Time Dependent Density Functional Theory - Study Materials | Arts 1, Study notes of Art

Download Time Dependent Density Functional Theory - Study Materials | Arts 1 and more Study notes Art in PDF only on Docsity! Time-dependent density-functional theory Carsten A. Ullrich University of Missouri-Columbia APS March Meeting 2008, New Orleans Neepa T. Maitra Hunter College, CUNY Outline 1. A survey of time-dependent phenomena 2. Fundamental theorems in TDDFT 3. Time-dependent Kohn-Sham equation 4. Memory dependence 5. Linear response and excitation energies 6. Optical processes in Materials 7. Multiple and charge-transfer excitations 8. Current-TDDFT 9. Nanoscale transport 10. Strong-field processes and control C.U. N.M. C.U. N.M. N.M. C.U. N.M. C.U. C.U. N.M. Start from ground state, evolve in time-dependent driving field: t=0 t>0 Nonlinear response and ionization of atoms and molecules in strong laser fields 1. Survey Real-time electron dynamics: second scenario 1. Survey Coupled electron-nuclear dynamics High-energy proton hitting ethene T. Burnus, M.A.L. Marques, E.K.U. Gross, Phys. Rev. A 71, 010501(R) (2005) ● Dissociation of molecules (laser or collision induced) ● Coulomb explosion of clusters ● Chemical reactions Nuclear dynamics treated classically For a quantum treatment of nuclear dynamics within TDDFT (beyond the scope of this tutorial), see O. Butriy et al., Phys. Rev. A 76, 052514 (2007). 1. Survey Linear response tickle the system observe how the system responds at a later time density response perturbation density-density response function 2. Fundamentals Runge-Gross Theorem Runge & Gross (1984) proved the 1-1 mapping: n(r t) vext(r t)   For a given initial-state ψ0, the time-evolving one-body density n(r t) tells you everything about the time-evolving interacting electronic system, exactly. Ψ0 This follows from : Ψ0, n(r,t)  unique vext(r,t)  H(t)  Ψ(t)  all observables For any system with Hamiltonian of form H = T + W + Vext , e-e interaction kinetic external potential Consider two systems of N interacting electrons, both starting in the same Ψ0 , but evolving under different potentials vext(r,t) and vext’(r,t) respectively: RG prove that the resulting densities n(r,t) and n’(r,t) eventually must differ, i.e. vext’(t), Ψ’(t) Ψο vext(t), Ψ(t) 2. Fundamentals Proof of the Runge-Gross Theorem (1/4) same Assume Taylor- expandability: The first part of the proof shows that the current-densities must differ. Consider Heisenberg e.o.m’s for the current-density in each system, ;t ) the part of H that differs in the two systems initial density  if initially the 2 potentials differ, then j and j’ differ infinitesimally later ☺ At the initial time: 2. Fundamentals Proof of the Runge-Gross Theorem (2/4)   n v for given Ψ0, implies any observable is a functional of n and Ψ0 -- So map interacting system to a non-interacting (Kohn-Sham) one, that reproduces the same n(r,t). All properties of the true system can be extracted from TDKS  “bigger-faster- cheaper” calculations of spectra and dynamics KS “electrons” evolve in the 1-body KS potential: functional of the history of the density and the initial states -- memory-dependence (see more shortly!)   If begin in ground-state, then no initial-state dependence, since by HK, Ψ0 = Ψ0[n(0)] (eg. in linear response). Then 2. Fundamentals The TDKS system   The KS potential is not the density-functional derivative of any action ! If it were, causality would be violated: Vxc[n,Ψ0,Φ0](r,t) must be causal – i.e. cannot depend on n(r t’>t) But if 2. Fundamentals Clarifications and Extensions   But how do we know a non-interacting system exists that reproduces a given interacting evolution n(r,t) ?   van Leeuwen (PRL, 1999) (under mild restrictions of the choice of the KS initial state Φ0) But RHS must be symmetric in (t,t’)  symmetry-causality paradox.  van Leeuwen (PRL 1998) showed how an action, and variational principle, may be defined, using Keldysh contours. then 2. Fundamentals Clarifications and Extensions   Restriction to Taylor-expandable potentials means RG is technically not valid for many potentials, eg adiabatic turn-on, although RG is assumed in practise. van Leeuwen (Int. J. Mod. Phys. B. 2001) extended the RG proof in the linear response regime to the wider class of Laplace-transformable potentials.   The first step of the RG proof showed a 1-1 mapping between currents and potentials  TD current-density FT In principle, must use TDCDFT (not TDDFT) for -- response of periodic systems (solids) in uniform E-fields -- in presence of external magnetic fields (Maitra, Souza, Burke, PRB 2003; Ghosh & Dhara, PRA, 1988) In practice, approximate functionals of current are simpler where spatial non- local dependence is important (Vignale & Kohn, 1996; Vignale, Ullrich & Conti 1997) … Stay tuned! 3. TDKS Time-dependent Kohn-Sham scheme (2) Only the N initially occupied orbitals are propagated. How can this be sufficient to describe all possible excitation processes?? Here’s a simple argument: Expand TDKS orbitals in complete basis of static KS orbitals, A time-dependent potential causes the TDKS orbitals to acquire admixtures of initially unoccupied orbitals. finite for 3. TDKS Adiabatic approximation depends on density at time t (instantaneous, no memory) is a functional of The time-dependent xc potential has a memory! Adiabatic approximation: (Take xc functional from static DFT and evaluate with time-dependent density) ALDA: 3. TDKS Time-dependent selfconsistency (1) time start with selfconsistent KS ground state propagate until here I. Propagate II. With the density calculate the new KS potential III. Selfconsistency is reached if for all 1 2 3 Prepare the initial state, usually the ground state, by a static DFT calculation. This gives the initial orbitals: Solve TDKS equations selfconsistently, using an approximate time-dependent xc potential which matches the static one used in step 1. This gives the TDKS orbitals: Calculate the relevant observable(s) as a functional of 3. TDKS Summary of TDKS scheme: 3 Steps 3. TDKS Example: two electrons on a 2D quantum strip periodic boundaries (travelling waves) hard walls (standing waves) z x C.A. Ullrich, J. Chem. Phys. 125, 234108 (2006) L Charge-density oscillations Δ initial-state density exact LDA ● Initial state: constant electric field, which is suddenly switched off ● After switch-off, free propagation of the charge-density oscillations Step 1: solve full 2-electron Schrödinger equation Step 2: calculate the exact time-dependent density Step 3: find that TDKS system which reproduces the density 3. TDKS Construction of the exact xc potential 1. A survey of time-dependent phenomena 2. Fundamental theorems in TDDFT 3. Time-dependent Kohn-Sham equation 4. Memory dependence 5. Linear response and excitation energies 6. Optical processes in Materials 7. Multiple and charge-transfer excitations 8. Current-TDDFT 9. Nanoscale transport 10. Strong-field processes and control Outline C.U. N.M. C.U. N.M. N.M. C.U. N.M. C.U. C.U. N.M. Almost all calculations today ignore this, and use an “adiabatic approximation” : vxc functional dependence on history, n(r t’<t), and on initial states Just take xc functional from static DFT and evaluate on instantaneous density But what about the exact functional? 4. Memory Memory dependence parametrizes density Hessler, Maitra, Burke, (J. Chem. Phys, 2002); Wijewardane & Ullrich, (PRL 2005); Ullrich (JCP, 2006) Any adiabatic (or even semi-local-in-time) approximation would incorrectly predict the same vc at both times. Eg. Time-dependent Hooke’s atom –exactly solvable 2 electrons in parabolic well, time-varying force constant k(t) =0.25 – 0.1*cos(0.75 t) •  Development of History-Dependent Functionals: Dobson, Bunner & Gross (1997), Vignale, Ullrich, & Conti (1997), Kurzweil & Baer (2004), Tokatly (2005) 4. Memory Example of history dependence 1. A survey of time-dependent phenomena 2. Fundamental theorems in TDDFT 3. Time-dependent Kohn-Sham equation 4. Memory dependence 5. Linear response and excitation energies 6. Optical processes in Materials 7. Multiple and charge-transfer excitations 8. Current-TDDFT 9. Nanoscale transport 10. Strong-field processes and control Outline C.U. N.M. C.U. N.M. N.M. C.U. N.M. C.U. C.U. N.M. 5. Linear Response TDDFT in linear response Poles at KS excitations Poles at true excitations adiabatic approx: no ω-dep Need (1) ground-state vS,0[n0](r), and its bare excitations (2) XC kernel Yields exact spectra in principle; in practice, approxs needed in (1) and (2). Petersilka, Gossmann, Gross, (PRL, 1996) Well-separated single excitations: SMA When shift from bare KS small: SPA Useful tools for analysis: “single-pole” and “small-matrix” approximations (SPA,SMA) Zoom in on a single KS excitation, q = i a Quantum chemistry codes cast eqns into a matrix of coupled KS single excitations (Casida 1996) : Diagonalize 5. Linear Response Matrix equations (a.k.a. Casida’s equations) q = (i  a)  Excitation energies and oscillator strengths Optical Spectrum of DNA fragments HOMO LUMO d(GC) π-stacked pair D. Varsano, R. Di Felice, M.A.L. Marques, A Rubio, J. Phys. Chem. B 110, 7129 (2006). Can study big molecules with TDDFT ! 5. Linear response Examples Circular dichroism spectra of chiral fullerenes: D2C84 F. Furche and R. Ahlrichs, JACS 124, 3804 (2002). 5. Linear response Examples 1. A survey of time-dependent phenomena 2. Fundamental theorems in TDDFT 3. Time-dependent Kohn-Sham equation 4. Memory dependence 5. Linear response and excitation energies 6. Optical processes in Materials 7. Multiple and charge-transfer excitations 8. Current-TDDFT 9. Nanoscale transport 10. Strong-field processes and control Outline C.U. N.M. C.U. N.M. N.M. C.U. N.M. C.U. C.U. N.M. 6. TDDFT in solids Excitations in bulk metals Quong and Eguiluz, PRL 70, 3955 (1993) Plasmon dispersion of Al ►RPA (i.e., Hartree) gives already reasonably good agreement ►ALDA agrees very well with exp. In general, (optical) excitation processes in (simple) metals are very well described by TDDFT within ALDA. Time-dependent Hartree already gives the dominant contribution, and fxc typically gives some (minor) corrections. This is also the case for 2DEGs in doped semiconductor heterostructures 6. TDDFT in solids Semiconductor heterostructures ●semiconductor heterostructures are grown with MBE or MOCVD ●control and design through layer-by-layer variation of material composition ●widely used class or materials: III-V compounds Interband transitions: of order eV (visible to near-IR) Intersubband transitions: of order meV (mid- to far-IR) CB lower edge VB upper edge ● Donor atoms separated from quantum well: modulation delta doping ● Total sheet density Ns typically ~1011 cm-2 6. TDDFT in solids n-doped quantum wells 6. TDDFT in solids Quantum well subbands Y €3 U\ pAf “ 6. TDDFT in solids Intersubband plasmon dispersions k (Å-1) ω (m eV ) C.A.Ullrich and G.Vignale, PRL 87, 037402 (2002) charge plasmon spin plasmon experiment 6. TDDFT in solids Optical absorption of insulators G. Onida, L. Reining, A. Rubio, RMP 74, 601 (2002) S. Botti, A. Schindlmayr, R. Del Sole, L. Reining, Rep. Prog. Phys. 70, 357 (2007) RPA and ALDA both bad! ►absorption edge red shifted (electron self-interaction) ►first excitonic peak missing (electron-hole interaction) Silicon Why does the ALDA fail?? 6. TDDFT in solids Optical absorption of insulators, again F. Sottile et al., PRB 76, 161103 (2007) Silicon Kim & Görling Reining et al. 6. TDDFT in solids Extended systems - summary ► TDDFT works well for metallic and quasi-metallic systems already at the level of the ALDA. Successful applications for plasmon modes in bulk metals and low-dimensional semiconductor heterostructures. ► TDDFT for insulators is a much more complicated story: ● ALDA works well for EELS (electron energy loss spectra), but not for optical absorption spectra ● difficulties originate from long-range contribution to fxc ● some long-range XC kernels have become available, but some of them are complicated. Stay tuned…. ● Nonlinear real-time dynamics including excitonic effects: TDDFT version of Semiconductor Bloch equations V.Turkowski and C.A.Ullrich, PRB 77, 075204 (2008) (Wednesday P13.7) 1. A survey of time-dependent phenomena 2. Fundamental theorems in TDDFT 3. Time-dependent Kohn-Sham equation 4. Memory dependence 5. Linear response and excitation energies 6. Optical processes in Materials 7. Multiple and charge-transfer excitations 8. Current-TDDFT 9. Nanoscale transport 10. Strong-field processes and control Outline C.U. N.M. C.U. N.M. N.M. C.U. N.M. C.U. C.U. N.M. χ  – poles at true states that are mixtures of singles, doubles, and higher excitations χ S -- poles only at single KS excitations, since one-body operator can’t connect Slater determinants differing by more than one orbital.   χ has more poles than χs ? How does fxc generate more poles to get states of multiple excitation character? Excitations of interacting systems generally involve mixtures of (KS) SSD’s that have either 1,2,3…electrons in excited orbitals. single-, double-, triple- excitations 7. Where the usual approxs. fail Double Excitations Now consider: Exactly Solve a Simple Model: one KS single (q) mixing with a nearby double (D) Strong non-adiabaticity! Invert and insert into Dyson-like eqn for kernel dressed SPA (i.e. ω-dependent): 7. Where the usual approxs. fail Double Excitations General case: Diagonalize many-body H in KS subspace near the double ex of interest, and require reduction to adiabatic TDDFT in the limit of weak coupling of the single to the double  NTM, Zhang, Cave,& Burke JCP (2004), Casida JCP (2004) 7. Where the usual approxs. fail Double Excitations usual adiabatic matrix element dynamical (non-adiabatic) correction Eg. Zincbacteriochlorin-Bacteriochlorin complex (light-harvesting in plants and purple bacteria) Dreuw & Head-Gordon, JACS 126 4007, (2004). TDDFT predicts CT states energetically well below local fluorescing states. Predicts CT quenching of the fluorescence. ! Not observed ! TDDFT error ~ 1.4eV TDDFT typically severely underestimates long-range CT energies Import ant pro cess in biomol ecules , large enoug h that T DDFT may be only feasibl e appr oach ! 7. Where the usual approxs. fail Long-Range Charge-Transfer Excitations We know what the exact energy for charge transfer at long range should be: Why TDDFT typically severely underestimates this energy can be seen in SPA -As,2 -I1 (Also, usual g.s. approxs underestimate I) Why do the usual approximations in TDDFT fail for these excitations? exact i.e. get just the bare KS orbital energy difference: missing xc contribution to acceptor’s electron affinity, Axc,2, and -1/R 7. Where the usual approxs. fail Long-Range Charge-Transfer Excitations ~0 overlap What are the properties of the unknown exact xc kernel that must be well- modelled to get long-range CT energies correct ?   Exponential dependence on the fragment separation R, fxc ~ exp(aR)   For transfer between open-shell species, need strong frequency-dependence. Gritsenko & Baerends (PRA, 2004), Maitra (JCP, 2005), Tozer (JCP, 2003) Tawada et al. (JCP, 2004) Step in Vxc re-aligns the 2 atomic HOMOs  near-degeneracy of molecular HOMO & LUMO  static correlation, crucial double excitations  frequency-dependence! (It’s a rather ugly kernel…) “LiH” 7. Where the usual approxs. fail Long-Range Charge-Transfer Excitations step Visualize electron dynamics as the motion (and deformation) of infinitesimal fluid elements: Nonlocality in time (memory) implies nonlocality in space! Dobson, Bünner, and Gross, PRL 79, 1905 (1997) I.V. Tokatly, PRB 71, 165105 (2005) 8. TDCDFT Nonlocality in space and time Zero-force theorem: Linearized form: If the xc kernel has a finite range, we can write for slowly varying systems: l.h.s. is frequency-dependent, r.h.s is not: contradiction! has infinitely long spatial range! 8. TDCDFT Ultranonlocality in TDDFT ● x x0 An xc functional that depends only on the local density (or its gradients) cannot see the motion of the entire slab. A density functional needs to have a long range to see the motion through the changes at the edges. 8. TDCDFT Ultranonlocality and the density uniform velocity oscillating velocity much better chance to capture the physics correctly! 8. TDCDFT Point of view of local current nonlocal nonlocal nonlocal local ● Continuity equation only gives the longitudinal current ● TDCDFT gives also the transverse current ● We can find a short-range current-dependent xc vector potential 8. TDCDFT Upgrading TDDFT: time-dependent Current-DFT generalization of RG theorem: Ghosh and Dhara, PRA 38, 1149 (1988) G. Vignale, PRB 70, 201102 (2004) uniquely determined up to gauge transformation full current can be represented by a KS system 8. TDCDFT Basics of TDCDFT ● automatically satisfies zero-force theorem/Newton’s 3rd law ● automatically satisfies the Harmonic Potential theorem ● is local in the current, but nonlocal in the density ● introduces dissipation/retardation effects xc viscoelastic stress tensor: velocity field 8. TDCDFT TDCDFT beyond the ALDA: the VK functional G. Vignale and W. Kohn, PRL 77, 2037 (1996) G. Vignale, C.A. Ullrich, and S. Conti, PRL 79, 4878 (1997) In contrast with the classical case, the xc viscosities have both real and imaginary parts, describing dissipative and elastic behavior: shear modulus dynamical bulk modulus reflect the stiffness of Fermi surface against defor- mations 8. TDCDFT XC viscosity coefficients GK: E.K.U. Gross and W. Kohn, PRL 55, 2850 (1985) NCT: R. Nifosi, S. Conti, and M.P. Tosi, PRB 58, 12758 (1998) QV: X. Qian and G. Vignale, PRB 65, 235121 (2002) 8. TDCDFT xc kernels of the homogeneous electron gas 1. A survey of time-dependent phenomena 2. Fundamental theorems in TDDFT 3. Time-dependent Kohn-Sham equation 4. Memory dependence 5. Linear response and excitation energies 6. Optical processes in Materials 7. Multiple and charge-transfer excitations 8. Current-TDDFT 9. Nanoscale transport 10. Strong-field processes and control Outline C.U. N.M. C.U. N.M. N.M. C.U. N.M. C.U. C.U. N.M. Koentopp, Chang, Burke, and Car (2008) two-terminal Landauer formula Transmission coefficient, usually obtained from DFT-nonequilibrium Green’s function 9. Transport DFT and nanoscale transport Problems: ● standard xc functionals (LDA,GGA) inaccurate ● unoccupied levels not well reproduced in DFT transmission peaks can come out wrong conductances often much overestimated need need better functionals (SIC, orbital-dep.) and/or TDDFT Current response: XC piece of voltage drop: Current-TDDFT Sai, Zwolak, Vignale, Di Ventra, PRL 94, 186810 (2005) dynamical resistance: ~10% correction 9. Transport TDDFT and nanoscale transport: weak bias In addition to an approximation for vxc[n;Ψ0,Φ0](r,t), also need an approximation for the observables of interest. Certainly measurements involving only density (eg dipole moment) can be extracted directly from KS – no functional approximation needed for the observable. But generally not the case. We’ll take a look at: High-harmonic generation (HHG) Above-threshold ionization (ATI) Non-sequential double ionization (NSDI) Attosecond Quantum Control Correlated electron-ion dynamics  Is the relevant KS quantity physical ? 10. Strong-field processes TDDFT for strong fields Erhard & Gross, (1996) Eg. He correlation reduces peak heights by ~ 2 or 3 TDHF 10. Strong-field processes High Harmonic Generation HHG: get peaks at odd multiples of laser frequency Measures dipole moment, |d(ω)|2 = ∫ n(r,t) r d3r so directly available from TD KS system L’Huillier (2002) Nguyen, Bandrauk, and Ullrich, PRA 69, 063415 (2004). Eg. Na-clusters 30 Up λ= 1064 nm I = 6 x 1012 W/ cm2 pulse length 25 fs •  TDDFT is the only computationally feasible method that could compute ATI for something as big as this! •  ATI measures kinetic energy of electrons – not directly accessible from KS. Here, approximate T by KS kinetic energy. • TDDFT yields plateaus much longer than the 10 Up predicted by quasi-classical one- electron models 10. Strong-field processes Above-threshold ionization ATI: Measure kinetic energy of ejected electrons L’Huillier (2002) Eg. Collisions of O atoms/ions with graphite clusters Isborn, Li. Tully, JCP 126, 134307 (2007) 10. Strong-field processes Coupled electron-ion dynamics Classical nuclei coupled to quantum electrons, via Ehrenfest coupling, i.e. http://www.tddft.org Castro, Appel, Rubio, Lorenzen, Marques, Oliveira, Rozzi, Andrade, Yabana, Bertsch Freely-available TDDFT code for strong and weak fields: 10. Strong-field processes Coupled electron-ion dynamics !! essentia l for photo chemistry, relaxation , electron tr ansfer, bra nching rat ios, reactions n ear surfac es... How about Surface-Hopping a la Tully with TDDFT ? Simplest: nuclei move on KS PES between hops. But, KS PES ≠ true PES, and generally, may give wrong forces on the nuclei. Should use TDDFT-corrected PES (eg calculate in linear response). But then, trajectory hopping probabilities cannot be simply extracted – e.g. they depend on the coefficients of the true Ψ (not accessible in TDDFT), and on non-adiabatic couplings. Craig, Duncan, & Prezhdo PRL 2005, Tapavicza, Tavernelli, Rothlisberger, PRL 2007, Maitra, JCP 2006 Classical Ehrenfest method misses electron-nuclear correlation (“branching” of trajectories) To learn more… Time-dependent density functional theory, edited by M.A.L. Marques, C.A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E.K.U. Gross, Springer Lecture Notes in Physics, Vol. 706 (2006) Upcoming TDDFT conferences: ● 3rd International Workshop and School on TDDFT Benasque, Spain, August 31 - September 15, 2008 http://benasque.ecm.ub.es/2008tddft/2008tddft.htm ● Gordon Conference on TDDFT, Summer 2009 http://www.grc.org (see handouts for TDDFT literature list)