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 LPMCN, Université Claude Bernard Lyon 1 and CNRS, France 3 European Theoretical Spectroscopy Facility September 18, 2007 Nanoquanta M. A. L. Marques (Coimbra) The ω-sternheimer equation Nanoquanta / 25

2 Outline 1 Introduction - the Sternheimer equation 2 Hyperpolarizabilities 3 van der Waals coefficients 4 Outlook resonant Raman scattering M. A. L. Marques (Coimbra) The ω-sternheimer equation Nanoquanta / 25

3 Outline Introduction - the Sternheimer equation 1 Introduction - the Sternheimer equation 2 Hyperpolarizabilities 3 van der Waals coefficients 4 Outlook resonant Raman scattering M. A. L. Marques (Coimbra) The ω-sternheimer equation Nanoquanta / 25

4 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 Nanoquanta / 25

5 Introduction - the Sternheimer equation Linear Response Dyson equation χ = χ 0 + χ 0 [v + f xc ] χ Casida s equation ˆRF q = Ω 2 qf q, where 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 Time propagation Sternheimer equation 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 Nanoquanta / 25

6 Introduction - the Sternheimer equation Linear response from the KS equations Apply a perturbation of the form δv ext σ (r, t) = κ 0 zδ(t) to the ground state of the system. At t = 0 + the Kohn-Sham orbitals are ϕ j (r, t = 0 + ) = e iκ 0z ϕ j (r). Propagate these KS wave-functions for a (in)finite time. The dynamical polarizability can be obtained from α(ω) = 1 d 3 r z δn(r, ω). κ 0 This prescription has been used with considerable success to calculate the photo-absorption spectrum of several finite systems. As it is based on the propagation of the Kohn-Sham equations, this approach can be easily extended to study non-linear response. M. A. L. Marques (Coimbra) The ω-sternheimer equation Nanoquanta / 25

7 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 Nanoquanta / 25

8 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 Nanoquanta / 25

9 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 Nanoquanta / 25

10 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 Nanoquanta / 25

11 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 Nanoquanta / 25

12 Outline Hyperpolarizabilities 1 Introduction - the Sternheimer equation 2 Hyperpolarizabilities 3 van der Waals coefficients 4 Outlook resonant Raman scattering M. A. L. Marques (Coimbra) The ω-sternheimer equation Nanoquanta / 25

13 Hyperpolarizabilities Optical Rectification 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 Nanoquanta / 25

14 Hyperpolarizabilities Second Harmonic Generation 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 Nanoquanta / 25

15 Hyperpolarizabilities Playtime: PNA in external electric field SHG of paranitroaniline in an external electric field β z ( 2ω,ω,ω) (a.u.) x10 3 a.u. zero field 0.5x10 3 a.u. β z ( 2ω,ω,ω) (a.u.) ω = 0 2ω = 0.2 ev ω [ev] Ex10 3 [a.u.] M. A. L. Marques (Coimbra) The ω-sternheimer equation Nanoquanta / 25

16 Outline van der Waals coefficients 1 Introduction - the Sternheimer equation 2 Hyperpolarizabilities 3 van der Waals coefficients 4 Outlook resonant Raman scattering M. A. L. Marques (Coimbra) The ω-sternheimer equation Nanoquanta / 25

17 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 Nanoquanta / 25

18 van der Waals coefficients Alternative Time Propagation 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 Nanoquanta / 25

19 van der Waals coefficients Alternative Time Propagation 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 Nanoquanta / 25

20 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 J. Chem. Phys. 127, (2007) M. A. L. Marques (Coimbra) The ω-sternheimer equation Nanoquanta / 25

21 van der Waals coefficients C 6 - The characteristic frequency London approximation which leads to α(iu) = α(0) 1 + (u/ω 1 ) 2 C 6 = 3ω 1 4 α2 (0) ω 1 = 0.34 IP (Ha) 0.24 ω 1 (a.u.) ω 1 (Ha) Number of Si atoms M. A. L. Marques (Coimbra) The ω-sternheimer equation Nanoquanta / 25

22 van der Waals coefficients C 3 Surface-cluster interaction 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 Nanoquanta / 25

23 Outline Outlook resonant Raman scattering 1 Introduction - the Sternheimer equation 2 Hyperpolarizabilities 3 van der Waals coefficients 4 Outlook resonant Raman scattering M. A. L. Marques (Coimbra) The ω-sternheimer equation Nanoquanta / 25

24 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 (1) Often, the same expression is used to interpret resonant scattering! M. A. L. Marques (Coimbra) The ω-sternheimer equation Nanoquanta / 25

25 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 Nanoquanta / 25

26 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 (2) (H 0 ɛ k ± ω I ) δψ(±ω I ) = V I 2 ψ0, (3) 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 Nanoquanta / 25

27 Outlook resonant Raman scattering Collaborators Xavier Andrade San Sebastián, Spain Alberto Castro Berlin, Germany Angel Rubio San Sebastián, Spain Silvana Botti Paris, France M. A. L. Marques (Coimbra) The ω-sternheimer equation Nanoquanta / 25