The frequency-dependent Sternheimer equation in TDDFT

Transcription

1 The frequency-dependent Sternheimer equation in TDDFT A new look into an old equation Miguel A. L. Marques 1 Centre for Computational Physics, University of Coimbra, Portugal 2 European Theoretical Spectroscopy Facility July 19, 2007 Gordon Conference M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

2 Homarus Why are lobsters BLUE? M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

3 Homarus Why are lobsters BLUE? Homarus gammarus (European lobster) M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

4 Astaxanthin (AXT) The red comes from the molecule astaxanthin, a cousin of beta carotene, which gives carrots their orange color and is a source of vitamin A. Astaxanthin, which looks red because it absorbs blue light, also colors shrimp shells and salmon flesh. The blue pigment in lobster shells also comes from crustacyanin, which is astaxanthin clumped together with a protein. (New York Times) M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

5 Molecule CIS TDDFT ZINDO/S Exp AXT AXTH AXT-His+ 623 AXT-His 473 AXT in α-crustacyanin: 632 nm B. Durbeej and L. A. Eriksson, Phys. Chem. Chem. Phys. 8, 4053 (2006). M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

6 Outline 1 Introduction - the Sternheimer equation 2 Some results Hyperpolarizabilities van der Waals coefficients 3 Visualizing Electronic Excitations 4 Outlook resonant Raman scattering M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

7 Outline Introduction - the Sternheimer equation 1 Introduction - the Sternheimer equation 2 Some results Hyperpolarizabilities van der Waals coefficients 3 Visualizing Electronic Excitations 4 Outlook resonant Raman scattering M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

8 Introduction - the Sternheimer equation Spectroscopy Spectroscopy: From the latin spectrum an appearance, an apparition, from spectare, to behold + the greek skopein to view. c unoccupied states v occupied states Examples: UV/Vis, IR, X-ray, Dichroism, NMR, Raman, etc. M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

9 Introduction - the Sternheimer equation Linear Response Time propagation Dyson equation i ϕ j(r, t) t Casida s equation where = ] [ v KS[n](r, t) ϕ j (r, t), (1) χ = χ 0 + χ 0 [v + f xc ] χ ˆRF q = Ω 2 qf q, R q,q = (ε aσ ε iσ ) 2 δ qq + 2 ε aσ ε iσ K q,q (ω n ) ε a σ ε i σ. Superoperators and Lanczos methods P 1 (ω L) 1 Q 1 = 1 ω a 1 + b 2 1 ω a 2 + c2 M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

10 Introduction - the Sternheimer equation The Sternheimer Equation Hartree-Fock: Coupled Hartree-Fock method DFT: Density functional perturbation theory M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

11 Introduction - the Sternheimer equation The Sternheimer Equation - Frequency with { } H (0) ɛ m ± ω + iη ψ (1) m (r, ±ω) = P c H (1) (±ω)ψ (0) m (r) H (1) (ω) = V (r) + m d 3 r n(1) (r, ω) r r + d 3 r f xc (r, r ) n (1) (r, ω) and occ. { [ ] n (1) (r, ω) = ψ (0) [ ] (1) m (r) ψ m (r, ω) + ψ (1) } (0) m (r, ω) ψ m (r) M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

12 Introduction - the Sternheimer equation The Sternheimer Equation - Frequency with { } H (0) ɛ m ± ω + iη ψ (1) m (r, ±ω) = P c H (1) (±ω)ψ (0) m (r) H (1) (ω) = V (r) + m d 3 r n(1) (r, ω) r r + d 3 r f xc (r, r ) n (1) (r, ω) and occ. { [ ] n (1) (r, ω) = ψ (0) [ ] (1) m (r) ψ m (r, ω) + ψ (1) } (0) m (r, ω) ψ m (r) Main advantages: (Non-)Linear system of equations solvable by standard methods Only the occupied states enter the equation Scaling is N 2, where N is the number of atoms Disadvantages: It is hard to converge close to a resonance M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

13 Introduction - the Sternheimer equation The Sternheimer Equation - Perturbations Different perturbations are possible: Electric V (r) = r i (e.g., polarizabilities, absorption, florescence...) Magnetic V (r) = L i (e.g., susceptibilities, NMR...) Atomic Displacements V (r) = v(r) R i (e.g., phonons...) σ (arb. units) Energy (ev) x y z M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

14 Introduction - the Sternheimer equation Code development - octopus Comput. Phys. Commun. 151, (2003) Phys. Stat. Sol. B 243, (2006) M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

15 Outline Some results 1 Introduction - the Sternheimer equation 2 Some results Hyperpolarizabilities van der Waals coefficients 3 Visualizing Electronic Excitations 4 Outlook resonant Raman scattering M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

16 First test: SHG Some results Hyperpolarizabilities Second harmonic generation of paranitroaniline: β( 2ω, ω, ω) β ( 2ω;ω,ω) [a.u.] exp. solv. This work β ( 2ω;ω,ω) [a.u.] exp. solv. exp. gas This work LDA/ALDA LB94/ALDA B3LYP CCSD ω [ev] ω [ev] JCP 126, (2007) M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

17 Optical Rectification Some results Hyperpolarizabilities Optical rectification of H 2 O: β(0, ω, ω) β (0;ω, ω) [a.u.] ω [ev] JCP 126, (2007) M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

18 Some results Van der Waals coefficients van der Waals coefficients Non-retarded regime Casimir-Polder formula ( E = C 6 /R 6 ): C AB 6 = 3 π Retarded regime ( E = K /R 7 ): 0 du α (A) (iu) α (B) (iu), K AB = 23c 8π 2 α(a) (0) α (B) (0) The polarizability is calculated from α ij (iu) = d 3 r n (1) j (r, iu)r i M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

19 Some results Alternative Time Propagation van der Waals coefficients Apply explicitly the perturbation: δv ext (r, t) = x j κδ(t t 0 ) The dynamic polarizability reads, at imaginary frequencies: α ij (iu) = 1 dt d 3 r x i δn(r, t)e ut κ M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

20 Some results Alternative Time Propagation van der Waals coefficients Apply explicitly the perturbation: δv ext (r, t) = x j κδ(t t 0 ) The dynamic polarizability reads, at imaginary frequencies: α ij (iu) = 1 dt d 3 r x i δn(r, t)e ut κ It turns out: Both Sternheimer and time-propagation have the same scaling Only a few frequencies are needed in the Sternheimer approach, but... 2 or 3 fs are sufficient for the time-propagation In the end, the pre-factor is very similar M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

21 Some results van der Waals coefficients C 6 - Polycyclic Aromatic Hydrocarbons C 6 AB (a.u./10 3 ) N A x N B C 6 /N α 2 /N 2 C 6 /N α 2 /N 2 M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

22 Some results C 6 - The characteristic frequency van der Waals coefficients London approximation α(iu) = α(0) 1 + (u/ω 1 ) 2 which leads to C 6 = 3ω 1 4 α2 (0) ω 1 = 0.34 IP (Ha) ω 1 (Ha) ω 1 (a.u.) Number of Si atoms M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

23 Some results C 3 Surface-cluster interaction van der Waals coefficients For a surface-cluster interaction, E = C 3 /R 3, where C 3 = 1 4π 0 du α(iu) ɛ(iu) 1 ɛ(iu) + 1 C 3 /N Si (a.u.) RPA TDLDA α/q 2 α + β ω 2 /q Number of Si atoms M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

24 Outline Visualizing Electronic Excitations 1 Introduction - the Sternheimer equation 2 Some results Hyperpolarizabilities van der Waals coefficients 3 Visualizing Electronic Excitations 4 Outlook resonant Raman scattering M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

25 Visualizing Electronic Excitations Visualizing Electronic Excitations We can look at, e.g., How to visualize and interpret electron bonds? Electronic density Quite featureless One-particle wave-functions Not uniquely defined Usually extend over large regions Electron localization function Bonds and lone-pairs are evident M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

26 Visualizing Electronic Excitations The time-dependent ELF The time-dependent electron localization function is defined as f elf = [ ] 2 C(r,t) C uni (r,t) For a Slater determinant: C det = N ϕ i (r, t) 2 1 [ n(r, t)] 2 [j(r, t)]2 4 n(r, t) n(r, t) i=1 M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

27 Visualizing Electronic Excitations TDELF C 2 H 2 in a strong laser field Laser: ω = 17 ev, T = 8 fs, I = Wcm 2 Phys. Rev. A (Rap. Comm.) 71, (2005) M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

28 Visualizing Electronic Excitations TDELF H + + OH H 2 O 0 fs 3.0 fs 6.0 fs 9.7 fs 13.3 fs 15.7 fs 18.1 fs 20.6 fs 23 fs 24.8 fs 27.2 fs 30.4 fs M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

29 Visualizing Electronic Excitations TDELF linear response The first order variation of the ELF is [ ψ (1) ψ (0) + ψ (0) ψ (1)] C (1) (r) = i 1 ρ (0) 2 ρ (0) ρ (1) + 1 ρ (0) 2 [ 4 ] ρ (0) 2 ρ(1) + 2 j(0) j (1) ρ (0) And the normalization: [ ρ(1) j (0) ] 2 [ ] ρ (0) 2 We obtain C (1) 0 = 6π 2 [ ρ (0)] 2/3 ρ (1) ( [ ] f (1) ELF = 2 f (0) 2 C (0) ELF C (0) 0 C (1) C (0) C(0) C (0) 0 C (1) ) 0 C (0) 0 M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

30 Visualizing Electronic Excitations LRELF An example, acrolein x y z σ(ω) [Å 2 ] ω [ev] M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

31 Visualizing Electronic Excitations LRELF An example, acrolein 5.64 ev 5.91 ev 6.14 ev M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

32 Visualizing Electronic Excitations LRELF Another example, pyrrole σ(ω) [Å 2 ] x 7 y z ω [ev] M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

33 Visualizing Electronic Excitations LRELF Another example, pyrrole 5.43 ev (y) 6.10 ev (z) 7.18 ev (x) M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

34 Outline Outlook resonant Raman scattering 1 Introduction - the Sternheimer equation 2 Some results Hyperpolarizabilities van der Waals coefficients 3 Visualizing Electronic Excitations 4 Outlook resonant Raman scattering M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

35 Outlook resonant Raman scattering Outlook Raman scattering I In non-resonant Stokes Raman spectra, the peak intensities are given by I ν ei A ν e j 1 ω ν (n ν + 1) where A ν lm = kγ 3 E E l E m u kγ w ν kγ Mγ (2) Often, the same expression is used to interpret resonant scattering! M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

36 Outlook resonant Raman scattering Outlook Raman scattering II Perturb the Hamiltonian: How to do resonant Raman correctly? H = H 0 + V E cos(ω E t) + V I cos(ω I t) Then the third derivative becomes: ( ) β(ω I + ω E ; ω I, ω E ) ψ 0 ˆp δψ(ω I + ω E ) + δψ( ω E ) ˆp δψ(ω I ) To recover the non-resonant expression, just put ω I = ω E = 0. M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

37 Outlook resonant Raman scattering Outlook Raman scattering III The perturbed wave-functions are solutions of the equations: Frequency: ɛ k ± ω E Frequency: ɛ k ± ω I (H 0 ɛ k ± ω E ) δψ(±ω E ) = V E 2 ψ0 (3) (H 0 ɛ k ± ω I ) δψ(±ω I ) = V I 2 ψ0, (4) Frequency: ɛ k + σ E ω E + σ I ω I with (σ i = ±, σ E = ±) (H 0 ɛ k + σ I ω I + σ E ω E ) δψ(σ I ω I + σ E ω E ) = V I 2 δψ(σ Eω E ) V E 2 δψ(σ Iω I ) M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35

38 Outlook resonant Raman scattering Collaborators Xavier Andrade San Sebastián, Spain Alberto Castro Berlin, Germany Angel Rubio San Sebastián, Spain Hardy Gross Berlin, Germany Silvana Botti Paris, France M. A. L. Marques (Coimbra) The ω-sternheimer equation Gordon / 35