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

1. Solving TD-DFT/BSE equations
nanoexcite2010
with
Lanczos- Haydock approch
C. Attaccalite and M. Grüning

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. 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. 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. 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. 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. LLaanncczzooss--HHaayyddoocckk mmeetthhoodd
nanoexcite2010
Main idea:
This allows to rewrite the dielectric function as:
R. Haydock, Comput. Phys. Commun. 20, 11 (1980)

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. LLaanncczzooss--HHaayyddoocckk ppeerrffoorrmmaannccee
nanoexcite2010

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. TTaammmm--DDaannccooffff bbrreeaakkddoowwnn 22
M. Gruning, A. Marini, X. Gonze
NanoLetters, 6, 257–265 (2010)
nanoexcite2010

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. 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. 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. 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. FFoorr tthhee HHeerrmmiittiiaann ccaassee
nanoexcite2010

17. For the pseudo-Hermitian case
Lanczos-Haydock for full TD-DFT/BSE
nanoexcite2010

18. nanoexcite2010
HHooww ddooeess iitt wwoorrkk??

19. LLeett''ss ppllaayy wwiitthh YYaammbboo
nanoexcite2010