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BSE and TDDFT at work

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BSE and TDDFT at work

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BSE and TDDFT at work

  1. 1. BSE and TDDFT at work Claudio Attaccalite a http://abbiinneeeell..ggrreennoobbllee..ccnnrrss..ffrr// CCEECCAAMM YYaammbboo SScchhooooll 22001133 ((LLaauussaannnnee))
  2. 2. Optical Absorption: Microscopic View Direct and indirect interactions between an e-h pair created by a photon Summing up all such interaction processes we get: L(r1 t1 ; r2 t2 ;r3 t3 ;r4 t4)=L(1,2,3,4) The equation for L is the Bethe Salpeter Equation. The poles are the neutral excitations.
  3. 3. Bethe Salpeter Equation Historical remarks… 1951 1970 First solution of BSE with dynamical effects: SShhiinnddoo aapppprrooxxiimmaattiioonn JPSJ 29, 278(1970) 1974 1995 Plane-waves implementation G. Onida et al. PRL 75, 818 (1995) First applications in solids: W. Hanke and L.J. Sham PRL 33, 582(1974) G. Strinati, H.J. Mattausch and W. Hanke PRL 45, 290 (1980)
  4. 4. Feynman's diagrams and Bethe-Salpeter equation L(1234)=L0(1234)+ L0 1256[v 575678−W565768] L7834 = + - Intrinsic 4-point equation. It describes the (coupled) propagation of two particles, the electron and the hole ! Quasihole and quasielectron L=L0+ L0 [ v−W ]L Retardation effects are W1,2=Wr1 , r2t1 , t2 neglected L1,2,3,4=Lr1, r2, r3, r4 ;t−t0=L1,2,3,4,
  5. 5. Construction of the BSE 1/2 We expand L in the independent particle basis L(r1, r2, r '1, r '2∣ω)= Σ k1, k2, k3,k4 ϕk1(r1)ϕk2(r2)Lk1, k2, k3, k 4ϕk3 (r '1)ϕk4(r '2) We start from L0 L0(r1, r2, r '1,r '2∣ω)=2∫ d ω' 2π G0(r1, r '2 ;ω+ω' )G0(r2, r '1 ;ω' ) G0(r1, r2 ;ω)=Σk , i where ∗ (r1)ϕk , i (r2) ϕk ,i ℏω−ϵk , i+i ηsign(ϵk , i−ϵF) ...integrating in the frequency we get... 0 =δk1, k2δk '1, k '2 Lk1, k2, k '1,k '2 2i ℏ f (ϵk2)−f (ϵk '2) ϵk '2−ϵk2+i δ+ℏω
  6. 6. Construction of the BSE 2/2 Time-dependent Hartree term: TD−harteee=2∫ϕi Hij , kl e(r )ϕj ∗ h(r )V (r−r ')ϕk e (r ')ϕl ∗ Screened Exchange Coulomb term: - h(r ') TD−SEX=∫ϕi Hij , kl e (r )ϕj ∗ h(r ' )W(r , r ')ϕk e (r )ϕl ∗ h(r ') ...and now let's solve the equation...
  7. 7. Bethe-Salpeter equation (4-points - space and time) - + - + - + Should we invert the equation for L for each frequency??? We work in transition space... exc A Hn1n2,n3n4  n3 n4=E A n1n2 L1,2,3,4=Lr1, r2, r3, r4 ;t−t0=L1,2,3,4, The frequency term can be separated and an-effective Hamiltonian can be derived without any frequency dependency 0 L=L0+−L0[v−W] L Original BSE [L1+[v−W]]=L−1 We solve the inverse Bethe-Salpeter eq., because it is easier
  8. 8. How to transform the BSE in an eigenvalues problem Using the definition of L0 That is diagonal in the e/h space
  9. 9. We can solve the equation once for all frequency!!!! ...with some linear algebra....
  10. 10. Absoprtion spectra and BSE
  11. 11. BSE calculation in practice
  12. 12. … Some results… Bruneval et al., PRL 97, 267601 (2006) V. Garbuio et al., PRL 97, 137402 (2006) Strinati et al., Rivista del Nuovo Cimento 11, 1 (1988) Tiago et al., PRB 70, 193204 (2004) Bruno et al., PRL 98, 036807 (2007) Albrecht et al., PRL 80, 4510 (1998)
  13. 13. Excitons in nanoscale systems Excitons in nanoscale systems Gregory D. Scholes, Garry Rumbles Nature Materials 5, 683 - 696 (2006) Nanotubes/Nanowires Colloidal quantum dots Frenkel excitons in photosynthesis
  14. 14. BSE for charge transfer excitons donor-acceptor complexes: benzene, naphthalene, and anthracene derivatives with the tetracyanoethylene acceptor X. Blase and C. Attaccalite Apl. Phys. Lett. 99, 171909 (2011)
  15. 15. Exciton analysis
  16. 16. Solving the equations in a smart way ...
  17. 17. The BSE can be large...... too large R
  18. 18. Tamm-Dancoff approximation
  19. 19. The dielectric constant doesn't require too much information
  20. 20. Let's come back to the original formula we can write the dielectric constant as ϵ2(ω, q)=4 πℑ[〈P∣ 1 ω−HEXC+i η∣P〉] ∣P〉= limq→0 eiqr q ∣0 〉 ...and ask the help of mathematicians...
  21. 21. Lanczos-Haydock method
  22. 22. Lanczos-Haydock algorithm
  23. 23. Lanczos-Haydock performance
  24. 24. What about TDDFT?
  25. 25. TDDFT versus BSE BSE L(1234)=L0(1234)+ +L0(1256)[ v (57)δ(56)δ(78)−W(56)δ(57)δ(68)] L(7834) TDDFT χ(12)=χ0(12)+χ0(13)[v (34)+f xc ]χ(42) BSE is a 4-points equation => unavoidable TDDFT is a 2-points equation => that can be rewritten as a 4-point equation
  26. 26. TDDFT in G-space Simple static fxc case: 0 (q ,ω)+χG,G2 χG,G'(q,ω)=χG,G' 0 (q ,ω)(vG2(q)+f G2,G3 xc (q))χG3 ,G(q ,ω) Microscopic dielectric constant: −1 (q ,ω)=δG,G'+vG(q)χG,G'(q ,ω) ϵG,G' Macroscopic dielectric constant: ϵM (q ,ω)= 1 −1 (q,ω) ϵG=0,G'=0 Advantages: 2-points eq. Disadvantages: the eqs. Has to be solved for each frequency
  27. 27. TDDFT in e/h space Time-dependent Hartree term: TD−hartree=2∫ϕi Hij , kl e(r )ϕj ∗ h(r )V (r−r ')ϕk e (r ')ϕl ∗ h(r ') Time-dependent exchange correla-tion function: TD−EXC=∫ϕi Hij , kl e(r)ϕj ∗ h(r)f xc (r , r ')ϕk e (r ' )ϕl ∗ h(r ' ) f xc (r , r ')= ∂V xc (r ) ∂ρ(r ' ) TD−SEX=∫ϕi Hij ,kl e (r )ϕj ∗ h(r ' )W(r , r ')ϕk e (r )ϕl ∗ h(r ') BSE
  28. 28. Beyond the Tamm- Dancoff approx.
  29. 29. Tamm-Dancoff breakdown 1
  30. 30. Tamm-Dancoff breakdown 2
  31. 31. Don't worry! Haydock method still works
  32. 32. SUMMARY ● Optical spectra can be calculated by mean of Green's function theory ●TDDFT and BSE can efficiently - be formulated in the e/h space ● By using Lanczos-Haydock approach we do not need to diagonalize the full matrix!
  33. 33. References!!! Reviews: ● Application of the Green’s functions method to the study of the optical properties of semiconductors Nuovo Cimento, vol 11, pg 1, (1988) G. Strinati ● Effects of the Electron–Hole Interaction on the Optical Properties of Materials: the Bethe–Salpeter Equation Physica Scripta, vol 109, pg 141, (2004) G. Bussi ● 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:

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