Linear response theory and TDDFT

Transcription

1 Linear response theory and TDDFT Claudio Attaccalite CECAM Yambo School 2013 (Lausanne)

2 Motivations: +- hν Absorption Spectroscopy Many Body Effects!!!

3 Motivations(II):Absorption Spectroscopy Absorption linearly related to the Imaginary part of the MACROSCOPIC dielectric constant (frequency dependent)

4 Outline Response of the system to a perturbation Linear Response Regime How can we calculate the response of the system? IP, local field effects and Time Dependent DFT Some applications and recent steps forward Conclusions

5 Spectroscopy

6 From Maxwell equation to the response function Materials equations: From Gauss's law: D(r,t )=ϵ 0 E (r, t)+ P (r, t) E (r, t)=4 π ρtot (r, t) D(r,t )=4 π ρext (r, t) Electric Field Electric Displacement Polarization In general: P(r,t )= χ (t t ',r, r ') E (t ' r ' )dt ' dr ' + dt 1 dt 2 χ 2 (...) E (t 1 ) E (t 2)+O ( E 3 ) For a small perturbation we consider only the first term, the linear response regime P(r,t )= χ (t t ',r, r ') E (t ' r ') dt ' dr ' In Fourier space: P(ω)=ϵ0 χ (ω) E (ω)=ϵ0 (ϵ(ω) 1) E (ω) D(ω)=ϵ0 ϵ(ω) E (ω)

7 Response Functions Moving from Maxwell equation to linear response theory we define D(ω) δ V ext (ω) ϵ(ω)= = ϵ0 E (ω) δ V tot (ω) δ V tot (ω) ϵ (ω)= δ V ext (ω) 1 where V tot ( r, t )=V ext ( r,t )+V ind ( r,t ' ) The induced charge density results in a total potential via the Poisson equation. V tot ( r t )=V ext ( r t )+ dt ' d r v ( r r )ρind ( r t ) ' δ ρind ϵ (ω)=1+ v δv ext δρind ϵ(ω)=1 v δ V tot 1 ' ' ' Our goal is to calculate the derivatives of the induced density respect to the external potential

8 The Kubo formula 1/2 H =H 0 + H ext (t)=h 0 + d r ρ(r) V ext (r, t) We star from the time-dependent Schroedinger equation: ψ i =[ H 0 + H ext (t)] ψ(t) t...and search for a solution as product of the solution for Ho plus an another function (interaction representation)... ih t ψ(t)=e ψ(t ) 0 ) ih t ψ(t ih t ext (t) ψ(t )= H ) i =e H ext (t)e ψ(t t and we can write a formal solution as: t i t H ext (t)dt ψ(t)=e 0 0) ψ(t

9 The Kubo formula 2/2 For a weak perturbation we can expand: t i t H ext (t)dt ψ(t)=e 0 1 t 0 )=[1+ t dt ' H ext (t ' )+O ( H 2ext )] ψ(t 0) ψ(t i 0 And now we can calculate the induced density: t ) ρ(t) ) ρ 0 i t [ρ(t), H ext (t ' )] +O( H 2ext ) ρ(t )= ψ(t ψ(t 0...and finally... t ρind (t )= i t 0 ρind (t )=ρ(t) ρ0 dr [ρ(r, t ),ρ(r ' t ')] V ext (r ', t ') Kubo Formula (1957) ind δ ρ (r, t ) ' ' χ ( r t, r t )= = i [δρ(r, t)δρ(r ' t ')] δ V ext (r ', t ' ) The linear response to a perturbation is independent on the perturbation and depends only on the properties of the sample

10 How to calculated the dielectric constant We want to calculate: ind δ ρ (r, t ) ' ' χ ( r t, r t )= = i [δ ρ(r,t )δ ρ(r ' t ' )] δ V ext (r ', t ' ) We expand X in an independent particle basis set χ ( r t, r ' t ' )= i, j,l,m k χ i, j, l, m, k ϕi, k (r )ϕ j, k (r ) ϕl, k (r ' )ϕ m, k (r ') ρ i, j, k χi, j, l, m, k= V l,m,k Quantum Theory of the Dielectric Constant in Real Solids Adler Phys. Rev. 126, (1962) The Von Neumann equation (see Wiki ρ k (t ) eff i =[ H k +V, ρ k ] t ρ k (t )= i f (ϵk, i ) ψi, k ψi, k What is Veff?

11 Independent Particle Independent Particle ρ i, j, k χi, j, l, m, k= V l,m,k ρ k (t ) eff i =[ H k +V, ρ k ] t Using: Veff = Vext i ρi, j, k = [ H +V eff, ρ ] k k i, j,k eff eff t V l,m,k V l,m, k { H i, j,k = ρ i, j, k δi, j ϵi (k) ρ k eff = δi, j f (ϵi, k )+ V +... eff V And Fourier transform respect to t-t', we get: f (ϵi, k ) f (ϵ j, k ) χ i, j, l, m, k (ω)= δ j,l δi, m ℏ ω ϵ j, k + ϵi, k +i η

12 Optical Absorption: IP δ ρni =χ 0 δ V tot 0 χ = ij ϕi (r) ϕ*j (r) ϕ*i (r ' ) ϕj (r ' ) ω (ϵi ϵ j )+ i η Hartree, Hartree-Fock, dft... Non Interacting System Absorption by independent Kohn-Sham particles =ℑ χ 0 = j D i 2 δ(ω (ϵ j ϵi )) ij 2 8 π '' 2 ϵ (ω)= 2 ϕi e v ϕ j δ(ϵi ϵ j ℏ ω) ω i, j Particles are interacting!

13 Time-dependent Hartree (local fields) Time-dependent Hartree (local fields effects) The induced charge density results in a total potential via the Veff = Vext + VH V tot r t =V ext r t dt ' d r ' v r r ' ind r ' t ' Poisson equation. r,t r,t V tot r ' ',t ' ' r, r ', t t ' = = V ext r ',t ' V tot r ' ',t ' ' V ext r ', t ' χ ( r t, r ' t ' )=χ 0 ( r t, r ' t ' )+ dt 1 dt 2 d r 1 d r 2 χ 0 ( r t, r 1 t 1 ) v ( r 1 r 2) χ ( r 2 t 2, r ' t ' ) ind Screening of the V ind V tot external perturbation ' 0 r, r = ind r, t V tot r ' t '

14 Time-dependent Hartree (local fields) PRB (2005)

15 Macroscopic Perturbation... δ ρind ϵ (ω)=1+ v δv ext δρind ϵ(ω)=1 v δ V tot 1 Which is the right equation?

16 ...microscopic observables Not correct!!

17 Macroscopic averages 1/3 In a periodic medium every function V(r) can be represented by the Fourier series: V (r)= dq G V (q+g) ei(q+g)r or V (r)= dq V (q, r) eiqr = dq G V (q+g) ei(q+g )r igr V (q,r )= V (q+g)e G Where: The G components describe the oscillation in the cell while the q components the oscillation larger then L

18 Macroscopic averages 2/4

19 Macroscopic averages 3/4

20 Macroscopic averages 4/4 The external fields is macroscopic, only components G=0

21 Macroscopic averages and local fields If you want the macroscopic δ ρind ϵ (ω)=1+ v δv ext then invert the dielectric constant δρind ϵ(ω)=1 v δ V tot response use the first equation and Local fields are not enough... 1

22 What is missing? Two particle excitations, what is missing? electron-hole interaction, exchange, higher order effects...

23 The DFT and TDDFT way

24 DFT versus TDDFT

25 DFT versus TDDFT

26 Time Dependent DFT [ ] 1 2 +V eff (r, t ) ψi (r, t)=i ψi (r,t ) 2 t V eff (r,t )=V H (r, t)+ V xc (r, t)+ V ext (r, t) Interacting System Petersilka et al. Int. J. Quantum Chem. 80, 584 (1996) I = V ext NI 0= V eff... by using... I = NI V ext = 0 V ext V H V xc V H V xc = 1 V ext V ext 0 v Non Interacting System TDDFT is an exact f xc theory for neutral excitations! q, = 0 q, 0 q, v f xc q, q,

27 Time Dependent DFT Choice of the xc-functional...with a good xc-functional you can get the right spectra!!!

28 Summary How to calculate linear response in solids and molecules The local fields effects: time-dependent Hartree Correlation problem: TD-Hartree is not enough! Correlation effects can be included by mean of TDDFT

29 29

30 References!!! On the web: Electronic excitations: density-functional versus many-body Green's-function approaches RMP, vol 74, pg 601, (2002 ) G. Onida, L. Reining, and A. Rubio Books: