Time-dependent density-functional theory APS March Meeting 2008, New Orleans Nee
Downs, Maggie, Features Reporter has reference to this Academic Journal, PHwiki organized this Journal Time-dependent density-functional theory 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 in addition to excitation energies 6. Optical processes in Materials 7. Multiple in addition to charge-transfer excitations 8. Current-TDDFT 9. Nanoscale transport 10. Strong-field processes in addition to control C.U. N.M. C.U. N.M. N.M. C.U. N.M. C.U. C.U. N.M. 1. Survey Time-dependent Schrödinger equation kinetic energy operator: electron interaction: The TDSE describes the time evolution of a many-body state starting from an initial state under the influence of an external time-dependent potential From now on, well (mostly) use atomic units (e = m = h = 1).
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Start from nonequilibrium initial state, evolve in static potential: t=0 t>0 1. Survey Real-time electron dynamics: first scenario Charge-density oscillations in metallic clusters or nanoparticles (plasmonics) New J. Chem. 30, 1121 (2006) Nature Mat. Vol. 2 No. 4 (2003) Start from ground state, evolve in time-dependent driving field: t=0 t>0 Nonlinear response in addition to ionization of atoms in addition to 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 1. Survey Optical spectroscopy Uses weak CW laser as Probe System Response has peaks at electronic excitation energies Marques et al., PRL 90, 258101 (2003) Green fluorescent protein 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 in addition to excitation energies 6. Optical processes in Materials 7. Multiple in addition to charge-transfer excitations 8. Current-TDDFT 9. Nanoscale transport 10. Strong-field processes in addition to control C.U. N.M. C.U. N.M. N.M. C.U. N.M. C.U. C.U. N.M.
2. Fundamentals Runge-Gross Theorem Runge & Gross (1984) proved the 1-1 mapping: n(r t) vext(r t) For a given initial-state y0, the time-evolving one-body density n(r t) tells you everything about the time-evolving interacting electronic system, exactly. This follows from : Y0, n(r,t) unique vext(r,t) H(t) Y(t) all observables For any system with Hamiltonian of as long as m H = T + W + Vext , e-e interaction kinetic external potential Consider two systems of N interacting electrons, both starting in the same Y0 , but evolving under different potentials vext(r,t) in addition to vext(r,t) respectively: RG prove that the resulting densities n(r,t) in addition to n(r,t) eventually must differ, i.e. 2. Fundamentals Proof of the Runge-Gross Theorem (1/4) Assume Taylor-exp in addition to ability: The first part of the proof shows that the current-densities must differ. Consider Heisenberg e.o.ms as long as the current-density in each system, the part of H that differs in the two systems initial density if initially the 2 potentials differ, then j in addition to j differ infinitesimally later At the initial time: 2. Fundamentals Proof of the Runge-Gross Theorem (2/4)
Take div. of both sides of in addition to use the eqn of continuity, As vext(r,t) vext(r,t) = c(t), in addition to assuming potentials are Taylor-exp in addition to able at t=0, there must be some k as long as which RHS = 0 The second part of RG proves 1-1 between densities in addition to potentials: 2. Fundamentals Proof of the Runge-Gross Theorem (3/4) If vext(r,0) = vext(r,0), then look at later times by repeatedly using Heisenberg e.o.m : 1st part of RG 1-1 mapping between time-dependent densities in addition to potentials, as long as a given initial state u(r) is nonzero as long as some k, but must taking the div here be nonzero Yes! By reductio ad absurdum: assume Then assume fall-off of n0 rapid enough that surface-integral 0 integr in addition to 0, so if integral 0, then contradiction 2. Fundamentals Proof of the Runge-Gross Theorem (4/4) i.e. n v as long as given Y0, implies any observable is a functional of n in addition to Y0 – 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 in addition to dynamics KS electrons evolve in the 1-body KS potential: functional of the history of the density in addition to the initial states – memory-dependence (see more shortly!) If begin in ground-state, then no initial-state dependence, since by HK, Y0 = Y0[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,Y0,F0](r,t) must be causal i.e. cannot depend on n(r t>t) But if 2. Fundamentals Clarifications in addition to 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 F0) But RHS must be symmetric in (t,t) symmetry-causality paradox. van Leeuwen (PRL 1998) showed how an action, in addition to variational principle, may be defined, using Keldysh contours. then 2. Fundamentals Clarifications in addition to Extensions Restriction to Taylor-exp in addition to able potentials means RG is technically not valid as long as 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-trans as long as mable potentials. The first step of the RG proof showed a 1-1 mapping between currents in addition to potentials TD current-density FT In principle, must use TDCDFT (not TDDFT) as long as – response of periodic systems (solids) in uni as long as m 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! 1. A survey of time-dependent phenomena 2. Fundamental theorems in TDDFT 3. Time-dependent Kohn-Sham equation 4. Memory dependence 5. Linear response in addition to excitation energies 6. Optical processes in Materials 7. Multiple in addition to charge-transfer excitations 8. Current-TDDFT 9. Nanoscale transport 10. Strong-field processes in addition to control Outline C.U. N.M. C.U. N.M. N.M. C.U. N.M. C.U. C.U. N.M.
3. TDKS Time-dependent Kohn-Sham scheme (1) Consider an N-electron system, starting from a stationary state. Solve a set of static KS equations to get a set of N ground-state orbitals: The N static KS orbitals are taken as initial orbitals in addition to will be propagated in time: Time-dependent density: 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 Heres a simple argument: Exp in addition to 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 as long as 3. TDKS Adiabatic approximation Adiabatic approximation: (Take xc functional from static DFT in addition to evaluate with time-dependent density) ALDA:
3. TDKS Time-dependent selfconsistency (1) 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 as long as all 3. TDKS Numerical time Propagation Propagate a time step Crank-Nicholson algorithm: Problem: must be evaluated at the mid point But we know the density only as long as times 3. TDKS Time-dependent selfconsistency (2) Predictor Step: nth Corrector Step: Selfconsistency is reached if remains unchanged as long as upon addition of another corrector step in the time propagation.
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 C.A. Ullrich, J. Chem. Phys. 125, 234108 (2006) 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
3. TDKS Construction of the exact xc potential Ansatz: 3. TDKS 2D quantum strip: charge-density oscillations The TD xc potential can be constructed from a TD density Adiabatic approximations get most of the qualitative behavior right, but there are clear indications of nonadiabatic (memory) effects Nonadiabatic xc effects can become important (see later) 1. A survey of time-dependent phenomena 2. Fundamental theorems in TDDFT 3. Time-dependent Kohn-Sham equation 4. Memory dependence 5. Linear response in addition to excitation energies 6. Optical processes in Materials 7. Multiple in addition to charge-transfer excitations 8. Current-TDDFT 9. Nanoscale transport 10. Strong-field processes in addition to control Outline C.U. N.M. C.U. N.M. N.M. C.U. N.M. C.U. C.U. N.M.
To learn more Time-dependent density functional theory, edited by M.A.L. Marques, C.A. Ullrich, F. Nogueira, A. Rubio, K. Burke, in addition to E.K.U. Gross, Springer Lecture Notes in Physics, Vol. 706 (2006) Upcoming TDDFT conferences: 3rd International Workshop in addition to 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 h in addition to outs as long as TDDFT literature list) Acknowledgments Harshani Wijewardane Volodymyr Turkowski Ednilsom Orestes Yonghui Li David Tempel Arun Rajam Christian Gaun August Krueger Gabriella Mullady Allen Kamal Giovanni Vignale (Missouri) Kieron Burke (Irvine) Ilya Tokatly (San Sebastian) Irene DAmico (York/UK) Klaus Capelle (Sao Carlos/Brazil) Meta van Faassen (Groningen) Adam Wasserman (Harvard) Hardy Gross (FU-Berlin) Collaborators: Students/Postdocs:
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