Introduction to TDDFT Linear-Response TDDFT in Frequency-Reciprocal space on a Plane-Waves basis: the DP (Dielectric Properties) code Valerio Olevano CNRS, Institut Néel, Grenoble, France 2 Résumé ● Motivation ● TDDFT ● Linear-Response TDDFT ● Frequency-Reciprocal space TDDFT ● TDDFT on a Plane Waves basis: the DP code ● Approximations: RPA, ALDA, new kernels ● Results ● Conclusions [email protected] Condensed Matter: 3 a Formidable Many-Body Problem ● Well known interaction: electromagnetic (like in QED) ● All our ignorance in Cond-Mat has roots into the many-body problem. [email protected] 4 the Many-Body Problem ● Already a problem in Classical Physics: 3-body problem (Euler, Lagrange, …) and on ● In Quantum Mechanics even 2 bodies is a problem [email protected] The Many-Body problem 5 kinetic e-ions many-body interaction e-e interaction where N can be up to the Avogadro number 1023!!! many-body wavefunction [email protected] 6 The Many-Body problem Factorizable Hamiltonian Solvable single-particle Schrödinger equation Many-body ground-state total energy Total many-body ground state wavefunction Condensed Matter 7 Theory Models Ab Initio (first principles) simplify and search (if possible) numerical solution but analytical solutions microscopic Hamiltonian ● Quantum Chemistry: ● Anderson CI ● Hubbard CC DMFT Subject Headings: … ● Kondo ● QMC ● Ising ● Density-Functional: ● Heisenberg DFT TDDFT ● Many-body Green function: adjustable GW parameters BSE approximations (U, J, ...) (LDA, ...) 8 Why do we need ab initio theories? 1) To understand and explain observed phenomena; 2) To offer experimentalists reference data; 3) To predict properties before the synthesis, the experiment. Purposes of the: [email protected] 9 Which ab initio theory for which properties/spectroscopy? [email protected] 10 Properties of Matter Electronic Ground State Electronic Excited States properties: properties: ● Total energy ● Electronic structure ● Atomic structure ● Band plot, gap ● Phase stability ● Metal/Insulator char. ● ● Electronic density Optical properties ● Dielectric properties ● Magnetic order ● Transport ● Elastic constant ● e-ph and Superconductivity ● Phonons 11 What DFT can predict (normal error 1~2% in the 99% of cond-mat systems) ● Atomic Structure, Lattice Parameters (XRD) ● Total Energy, Phase Stability ● Electronic Density (SEM, XRS, STM) ● Elastic Constants ● Phonon Frequencies (IR, Neutron scattering) that is, all Ground State Properties! and this also in Strongly Correlated systems!: Vanadium Oxide, VO DFT-LDA nlcc DFT-LDA semic EXP [Longo et al.] 2 a 5.659 Å 5.549 Å 5.7517 ± 0.0030 Å lattice b 4.641 Å 4.522 Å 4.5378 ± 0.0025 Å parameters c 5.420 Å 5.303 Å 5.3825 ± 0.0025 Å M. Gatti et al., PRL 2007 121.46 121.73˚ 122.646˚ ± 0.096 12 What DFT cannot predict ● Electronic Structure, Bandplot ● Bandgap, Metal/Insulator/Semiconductor ● Optical and Dielectric properties that is, all Excited State Properties! You can use DFT to predict these properties, but don't blame DFT if it fails! [email protected] 13 Which ab initio theory for which excited state properties/spectroscopy [email protected] 14 Excitations: Charged vs Neutral Charged Excitations Neutral Excitations N → N+1 (or N-1) N → N (Photoemission Spectroscopy) (Optical and Dielectric e h Spectroscopy) c h c v v Inverse Photoemission Optical Absorption [email protected] 15 Excitations: Charged vs Neutral Charged Excitations Neutral Excitations N → N+1 (or N-1) N → N c c electron-electron electron-electron interaction interaction + electron-hole interaction v v Inverse Photoemission Optical Absorption [email protected] 16 Possible ab-initio Theories for the Excited States ● TDDFT (Time-Dependent Density-Functional Theory) in the Approximations: ● RPA Neutral ● TDLDA Excitations ● beyond ● MBPT (Many-Body Theory) in the Approximations: ● GW Charged Excitations ● BSE (Bethe-Salpeter Equation) Neutral Excitations [email protected] 17 Neutral Excitations Spectroscopies (for TDDFT) [email protected] 18 Optical Properties incident reflected hn photon photon c wcv v sample transmitted photon hn d e t e c t o r [email protected] 19 Optical Spectroscopies Aspnes and Studna Lautenschlager et al. PRB 27, 985 (1983). PRB 36, 4821 (1987). Optical Absorption Refraction Index Aspnes and Studna PRB 27, 985 (1983). [email protected] Reflectance 20 Dielectric Properties (EELS) - e c e- electrons wcv e- v - sample e c e- e- d e v wp t e c plasmon t o r [email protected] 21 Synchrotron Radiation (IXSS) X X rays c wcv v sample c d e t e c t v p o w r ESRF, Grenoble plasmon [email protected] 22 Energy-Loss Spectroscopies Stiebling Z. Phys. B 31, 355 (1978). Energy Loss (EELS) Sturm et al. Schuelke and Kaprolat PRL 48, 1425 (1982), PRL 67, 879 (1991) (ESRF, Grenoble) (ESRF, Grenoble). Inelastic X-ray Scattering Coherent Inelastic Spectroscopy (IXSS) Scattering Spectroscopy (CIXS) [email protected] 23 TDDFT [email protected] 24 What is TDDFT? ● TDDFT is an extension of DFT; it is a DFT with time-dependent external potential: Static Ionic Potential Hohenberg-Kohn Static Ionic Potential theorem + External Perturbation Runge-Gross (E.M. field) theorem ● Fundamental degree of freedom: Time-Dependent electronic Density (r,t) (instead of the total many-body wavefunction (r1,...,rN,t)) [email protected] 25 TDDFT milestones ● Runge and Gross (1984): rigorous basis of TDDFT. ● Gross and Kohn (1985): TDDFT in Linear Response. [email protected] 26 The Runge Gross theorem PRL 52, 997 (1984) i∂t 1 t=T V 1 t W 1 t i∂t 2 t =T V 2 tW 2 t v1 t≠v 2 t ct ⇒ 1t ≠2 t (t) Any observable is functional v of the (time-dependent) density: 1 o t=〈 t∣o∣ t〉=o[]t v 2 t [email protected] 27 Runge Gross theorem caveats 1) Any observable is functional of the o t=o [, 0]t density and of the initial state 0=(t0) 2) The Runge-Gross theorem is proven under the requirement of v(t) expanded v(t) What about this case? around t0. And if you run into problems already in describing the (non- equilibrium) initial state, you can't go on with TDDFT. Previous demonstrations required periodic v(t) or a small td t t perturbation (linear response), or later 0 (Laplace transformable switch-on v(t) This also is not potentials+initial ground-state). But there Runge-Gross described is no general proof of the Runge-Gross theorem. t t 0 [email protected] 28 TDDFT vs DFT Zoological Comparative Anatomy [email protected] 29 DFT vs TDDFT Hohenberg-Kohn: Runge-Gross: vr⇔ r vr ,t ⇔ r , t The Total Energy: The Action: t1 ∣ ∣ 〈∣H∣〉=E [] ∫ dt 〈t i∂t−H t t〉= A[] t0 are unique functionals of the density. The extrema of the: The stationary points of the: Total Energy Action E [] A[] =0 =0 r r ,t give the exact density of the system: r r ,t 30 DFT vs TDDFT Kohn-Sham: Runge-Gross: N N KS 2 KS 2 r =∑∣i r ∣ r ,t =∑∣i r ,t∣ i=1 i=1 KS r ' Exc [] r ' ,t A [] v r=v r ∫ dr ' v KS r , t=vr , t∫ dr ' xc ∣r−r '∣ r ∣r−r '∣ r , t KS KS KS KS KS KS KS H r i r=i i r i∂t i r ,t =H r , ti r , t Kohn-Sham equations (t) W (t) W=0 v vKS real system Kohn-Sham system t t TDDFT further caveats 31 1) The functional Axc[] is defined only for v- representable densities. It is undefined for which do not correspond to some A[] A [] potential v → problems when variations =0 xc r ,t r ,t A[] with respect to arbitrary densities are required in order to search for δ AKS [ρ] If exists =v KS (r , t) stationary points. δρ(r ,t) Then we can define the 2) A functional A[] with (t) on the real time, Legendre transform: is ill defined, since we would run in a A [v ]=− A[]∫ dr dt r ,tvr , t causality-symmetry paradox → we can A [v] =r , t solve the problem by working with t on the vr , t Keldysh contour. Response functions are symmetric on the contour and become A r ,t causal on physical time. = vr , t vr 't ' vr 't ' t 0 t symmetric causal Van Leeuwen, PRL 80, 1280 (1998). (=0 for t'>t) [email protected] 32 TDDFT in Linear Response [email protected] TDDFT in Linear Response 33 Gross and Kohn (1985) ● If: · with: Strong (Laser) perturbations excluded! TDDFT = DFT + Linear Response (to the time-dependent perturbation) [email protected] 34 Runge-Gross Theorem for Linear Response TDDFT DFT: TDDFT: LR-TDDFT: [email protected] 35 Calculation Scheme DFT calculation (ABINIT code) LR-TDDFT calculation (DP code) [email protected] 36 Polarizability External Perturbation (variation in the external potential) Induced Density (variation in the density induced by v) implicit definition of the polarizability Polarizability Fundamental quantity associated to the cond-mat system - v is also indicated as v = v purpose of the TDDFT calculation d d d ext [email protected] LR-TDDFT Kohn-Sham scheme: 37 the Independent Particle Polarizability (0) Let's introduce a ficticious, Kohn-Sham non-interacting system, such that: definition of the polarizability of the Kohn-Sham system: Kohn-Sham polarizability (or independent-particle polarizability) It is also indicated as KS = = (0) = 0 s [email protected] Variation in the Kohn-Sham 38 effective Total Potential We write the Kohn-Sham effective potential variation KS (= ) v