Linear-response excitations. Silvana Botti

Transcription

1 from finite to extended systems 1 LSI, CNRS-CEA-École Polytechnique, Palaiseau, France 2 LPMCN, CNRS-Université Lyon 1, France 3 European Theoretical Spectroscopy Facility September 3, 2008 Benasque, TDDFT school

2 Outline 1 Electronic excitations within linear response 2 From KS equations to band structures 3 Optical absorption: different computational schemes 4 More on extended systems

3 Outline 1 Electronic excitations within linear response 2 From KS equations to band structures 3 Optical absorption: different computational schemes 4 More on extended systems

4 Spectroscopy Electronic excitations: Optical absorption Electron energy loss Inelastic X-ray scattering Photoemission Inverse photoemission...

5 Linear response For a sufficiently small perturbation, the response of the system can be expanded into a Taylor series with respect to the perturbation. The linear coefficient linking the response to the perturbation is called response function. It is independent of the perturbation and depends only on the system. Example We will consider only the first order (linear) response. We will not consider strong field interaction (e.g. intense lasers). We will consider non-magnetic materials. Density-density response function: δρ(r, t) = dt dr χ ρρ (r, t, r, t )v ext (r, t ) Dielectric tensor: D(r, t) = dt dr ɛ(r, t, r, t )E(r, t )

6 Microscopic dielectric Function Question How can we calculate the microscopic dielectric functions? Answer They are determined by the elementary excitations of the medium: interband and intraband transitions, as well as collective excitations.

7 Outline 1 Electronic excitations within linear response 2 From KS equations to band structures 3 Optical absorption: different computational schemes 4 More on extended systems

8 Kohn-Sham equations ] [ v KS (r) ϕ i (r) = ε i ϕ i (r) n (r) = N i ϕ i (r) 2 = electron density ε i = KS eigenvalues, ϕ i (r) = KS single-particle orbitals Hartree potential v KS (r) = v (r) + v H (r) + v xc (r) unknown exchange-correlation (xc) potential

9 Can we calculate spectra within static DFT? DFT allows calculations of electronic GROUND STATE properties in an accurate (usually within 1-2%) and efficient way: total energy, phase stability, atomic structure, lattice parameters (X-ray diffraction), electronic density (Scanning tunneling microscopy), elastic constants, phonon frequencies (IR, Raman, Neutron scatterning). Ground-state DFT is not designed to access EXCITED STATES: quasiparticle electronic structures (photoemission spectra), band gap (photoemission gap and optical gap), electronic excitations spectra (absorption, EELS, IXSS, etc.).

10 Can we calculate spectra within static DFT? DFT allows calculations of electronic GROUND STATE properties in an accurate (usually within 1-2%) and efficient way: total energy, phase stability, atomic structure, lattice parameters (X-ray diffraction), electronic density (Scanning tunneling microscopy), elastic constants, phonon frequencies (IR, Raman, Neutron scatterning). Ground-state DFT is not designed to access EXCITED STATES: quasiparticle electronic structures (photoemission spectra), band gap (photoemission gap and optical gap), electronic excitations spectra (absorption, EELS, IXSS, etc.).

11 Can we calculate spectra within static DFT? DFT allows calculations of electronic GROUND STATE properties in an accurate (usually within 1-2%) and efficient way: total energy, phase stability, atomic structure, lattice parameters (X-ray diffraction), electronic density (Scanning tunneling microscopy), elastic constants, phonon frequencies (IR, Raman, Neutron scatterning). Ground-state DFT is not designed to access EXCITED STATES: quasiparticle electronic structures (photoemission spectra), band gap (photoemission gap and optical gap), electronic excitations spectra (absorption, EELS, IXSS, etc.).

12 Can we calculate spectra within static DFT? DFT allows calculations of electronic GROUND STATE properties in an accurate (usually within 1-2%) and efficient way: total energy, phase stability, atomic structure, lattice parameters (X-ray diffraction), electronic density (Scanning tunneling microscopy), elastic constants, phonon frequencies (IR, Raman, Neutron scatterning). Ground-state DFT is not designed to access EXCITED STATES: quasiparticle electronic structures (photoemission spectra), band gap (photoemission gap and optical gap), electronic excitations spectra (absorption, EELS, IXSS, etc.).

13 Kohn-Sham one-electron band structure The one-electron band structure is the dispersion of the energy levels as a function of k in the Brillouin zone. The Kohn-Sham eigenvalues and eigenstates are not one-electron energy states for the electron in the solid. However, it is common to interpret the solutions of Kohn-Sham equations as one-electron states: the result is often a good representation, especially concerning band dispersion. Gap problem: the KS band structure underestimates systematically the band gap (often by more than 50%).

14 GaAs band structure 8 6 L 4 3,c 2 L 1,c 0 X 3,c X 1,c Γ 15,c Γ 1,c Γ 15,v Experimental gap: 1.53 ev DFT-LDA gap: 0.57 ev Energy (ev) L 3,v L 2,v X 5,v X 3,v X 1,v GaAs Applying a scissor operator (0.8 ev) we can correct the band structure. See the lectures of Rex Godby! L 1,v -12 Γ 1,v L Γ X K Γ

15 Outline 1 Electronic excitations within linear response 2 From KS equations to band structures 3 Optical absorption: different computational schemes 4 More on extended systems

16 An intuitive Picture: Absorption c unoccupied states v occupied states Independent particle KS picture using ε KS i and ϕ KS i

17 The simplest way: independent-particle transitions c unoccupied states v occupied states Fermi s golden rule: χ KS v,c c D v 2 δ(ɛ c ɛ v ω)

18 The simplest way c unoccupied states v occupied states ɛ 2 (ω) = 2 4π2 ΩN k ω 2 lim 1 q 0 q 2 m v,c,k 2 δ(ɛ ck ɛ vk ω) v,c,k m v,c,k = c q v v velocity matrix elements

19 Joint density of states In the independent-particle approximation the dielectric function is determined by two contributions: optical matrix elements and energy levels. ɛ 2 (ω) = 2 4π2 ΩN k ω 2 lim 1 q 0 q 2 m v,c,k 2 δ(ɛ ck ɛ vk ω) v,c,k If m v,c,k can be considered constant then the spectrum is essentially given by the joint density of states: ɛ 2 JDOS/ω 2 = 1 N k ω 2 v,c,k δ(ɛ ck ɛ vk ω)

20 An intuitive Picture: Absorption c unoccupied states v occupied states Photoemission process hν (E kin + φ) = E N 1,v E N,0 = ε v

21 An intuitive Picture: Absorption c unoccupied states v occupied states Electron-hole interaction: Excitons!

22 Beyond independent-particle picture one way c unoccupied states v occupied states Link to other lectures! GW correction to energy levels (Rex Godby) Excitons from Bethe-Salpeter equation (Matteo Gatti)

23 Beyond independent-particle picture another way TDDFT and linear response (Hardy Gross) Vext macroscopic V ind macroscopic and microscopic δn = χv ext χ = χ KS + χ KS (v + f xc ) χ v = Coulomb potential, related to local field effects f xc = quantum exchange-correlation effects

24 Dealing with excitations: absorption Here we consider TDDFT within linear response. We want to calculate the absorption cross-section (or equivalently for a solid the dielectric function), σ(ω) = 4πω Im {α(ω)} (1) c where α(ω) is the dynamical polarizability. We have seen that all we need is the density-density response function: α(ω) = d 3 r d 3 r z χ(r, r, ω) z χ measures how the density changes when we perturb the potential δn(r, ω) = d 3 r χ(r, r, ω)δv ext (r, ω)

25 Changing the perturbation Different perturbations different terms but similar equations! Electric V (r) = r i (e.g., polarizabilities, absorption, luminescence...) Magnetic V (r) = L i (e.g., susceptibilities, NMR...) Atomic Displacements V (r) = v(r) R i (e.g., phonons...)

26 I) Time propagation of the KS equations Apply a perturbation of the form δv ext σ (r, t) = κ 0 zδ(t) to the ground state of a finite system. At t = 0 + the Kohn-Sham orbitals are ϕ j (r, t = 0 + ) = e iκ 0z ϕ j (r). Propagate these occupied KS wave-functions for a (in)finite time. The dynamical polarizability can be obtained from α(ω) = 1 κ 0 d 3 r z δn(r, ω).

27 Time propagation of the KS equations Main advantages: Accurate photo-absorption spectrum of several finite systems. Can be easily extended to study non-linear response. Decreases storage requirements. It allows the entire frequency-dependent dielectric function to be calculated at once. The scaling with the number of atoms is quite favorable. Disadvantages: The prefactor is fairly large as such calculations typically require time-steps with a time-step of 10 3 fs.

28 II) The response function: TDDFT equations The response function yields δn upon a infinitesimal change in the potential: δn(r, ω) = d 3 r χ(r, r, ω)δv ext (r, ω) By construction of the time-dependent KS system, δn can also be calculated from δn(r, ω) = d 3 r χ KS (r, r, ω)δv KS (r, ω) χ KS is the density-density response function of the non-interacting KS electrons. In terms of the stationary KS orbitals it reads χ KS (r, r, ω) = lim η 0 + jk (f k f j ) ϕ j(r)ϕ j (r )ϕ k (r )ϕ k (r) ω (ɛ j ɛ k ) + iη

29 The Dyson equation Using the definition of the KS potential, we have δv KS (r, ω) = δv ext (r, ω) + d 3 r δn(r, ω) r r + d 3 r f xc (r, r, ω)δn(r, ω), where the xc kernel f xc is the functional derivative f xc [n](r, r, t t ) = δv xc[n](r, t) δn(r, t ) n=n0 Combining the previous results, we arrive at the Dyson equation χ(r, r, ω) = χ KS (r, r, ω) [ ] + d 3 x d 3 x 1 χ(r, x, ω) x x + f xc(x, x, ω) χ KS (x, r, ω)

30 xc kernels: approximations The key ingredient of linear-response within TDDFT is the xc kernel. This is a very complicated functional of the density that encompasses all non-trivial many-body effects of the system. Example of common approximations: Adiabatic LDA Adiabatic GGAs f ALDA xc (r, r, ω) = δ(r r ) d dn v HEG (n(r))... (many more kernels in the TDDFT book) Adiabatic kernels are local in time. xc

31 Common implementation for solids The TDDFT equations in the linear-response regime can be cast in numerous different forms. For solids, in most implementations: The Dyson equation is solved routinely by projecting all quantities onto a suitable set of basis functions (planewaves within pseudotentials, or localized basis for all-electron calculations). We get the matrix equation: with χ(ω) = χ G,G (ω) χ(ω) = χ KS (ω) + χ KS (ω) (v + f xc (ω)) χ(ω) To obtain an absorption spectrum or the loss function at a given momentum transfer only one component of the matrix is required, obtained by solving a linear system of equations, thus avoiding the matrix inversion.

32 The poles of the response function Lehmann representation of the density response function: [ 0 ˆn(r) m m ˆn(r χ(r, r ) 0, ω) = lim η 0 + ω (E m m E 0 ) + iη 0 ˆn(r ) m m ˆn(r) 0 ω + (E m E 0 ) + iη χ (for a finite system) has poles at the excitation energies, Ω = E m E 0, while χ KS has poles at the KS eigenvalue differences, ω jk = ɛ j ɛ k. After some pages of algebra: j k [ δjj δ kk ω jk + ( ) f k f j Kjk,j k (Ω)] β j k = Ωβ jk, where we have defined the matrix element [ ] K jk,j k (ω) = d 3 r d 3 r ϕ 1 j (r)ϕ k (r) r r + f xc(r, r, ω) ϕ j (r )ϕ k (r ) ]

33 Single pole approximation If the excitation is well described by one single-particle transition we can neglect the off-diagonal terms of K jk,j k : Ω SPA = ω RK 12,12. The SPA also describes the spin-multiplet structure of otherwise spin-unpolarized ground states through the spin-dependence of f xc : ω sing = ω R ω trip = ω R d 3 r d 3 r [ d 3 r ϕ 1(r)ϕ 2 (r) 1 r r + f (1) xc (r, r, ω) d 3 r ϕ 1(r)ϕ 2 (r)f (2) xc (r, r, ω)ϕ 1 (r )ϕ 2(r ), ] ϕ 1 (r )ϕ 2(r ) where f (1) xc = (f xc + f xc ) /2 f (2) xc = (f xc f xc ) /2

34 Casida s formulation Parametrization of the linear change of the density: δn(r, ω) = ia [ξ ia (ω)ϕ a(r)ϕ i (r) + ξ ai (ω)ϕ a (r)ϕ i (r)], where i denotes an occupied and a a virtual state. After some algebra, one can prove that the excitation energies can be determined as solutions of the pseudo-eigenvalue problem a i [ δaa δ ii (ɛ j ɛ k ) ɛ a ɛ i K ai,a i (Ω) ] ɛ a ɛ i βa i = Ω2 β ai. The eigenvalues of this equation are the square of the excitation energies, while the eigenvectors can be used to calculate the oscillator strengths. This is the formula implemented in quantum-chemistry codes.

35 III) The Sternheimer equation Hartree-Fock: Coupled Hartree-Fock method DFPT: Density functional perturbation theory

36 The Sternheimer equation { } H (0) ɛ m ± ω + iη ψ m (1) (r, ±ω) = P c H (1) (±ω)ψ m (0) (r) with and 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) occ. { [ ] n (1) (r, ω) = ψ m (0) [ ] (1) (r) ψ m (r, ω) + ψ m (1) } (0) (r, ω) ψ m (r) X. Andrade et al., J. Chem. Phys. 126, (2007).

37 The Sternheimer equation 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. Easy to use to extend to hyperpolarizabilities. Disadvantages: It is hard to converge close to a resonance.

38 IV) Superoperators and Lanczos methods The dynamical polarizability is represented by a matrix continued fraction whose coefficients can be obtained from a Lanczos method: P 1 (ω L) 1 Q1 = This method uses only occupied states. 1 ω a 1 + b 2 1 ω a c 2 Calculations scale favorably with the system size. Up to now only test applications. B. Walker et al., Phys. Rev. Lett. 96, (2006). D. Rocca et al., J. Chem. Phys. 128, (2008).

39 Summary for finite systems A large majority of the calculations is done using the ALDA approximation. In many cases (low Z atoms) ALDA gives essentially the same results as AGGA. The results are quite good, often of the same quality of more demanding approaches, like CI using only single-excitations, or BSE. In some systems the ALDA underestimates the onset of absorption mainly due to the incorrect asymptotics of the LDA potential. The ALDA also fails to reproduce the excitations of stretched H 2, Na 2, etc.

40 Outline 1 Electronic excitations within linear response 2 From KS equations to band structures 3 Optical absorption: different computational schemes 4 More on extended systems

41 Absorption spectra Question Which level of approximation should I use if I am interested in comparing to experiments? Answer There is not a unique answer. It depends on the system and on the kind of spectroscopy. In some cases, the independent-particle approximation already gives results good enough. It is often necessary to go beyond the independent particle approximation. See the lectures of Matteo Gatti!

42 EELS of graphite For a q in the plane the independent-particle approximation agrees with experiment. What happens when q 90? Remind: ɛ M (q) = 1 ɛ 1 00 (q) A. G. Marinopoulos et al., Phys. Rev. Lett. 89, (2002).

43 Absorption spectrum of silicon 50 Si ε 2 (ω) E 1 E 2 EXP RPA SO E 1 and E 2 peaks are red-shifted. excitonic effects on E 1 are missing. Scissor operator does not help: blue-shifted peaks ω [ev]

44 Absorption spectrum of silicon Si EXP BSE TDLDA TDDFT - LRC Adiabatic LDA for xc kernel does not improve the spectrum. ε 2 (ω) 30 BSE gives the correct result BSE-derived kernels give the correct results ω [ev] S. Botti et al., Rep. Prog. Phys. 70, 357 (2007).

45 Independent-particle absorption spectrum of silicon 60 Si 50 E 1 RPA SO E 2 EXP ε 2 (ω) Comparison with Hartree-Fock calculation ω [ev] F. Bruneval et al., J. Chem. Phys. 124, (2006)

46 For extended systems The ALDA approximation does not improve the independent-particle absorption spectra. In many cases ALDA gives essentially the same results as RPA. BSE and BSE-derived xc kernels + GW energies correct the spectra. The main problem of ALDA is the wrong asymptotic limit.