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# Linear response theory and TDDFT

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Introduction to linear response theory and TDDFT

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### Linear response theory and TDDFT

1. 1. Claudio Attaccalite http://abineel.grenoble.cnrs.fhttp://abineel.grenoble.cnrs.f r/r/ Linear response theory and TDDFT CECAM Yambo School 2013 (Lausanne)CECAM Yambo School 2013 (Lausanne)
2. 2. Motivations: Absorption Spectroscopy -+ - Many Body Effects!! ! h ν
3. 3. Motivations(II):Absorption Spectroscopy Absorption linearly related to the Imaginary part of the MACROSCOPIC dielectric constant (frequency dependent)
4. 4. Outline How can we calculate the response of the system? IP, local field effects and Time Dependent DFT Some applications and recent steps forward Conclusions Response of the system to a perturbation → Linear Response Regime
5. 5. Spectroscopy
6. 6. From Maxwell equation to the response function D(r ,t)=ϵ0 E(r ,t)+P(r ,t) ∇⋅E(r ,t)=4 πρtot (r ,t) ∇⋅D(r ,t)=4 πρext (r ,t) From Gauss's law:Materials equations: Electric Displacemen t Electric Field Polarization 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 ) In general: For a small perturbation we consider only the first term, the linear response regimeP(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. 7. Response Functions ϵ(ω)= D(ω) ϵ0 E(ω) = δV ext (ω) δV tot (ω) Moving from Maxwell equation to linear response theory we define ϵ −1 (ω)= δVtot (ω) δVext (ω) Vtot (⃗r t)=V ext (⃗r t)+∫dt '∫d ⃗r ' v(⃗r−⃗r ' )ρind (⃗r ' t ' ) Vtot (⃗r ,t)=V ext (⃗r ,t)+Vind (⃗r ,t ') where The induced charge density results in a total potential via the Poisson equation. ϵ −1 (ω)=1−v δρind δV ext ϵ(ω)=1+v δρind δVtot Our goal is to calculate the derivatives of the induced density respect to the external potential
8. 8. The Kubo formula 1/2 H=H0+Hext (t)=H0+∫d rρ(r) V ext (r ,t) We star from the time-dependent Schroedinger equation: i ∂ ψ ∂t =[H0+Hext (t)] ψ(t) ...and search for a solution as product of the solution for Ho plus an another function (interaction representation)... ̃ψ(t)=e i H0 t ψ(t) i ∂ ̃ψ(t) ∂t =e iH 0 t Hext(t)e −iH0 t ̃ψ(t)= ̃Hext (t) ̃ψ(t) ...and we can write a formal solution as: ̃ψ(t)=e −i∫t 0 t ̃H ext(t)dt ̃ψ(t0)
9. 9. Kubo Formula (1957) r t ,r' t' = ind r ,t ext r' ,t ' =−i〈[ r ,t r' t ']〉 The Kubo formula 2/2 ̃ψ(t)=e −i∫t0 t ̃H ext(t)dt ̃ψ(t0)=[1+ 1 i ∫t0 t dt ' Hext (t ')+O(Hext 2 )] ̃ψ(t0) For a weak perturbation we can expand: And now we can calculate the induced density: ρ(t)=〈 ̃ψ(t)∣̃ρ(t)∣̃ψ(t)〉≈〈ρ〉0−i∫t0 t 〈[ρ(t), Hext (t ')]〉+O(Hext 2 ) ρind (t)=−i∫t0 t ∫dr〈[ρ(r ,t),ρ(r' t ')]〉ϕext (r' ,t ') ...and finally......and finally... The linear response to a perturbation is independent on the perturbation and depends only on the properties of
10. 10. How to calculated the dielectric constant i ∂ ̂ρk (t) ∂t =[Hk +V eff , ̂ρk ] ̂ρk (t)=∑i f (ϵk ,i)∣ψi,k 〉〈 ψi,k∣ The Von Neumann equation (see Wiki http://en.wikipedia.org/wiki/Density_matrix) r t ,r ' t ' =  ind r ,t ext r' ,t ' =−i〈[ r ,t r' t ']〉We want to calculate: We expand X in an independent particle basis set χ(⃗r t ,⃗r' t' )= ∑ i, j,l,m k χi, j,l,m, k ϕi, k (r)ϕj ,k ∗ (r)ϕl,k (r')ϕm ,k ∗ (r') χi, j,l,m, k= ∂ ̂ρi, j, k ∂Vl,m ,k Quantum Theory of the Dielectric Constant in Real Solids Adler Phys. Rev. 126, 413–420 (1962) What is Veff ?
11. 11. Independent Particle Independent Particle Veff = Vext ∂ ∂Vl ,m,k eff i ∂ρi, j ,k ∂t = ∂ ∂Vl ,m, k eff [Hk+V eff , ̂ρk ]i, j, k Using: { Hi, j ,k = δi, j ϵi(k) ̂ρi, j, k = δi, j f (ϵi,k)+ ∂ ̂ρk ∂V eff ⋅V eff +.... And Fourier transform respect to t-t', we get: χi, j,l,m, k (ω)= f (ϵi,k)−f (ϵj ,k) ℏ ω−ϵj ,k+ϵi ,k+i η δj ,l δi,m i ∂ ̂ρk (t) ∂t =[Hk +V eff , ̂ρk ] χi, j,l,m, k= ∂ ̂ρi, j, k ∂Vl,m ,k
12. 12. Optical Absorption: IP Non Interacting System δρNI=χ 0 δVtot χ 0 =∑ ij ϕi(r)ϕj * (r)ϕi * (r')ϕj(r ') ω−(ϵi−ϵj)+ i η Hartree, Hartree-Fock, dft... =ℑχ0=∑ ij ∣〈 j∣D∣i〉∣2 δ(ω−(ϵj −ϵi)) ϵ '' (ω)= 8 π2 ω2 ∑ i, j ∣〈ϕi∣e⋅̂v∣ϕj 〉∣ 2 δ(ϵi−ϵj−ℏ ω) Absorption by independent Kohn-Sham particles Particles are interacting!
13. 13. Time-dependent Hartree (local fields) Time-dependent Hartree (local fields effects) Veff = Vext + VH Vtot r t=V ext r t∫dt '∫d r' v r−r' ind r' t'  The induced charge density results in a total potential via the Poisson equation. r ,r' ,t−t '=  r ,t V ext r ' ,t ' =  r ,t  Vtot r ' ' ,t ' '  Vtot r' ' ,t ' '  V ext r' ,t ' χ(⃗r t ,⃗r ' t ' )=χ0(⃗r t ,⃗r ' t ' )+∫dt1 dt2∫d ⃗r1 d ⃗r2 χ0 (⃗r t , ⃗r1 t1)v(⃗r1−⃗r2)χ(⃗r2 t2 ,⃗r ' t ' ) ind Vind Vtot 0r ,r ' =  ind r ,t V tot r ' t '  Screening of the external perturbation
14. 14. Time-dependent Hartree (local fileds) PRB 72 153310(2005)
15. 15. Macroscopic Perturbation.... ϵ −1 (ω)=1+v δρind δV ext ϵ(ω)=1−v δρind δV tot Which is the right equation?
16. 16. ...microscopic observables Not correct!!
17. 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)=∫dqV (q,r)eiqr =∫dq∑G V (q+G)ei(q+G)r Where: V (q ,r)=∑G V (q+G)eiGr The G components describe the oscillation in the cell while the q components the oscillation larger then L
18. 18. Macroscopic averages 2/4
19. 19. Macroscopic averages 3/4
20. 20. Macroscopic averages 4/4 The external fields is macroscopic, only components G=0
21. 21. Macroscopic averages and local fields If you want the macroscopic response use the first equation and then invert the dielectric constant ϵ −1 (ω)=1+v δρind δV ext ϵ(ω)=1−v δρind δV tot Local fields are not enough....
22. 22. What is missing? Two particle excitations, what is missing?Two particle excitations, what is missing? electron-hole interaction, exchange, higher order effects......
23. 23. The DFT and TDDFT way
24. 24. DFT versus TDDFT
25. 25. DFT versus TDDFT
26. 26.  V ext=0  V extV HV xc q ,= 0 q , 0 q,vf xc q ,q , TDDFT is an exact theory for neutral excitations! Time Dependent DFT V eff (r ,t)=V H (r ,t)+ V xc (r ,t)+ V ext (r ,t) Interacting System Non Interacting System Petersilka et al. Int. J. Quantum Chem. 80, 584 (1996)  I= NI=  I  Vext 0=  NI  V eff ... by using ... = 0 1 V H  V ext   V xc V ext  v f xc  i ∂ ̂ρk (t) ∂t =[ HKS , ̂ρk ]=[ Hk 0 +V eff , ̂ρk ]
27. 27. Time Dependent DFT Choice of the xc- functional ...with a good xc-functional you can get the right spectra!!!
28. 28. Summary ● How to calculate linear response in solids 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. 29
30. 30. References!!! Electronic excitations: density-functional versus many-body Green's-function approaches RMP, vol 74, pg 601, (2002 ) G. Onida, L. Reining, and A. Rubio On the web: ● http://yambo-code.org/lectures.php ● http://freescience.info/manybody.php ● http://freescience.info/tddft.php ● http://freescience.info/spectroscopy.php Books: