫v P WP = - i ∫GGΓΓ = δ + ∫ δΣ/δG GGΓ(G -1 = G H-1– Σ)Hedin’s equations

Σ = i ∫GWΓW = v + ∫v P WP = - i ∫GGΓΓ = δ + ∫ δΣ/δG GGΓ(G -1 = G H-1– Σ)Γ = δ

Σ = i ∫GWΓW = v + ∫v P WP = - i ∫GGΓΓ = δ + ∫ δΣ/δG GGΓ(G -1 = G H-1– Σ)Γ = δ

Σ = i ∫GWΓ = i GWW = v + ∫v P WP = - i ∫GGΓ = -iGGΓ = δ + ∫ δΣ/δG GGΓ(G -1 = G H-1– Σ)Γ = δGW Approximation, Hedin 1965

E c -E v (minimum, eV)DFT-KSΣ=iGWExp.SiliconDiamondMgO0.554.265.31.195.647.81.175.487.83(HF much bigger! Even Silicon……)….this makes the solid so difficult!!!

W = v + ∫v P W(1 -vP) W = vW=(1 -vP) -1 v = ε -1 v = (1+vχ)v“χ” = P + P v χP = - i ∫GGΓ

Γ = δΣ = i ∫GWΓW = v + ∫v P WP = - i ∫GGΓRPA formΓ = δ + ∫ δΣ/δG GGΓ

Γ = δΣ = i ∫GWΓW = v + ∫v P WP = - i ∫GGΓΓ = δ + ∫ δΣ/δG GGΓ-i ∫GGΓ = -i GG -i ∫GG ∫ δΣ/δG GGΓP = P 0 + P 0 δΣ/δG iP Bethe SalpeterΣ = i GWP ≅ P 0 -P 0 W P χ = P 0 + P 0 (v-W) χ

χ = P 0 + P 0 (v-W) χδV H /δρ

χ = P 0 + P 0 (v-W) χδV H /δρδΣ/δG

χ = P 0 + P 0 (v-W) χδV H /δρ δΣ/δG δΣ(12)/δG(34)

Absorption ?cv

Absorption ?cv

Absorption ?cvElectron-hole interactionExcitonic effectsBethe-Salpeter Equation

Absorption ?cVariation of potentialvElectron-hole interactionExcitonic effectsBethe-Salpeter Equation

χ=χ 0 + χ 0 (v-W) χ =

(H el + H hole + H el-hole ) A λ = E λ A λVariation of potentialIm [ε] ~ Σ vc || 2δ (E c -E v -ω)Im [ε] ~ Σ λ | Σ vc A λvc| 2 δ (E λ -ω)->Mixing of transitions->Modification of excitation energies

RPABSE

BSE and TDDFT: different worlds, same physicsTDDFT:χ = χ 0 + χ 0 (v+f xc ) χBSE:(H el + H hole + H el-hole ) A λ = E λ A λ

BSE and TDDFT: different worlds, same physicsTDDFT:χ = χ 0 + χ 0 (v+f xc ) χδV H /δρδV xc /δρBSE:(H el + H hole + H el-hole ) A λ = E λ A λ

BSE and TDDFT: different worlds, same physicsTDDFT:χ = χ 0 + χ 0 (v+f xc ) χδV H /δρδV xc /δρBSE:(H el + H hole + H el-hole ) A λ = E λ A λ(∆E GW + - ) A λ = E λ A λ

BSE and TDDFT: different worlds, same physicsTDDFT:χ = χ 0 + χ 0 (v+f xc ) χδV H /δρδV xc /δρBSE:(H el + H hole + H el-hole ) A λ = E λ A λ(∆E GW + - ) A λ = E λ A λδV H /δρδΣ xc /δG

BSE and TDDFT: different worlds, same physicsTDDFT:χ = χ 0 + χ 0 (v+f xc ) χδV H /δρδV xc /δρBSE:4 χ = 4 χ 0 + 4 χ 0 (v+δΣ xc/ δG) 4 χ(∆E GW + - ) A λ = E λ A λδV H /δρδΣ xc /δG

BSE and TDDFT: different worlds, same physicsTDDFT:χ = χ 0 + χ 0 (v+f xc ) χ(∆E KS + - ) A λ = E λ A λBSE:4 χ = 4 χ 0 + 4 χ 0 (v+ δΣ xc/ δG ) 4 χ(∆E GW + - ) A λ = E λ A λ

BSE and TDDFT: different worlds, same physics…….but TDDFT direct + fast!!!TDDFT:χ = χ 0 + χ 0 (v+f xc ) χ(∆E KS + - ) A λ = E λ A λBSE:4 χ = 4 χ 0 + 4 χ 0 (v+ δΣ xc/ δG ) 4 χ(∆E GW + - ) A λ = E λ A λ

BSE and TDDFT: different worlds, same physics…….but TDDFT direct + fast!!!TDDFT:χ = χ 0 + χ 0 (v+f xc ) χ(∆E KS + - ) A λ = E λ A λf xc = 0 : RPAf xc = δV xcLDA/δρ : TDLDA

TDLDA

Solids: TDLDA ~ RPACan we get f xc from BSE?

BSE and TDDFT: different worlds, same physics…….but TDDFT direct + fast!!!TDDFT:χ = χ 0 + χ 0 (v+f xc ) χ(∆E KS + - ) A λ = E λ A λBSE:(∆E GW + - ) A λ = E λ A λ4 χ = 4 χ 0 + 4 χ 0 (v+ δΣ xc/ δG ) 4 χ

Comparison in“Transition” spaceΦ(n1,n2;r)≡ψ (r)ψ* (r)n1 n2FBSE(n1n2)(n3n4)=−∫drdr'Φ(n1,n3;r)W(r',r)Φ * (n2,n4;r')FTDDFT(n1n2)(n3n4)= ∫ drdr'Φ(n1,n2;r)fxc(r,r')Φ * (n3,n4;r')

Comparison in“Transition” spaceΦ(n1,n2;r)≡ψ (r)ψ* (r)n1 n2FBSE(n1n2)(n3n4)=−∫drdr'v,v’c,c’Φ(n1,n3;r)W(r',r)Φ * (n2,n4;r')FTDDFT(n1n2)(n3n4)= ∫ drdr'Φ(n1,n2;r)fxc(r,r')Φ * (n3,n4;r')v,c v’,c’

Asymptotic long-range behaviorof the xc KernelΦ (v,k (v,k,c,k ,c,k +q +q ;G ;G =0) =0) q → q 0 0O (q (q))→constq→0F (n1n2)(n3n4) q→0(n1n2)(n3n4)( )ffxc ( ,G G' 0) q 02xc ( q ,G = G' = 0) → q 0 O 1 q 2

Reining, Olevano, Rubio, Onida (PRL 2002)

Botti et al (2002)Refraction Index

Botti, Vast, Reining, Olevano, Andreani (PRL, 2002)

Botti et al (2002)

•The lack of a long range term in f xcLDAisrelatively weightless in the EEL but iscrucial in Absorption spectra!Our Explanation⎧ EEL ∝ ε −1 =1+vχ⎨⎩ χ=χ (0) +χ (0) (v+ f xc )χ• In the EEL f xc is added to the full Coulombian that already contains along range contribution⎧ ε M =1−vχ⎨⎩ χ =χ (0) +χ (0) (v + f xc )χCoulombianwithout long range termAbsorption• In the Absorption spectra f xc is added to a Coulombian thatdoes not contain a long range contribution

BSE and TDDFT: merge!TDDFT:χ = χ 0 + χ 0 (v+f xc ) χ

BSE and TDDFT: merge!TDDFT:χ = χ 0 + χ 0 (v+f xc ) χχ = (1- χ 0 (v+f xc )) -1 χ 0

BSE and TDDFT: merge!TDDFT:χ = χ 0 + χ 0 (v+f xc ) χχ = (1- χ 0 (v+f xc )) -1 χ 0χ =X (X - χ 0 (v+f xc )X) -1 χ 0X =Σ vc |vc>

BSE and TDDFT: merge!TDDFT:χ = χ 0 + χ 0 (v+f xc ) χχ =X (X - χ 0 (v+f xc )X) -1 χ 0X =Σ vc |vc>

BSE and TDDFT: merge!TDDFT:χ = χ 0 + χ 0 (v+f xc ) χχ =X (X - χ 0 (v+f xc )X) -1 χ 0X =Σ vc |vc>

BSE and TDDFT:”Mapping theory”TDDFT:(∆E KS + - ) A λ = E λ A λBSE:(∆E GW + - ) A λ = E λ A λ

BSE and TDDFT :”Mapping theory”TDDFT:χ = χ 0 + χ 0 (v+f xc ) χχ =X (X - χ 0 (v+f xc )X) -1 χ 0χ =X (X - χ 0 v X + Τ) -1 χ 0χ 0 with GW energiesT= Σ vc,v’c’ 1/(E c -E v -ω) |vc> > >

Silicon: Optical AbsorptionF. Sottile et al.,Phys Rev. Lett91, 056402 (2003).Parameter-free!!!

Solid Argon: Bound Excitons• Francesco Sottile, PhD thesis (2003)

BSE and TDDFT :”Mapping theory”TDDFT:χ = χ 0 + χ 0 (v+f xc ) χχ =X (X - χ 0 (v+f xc )X) -1 χ 0χ =X (X - χ 0 v X + Τ) -1 χ 0χ 0 with GW energiesT= Σ vc,v’c’ 1/(E c -E v -ω) |vc> > >

Advantages over BSE: Diago,inversion SumAdvantages of this approach:- interpolation,……- change order of operations no 4point- modelisation (long range etc)Reining et al, PRL 88, 066404 (2002); Sottile et al, PRL 91, 056402 (2003)Perturbation theory:- work with χ 0 f xc , not f xc- flexible: χ 0 XAdragna PhD thesis and PRB 68, 165108 (2003),Marini et al PRL 91, 256402 (2003) :X χ 0 )

All of us:•Potentially efficient 2-point simulation of BSE results•Inspired by TDDFT•Is it TDDFT? (We have still QP energies)χ= χ 0QP+ χ 0QP(v+f xcmb) χχ= χ 0KS+ χ 0KS(v+f xc ) χ f xc = f xcmb+ (χ 0QP) -1 -(χ 0KS) -1

ConclusionWe can get DF quantities from MBPT (well known)The aim is to simplify our life:MBPT is more “understandable”DFT is more “efficient”.The difficulty: get the DF answer without solving the wholemany-body problem first!In the case of the BSE – TDDFT this has worked out fine.

How to describe exchange and correlation effectsin the response properties of valence electrons?Answers from TDDFT and MBPTSilvana BottiFabien BrunevalApostolos MarinopoulosValerio OlevanoFrancesco SottileNathalie VastLucia ReiningLaboratoire des Solides IrradiésPalaiseauNANOPHASE-NANOQUANTA