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Solving TD-DFT/BSE equations with Lanczos-Haydock approach

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Solving TD-DFT/BSE equations with Lanczos-Haydock approach

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Solving TD-DFT/BSE equations with Lanczos-Haydock approach

  1. 1. Solving TD-DFT/BSE equations nanoexcite2010 with Lanczos- Haydock approch C. Attaccalite and M. Grüning
  2. 2. LLiinneeaarr rreessppoonnssee TTDD--DDFFTT//BBSSEE Solid State approach: Dyson-like equation Casida approach: Eq. rewritten in a basis of e-h pairs nanoexcite2010
  3. 3. CCaassiiddaa aapppprrooaacchh ttoo TTDD--DDFFTT//BBSSEE nanoexcite2010 The matrix in term of e-h pairs: ● Can be very large, e.g.106x106 (diagonalization: N3) ● Is not Hermitian (less efficient/stable algorithms)
  4. 4. TTaammmm--DDaannccooffff aapppprrooxxiimmaattiioonn Only positive e­h pairs are considered and coupling between e­h pair at positive nanoexcite2010 and negative energiesis neglected X X 1) Successful to describe optics absorption of many systems 2) The non­Hermitian BSE reduces to a Hermitian one 3) BSE can be solved using efficient iterative schemes
  5. 5. BBSSEE aanndd ddiieelleeccttrriicc ffuunnccttiioonn dielectric function can be expressed in terms of the ground state |0> and the eigenstates of the BSE  〉 ∣vacuum 〉 ∣f 〉=Σc , v c , v  ac nanoexcite2010 2, q=4Σ∣〈f eiqr ∣f q ∣0〉 2∣Ef−Ei− where ∣0 〉=Πv av   av   av∣0 〉
  6. 6. BBSSEE aanndd ddiieelleeccttrriicc ffuunnccttiioonn nanoexcite2010 2,q =4Σf 〈0∣eiqr q ∣f 〉 HBSE−〈 f ∣eiqr q ∣0〉 We can eliminate the sum over |f> Using the Dirac idensity: 1 xi =P1 x −i x  2, q =−4ℑ[〈 P∣ 1 −HBSEi ∣P〉 ] Where: ∣P〉=limq0 eiqr q ∣0 〉 This formula involves only The ground state L. X. Benedict and E. L. Shirley PRB, 59, 5441 (1999) M. Marsili, Ph.D. thesis, Universita di Roma "Tor Vergata", 2006
  7. 7. LLaanncczzooss--HHaayyddoocckk mmeetthhoodd nanoexcite2010 Main idea: This allows to rewrite the dielectric function as: R. Haydock, Comput. Phys. Commun. 20, 11 (1980)
  8. 8. LLaanncczzooss--HHaayyddoocckk aallggoorriitthhmm Matrix Elements: Basis: a ∣1 〉=∣P 〉 1=〈 1∣H∣1 〉 nanoexcite2010 a2=〈 2∣H∣2 〉 b2=〈 1∣H∣2 〉 ∣2〉= H∣1 〉−a1∣1 〉 〈2∣2 〉 ∣3 〉= H∣2 〉−a2∣2 〉−b1∣1 〉 ∣〈 3∣3 〉∣ 〈1∣H∣3 〉=0 ! ...... ...... ∣i1 〉= H∣i 〉−ai∣i 〉−bi−1∣i−1 〉 ∣〈 i1∣i1 〉∣ bi=〈 i−1∣H∣i 〉 ai=〈 i∣H∣i 〉
  9. 9. LLaanncczzooss--HHaayyddoocckk ppeerrffoorrmmaannccee nanoexcite2010
  10. 10. TTaammmm--DDaannccooffff bbrreeaakkddoowwnn 11 Plasmons in bulk materials V. Olevano and L. Reining PRL 86, 5962 (2001) nanoexcite2010 Cromophores Y. Ma, M. Rohfling and C. Molteni J. Chem. Theory Comput. 6, 257–265 (2010)
  11. 11. TTaammmm--DDaannccooffff bbrreeaakkddoowwnn 22 M. Gruning, A. Marini, X. Gonze NanoLetters, 6, 257–265 (2010) nanoexcite2010
  12. 12. TTaammmm--DDaannccooffff bbrreeaakkddoowwnn 33 Mixed excitonic­plasmonic excitation nanoexcite2010 Nanostructures M. Gruning, A. Marini, X. Gonze NanoLetters, 6, 257–265 (2010) Quasiparticle band gap
  13. 13. NNoonn--HHeerrmmiittiiaann aallggoorriitthhmmss nanoexcite2010 Standard non­Hermitian case: Arnoldi: Bi-Lanczos: Standard Lanczos is unstable for non-Herminitiam matrices (see J. H. Wilkison, “The Algebrica Eigenvalue Problem”)
  14. 14. DDeeffiinniittiioonn ppsseeuuddoo--HHeerrmmiicciittyy nanoexcite2010 Hermitian: 〈∣H∣'〉=〈'∣H∣〉 Pseudo-Hermitian: If H is Herminitian with respect 〈⋅∣∣⋅〉H inner product we can reduce it in the form
  15. 15. TTDD--DDFFTT ppsseeuuddoo--HHeerrmmiittiiaann In case of the TD-DFT or BSE Hamiltonian we have: nanoexcite2010 Using H we can define the inner product: Then we rewrite our expectation value a complete basis set orthonormal respect to this inner product
  16. 16. FFoorr tthhee HHeerrmmiittiiaann ccaassee nanoexcite2010
  17. 17. For the pseudo-Hermitian case Lanczos-Haydock for full TD-DFT/BSE nanoexcite2010
  18. 18. nanoexcite2010 HHooww ddooeess iitt wwoorrkk??
  19. 19. LLeett''ss ppllaayy wwiitthh YYaammbboo nanoexcite2010

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