Fundamentals of Time-Dependent Density Functional Theory II - PowerPoint Presentation

Fundamentals of Time-Dependent Density Functional TheoryII Neepa T. MaitraHunter College and the Graduate Center of the City University of New York

Plan – introduction to what is memory -- adiabatic & non-adiabatic approximations and some exact conditions -- initial-state dependence example -- history-dependence example -- time-dependent spectra, if time permits..

Whence Memory? History: n(r t’<t), and, initial states Y 0 , and F 0 of true and KS systems Hartree is naturally adiabatic – depends only on instantaneous density Actually, v ext [n,Y0] (rt) but vext is prescribed by problem at hand, so functional dependence not directly important. Y0 : n vext Recall Runge-Gross: Y0: the true initial stateF0: the initial state to start the KS calculation -- can choose any state that has same n(r,0) and n(r,0) as Y0 . Vxc is different for each choice   true system KS system usually choose Slater determinant but not necessary 1-1 F0 : n vS 1-1

Memory Memory can be thought to arise from using a reduced variable, n( r,t): tracing over N-1 spatial variables  memory-dependence. functional depend s on history, n(r t’<t), and on initial states of true and KS systems Also, for a general observable: A[ n ; F 0] Special, and common, case: Y0 = Y GS F0 = FGSThen, by the Hohenberg -Kohn theorem, Y0 = Y0[n(0)] and F0 = F0 [n(0)] -- no explicit initial-state-dependence  vxc[n]( r,t) e.g. linear response regime.

The Adiabatic Approximation Almost all calculations today ignore memory, and use an adiabatic approximation : input instantaneous density into a ground-state approximation Example:

The Adiabatic Approximation Almost all calculations today ignore memory, and use an adiabatic approximation : input instantaneous density into a ground-state approximation Two sources of error: Adiabatic approximation itself Ground-state functional approximation To disentangle, study “adiabatically-exact” potential: v xcA-ex ( r,t) = vxcexact-gs[n(t)](r)

Development of Memory-Dependent Functionals …Gross-Kohn (1985) Phys. Rev. Lett. 55, 2850 (1985) linear-response kernel of the uniform electron gas at finite frequency Non-adiabatic -- time-non-local although spatially local; “finite-frequency LDA” Violates exact conditions: harmonic potential theorem, zero-force theorem

A couple of exact conditions in TDDFT: Harmonic Potential Theorem (Dobson (PRL 73, 2244, (1994); Vignale PRL 74, 3233, (1995)) N electrons in a harmonic well subject to a TD uniform electric field, E(t)  density rigidly sloshes back and forth following classical center of mass oscillations n( r ,t ) = n GS ( r – r CM (t)) Vxc( r,t) = VxcGS( r – rCM(t)) Instead, GK finds an n-dependent shift in the frequency of the CM motion, and a damping of the oscillations. One way to think about why is that when you only look locally at the density at r, you can’t tell difference between sloshing motion and local compression/rarefaction ….(whiteboard sketch)

A couple of exact conditions in TDDFT: Zero Force Theorem (Vignale PRL 74, 3233, (1995 ); Phys. Lett. A, 209, 206 (1995) ) xc field cannot exert a net force on itself Exercise! Prove this! Hint: evaluate <r> using Y (t) and then F (t); then subtract…   L inear response regime: Another exercise! Prove this too! Using GK: f xc unif[ n0(r)](q=0,w) w-dependentw-independent The exact conditions imply time-non-locality  spatially non-local n-dependence, i.e. a local density approximation with memory does not exist. even in limit of slowly-varying densities  “ultra-non-locality”

… Development of Memory-Dependent Functionals Vignale-Kohn (VK) (1996) – spatially local approx in terms of the current-density, j( r ,t )  TD-current-density-FT Phys . Rev. Lett . 77, 2037 (1996) Dobson- B ünner-Gross (1997)Phys. Rev. Lett. 79, 1905 (1997)Apply Gross-Kohn in frame that moves along with local velocity of electron fluid: memory resides with the fluid element.Spatially-local relative to where a fluid element at (r,t) was at earlier times t’, R’( t’|r,t) r,t R’,t’Non-adiabatic, and satisfies harmonic potential theorem, zero-force theorem j A ext 1-1 Y 0 Based on map: Non-adiabatic, and satisfies harmonic potential theorem, zero-force theorems…VK constructed from dynamical longitudinal and transverse responses to slowly-varying perturbations of uniform electron liquid; involves Navier-Stokes-like eqn with complex viscosity coefficients . Andre Schleife’s lecture Saturday

… A little more about Vignale-Kohn and TDCDFT…Vignale -Ullrich-Conti (1997) – extended VK to non-linear regime. G. Vignale, C.A. Ullrich, and S. Conti, PRL 79, 4878 (1997 ) Note that RG’s 1 st step was j  v ext Using A instead of v makes it easier to satisfy non-interacting representability: many currents of interacting systems in scalar potentials can only be reproduced by a non-interacting systems in vector potentials Note that spatially local knowledge of current j  spatially ultra-nonlocal dependence on density n Seen some success for: correcting overestimate of LDA polarizabilities in long-chain polymers, dissipation in extended systems, spin-Coulomb drag, stopping power in metals….BUT problems for finite systems due to spurious dampingE.g. So even for static response (no memory), VK can help when spatial-non-locality important.

… Other Memory-Dependent Functionals Tokatly (2005, 2007) –TD-deformation-FT Ch. 25 in “Fundamentals of TDDFT” book, I.V. Tokatly , PRB 71, 165104 and 165105 (2005); PRB 75 , 125105 (2007) Formulate density & current dynamics in a Lagrangian frame. Since moving with the flow, spatially local xc is sensible & all complications including memory are contained in Green’s deformation tensor g i j Orbital functionals v xc[{fi(t)}]– instantaneous KS orbitals incorporate “infinite KS memory”Computationally more involved: TDOEPDevelopment of true ISD-Functionals? none yet! Nevertheless, ISD and history-dependence are intimately entangled….next slide.. Kurzweil & Baer (2004, 2005, 2006) – Galilean- invariant “memory action functional”, J. Chem. Phys. 121 , 8731 (2004).

“Memory” condition History and initial-state dependence are entangledVxc [n t’ ; Y 0 = Y (t’), F 0 = F (t’) ] ( r,t ) independent of t’ (for t>t’) t t ’ t ’ t ’ density n t’ defined only from t’ onwardThis is a very hard condition to satisfy for non-adiabatic functionals.Exercise! Does ALDA satisfy this? Do you think VK satisfies this? n (t) Maitra, Burke, Woodward, PRL 89, 023002 (2002)

Trading ISD for more history Evolve initial states backward in time, in some potential, to a ground-state  no ISD due to Hohenberg -Kohn  instead, must tack on extra piece of “pseudo pre-history” V xc [n; Y 0 , F 0 ]( r t) = V xc[n](r t) ~ n (r t) t -T -T’ t n (r t) Starts at t=0 in initial true state Y 0 and KS evolves from initial state F 0 ~ Starts at some time –T from some ground state: “initial” ground-state (any) pseudoprehistory The pseudoprehistory is not unique – may find many ground-states that evolve to the same state at t=0, in different amounts of time, in different v’s. Eqn applies to all – gives a strict exact test for approximate functionals . “memory condition”

So far, have done a lot of formalism. Now let’s take a look at some examples where memory is at play.

Implications for the Adiabatic Approximation: v xc [n;Y0,F0 ]( r,t ) Adiabatic approx designed to work for initial ground- states -- For initial Y 0 an excited state, say, these use v xc derived for a ground-state of the same initial n(r). Important in photoinduced dynamics generally: start the actual dynamics simulation after initial photo-excitation. d epends on the choice of the initial KS state How big is the resulting error?n(r,t) does not uniquely define the potential, need ISD Initial-State Dependence (ISD)

For a given initial excited Y0, how different are the exact Vxc ’ s for different choices of KS initial F 0 ? Is there a “best choice” of KS initial state to use if stuck with an adiabatic approx ? e.g. Start in 1st interacting excited state of 1D He d ensity, n* Initial exact v xc for F gs for F* vc A-ex quite wrongBut it’s closer to choice of F* than Fgs P. Elliott and N. T. Maitra , PRA 85, 052510 (2012) Many different choices of initial KS state: e.g :F* = singlet excited state (two orbitals) Fgs = singlet ground state (one orbital)All must have density n* KS potential x c potential

More examples, including ensuing time-development, recently demonstrated using a ne w fix ed-point iteration method to find the v(r,t ) for a given n( r,t ) & Y 0 and different choices of F 0 : M. Ruggenthaler, S. E. B. Nielsen, R. van Leeuw en, PRA 88 , 022512 (2013 ).S. E. B. Nielsen, M. Ruggenthaler, R. van Leeuwen , EPL, 101 (2013) 33001J. I. Fuks , S. E. B. Nielsen, M. Ruggenthaler, N. T. Maitra, PCCP 18, 20976 (2016)

An example of history dependence2 soft-Coulomb interacting fermions living in and subject to exact v c “adiabatically-exact” v c adia -ex = v c exact-gs [n(t)] densityMovie d ensity Vc Weak on-resonant driving of 1D He atom:A = 0.00667 auw = 0.533 au (1st excn.)Non-adiabatic step-peak features appear generically in non-perturbative dynamics. E.g. Elliott, Fuks, Rubio, Maitra, PRL 109, 266404 (2012)Ramsden, Godby, PRL 109, 036402 (2012)For more references, see Perspective in J. Chem. Phys. 144 , 220901(2016).

Finding the exact xc potential for a given known density-evolution One can show: where So, problem becomes finding the exact KS orbitals -- generally difficult, but possible, Nielsen , Ruggenthaler , van Leeuwen , Europhys . Lett. 101, 33001 (2013) O ne easy case: 2 electrons spin-singlet in 1D, in a doubly-occupied KS orbitalni(x,t)  ½ n( x,t), the exact density, and ui( x,t) j(x,t)/n(x,t ), where j is the exact current-density Generally, not so easy. orbital-density and orbital-phase of any one of the occupied orbitals In 1D, can express in terms of orbital-density and orbital-”velocity”, ui = ji /n i into TDKS Exercise: show this!

Expression directly for the exact exchange-correlation potential … TD one-body density-matrix: interacting, KS Equate equation of motion for n( r,t ) / coming from interacting system,   with that of the KS system, and then subtract  TD exchange-correlation hole Those expressions are directly for v s ; to find v xc we must subtract Hartree and v extNote there is also an expression directly for vxc: Exact expression for the TD exchange-correlation potential

A class of phenomena where the lack of memory-dependence really screws up the dynamics -- Resonantly-driven dynamics-- Time-resolved pump-probe spectroscopy …and the “explanation” in terms of their violation of another exact condition Finally, if time…

Spectrum depends on pump-probe delay, but if the nuclei don’t move, then the peak positions don’t change. Approximate TDDFT functionals violate this  muddled interpretation of spectra Violation of exact condition explains erroneously “t-dep electronic structure” observed, e.g. de Giovannini et al. PCCP 14 , 1363 (2014); Raganuthan & Nest JCTC 8 , 806 (2012); Habenicht et al. JCP 141, 184112 (2014); Fischer et al. JCTC 11, 4294 (2015) Time-Resolved Pump-Probe Spectroscopy T aken from Krausz & Ivanov, Rev. Mod. Phys. 81, 163 (2009)

 Can be formalized as a new exact condition on the generalized xc kernel. The ATDDFT frequencies typically change in time, unlike the exact, even if Vext(r) only time-dep. n(t)  v xc [n](t) time-dep Bare KS frequencies change in time & fxc correction also time-dependent These two time-dependences cancel with exact functional, but not with approximate functionals! Y gs f ield-free evolution T n(0) Measure response of to a perturbation  Generalized density-density response functions around the state at time when field is turned off, T :

J. I. Fuks , K. Luo. E. Sandoval, N. T. Maitra, Phys. Rev. Lett. 114, 183002 (2015 ).K. Luo, J. I. Fuks , N. T. Maitra, J. Chem. Phys. 145 , 044101 (2016). L et w i be a pole of the Fourier transform w.r.t. (t-t’) of then Y gs f ield-free evolution T n (0) Measure response of to a perturbation  “resonance condition”

E.g. Resonantly-driven dynamics: Apply a weak field resonant with the charge-transfer excitation shown in inset (Rabi) --> large change in dipole moment as the charge transfers Most functionals fail because of their violation of the resonance condition. In this case (not generally), Adiabatic EXX fully charge-transfers the photo-excited electron: Since hardly changes during the evolution, this is basically static.  

Time-Dependent Spectra . J. I. Fuks , K. Luo. E. Sandoval, N. T. Maitra, Phys. Rev. Lett. 114, 183002 (2015 );K. Luo, J. I. Fuks , N. T. Maitra, J. Chem. Phys. 145 , 044101 (2016 ). Spectra taken at different times during the evolution with different functionals Peak positions shouldn’t change -- but SIC-LSD ones do  poor dynamics

A slew* of related exact conditions… We had, for a given transition between two states, wi, pole of satisfies Relatedly, consider response of stationary states: (1) Consider a given V ext (0), with a set of interacting eigenstates F requency for given transition k k ’ must be independent of which of these states Y k or Yk ’ chosen to perturb around. And, “Consistency” for other transitions. (2) Let be a set of stationary states of different KS potentials, all with the same density nk of a particular stationary state of interacting potential Vext(0). Frequency for a given transition must be independent of choice of F i in the response function -- Can express these 2 conditions in Casida-like matrix equations *slew: noun, 1. multitude, 2. a violent or uncontrollable sliding movement

Thanks for your attention! ● N. T. Maitra, Perspective in J. Chem. Phys. 144, 220901 (2016).● M. R. Provorse and C. M. Isborn, Int. J. Quant. Chem. 116, 739 (2016). ● C. A. Ullrich and Zeng-hui Yang Brazilian J. of Phys. 44 , 154 (2014). http://arxiv.org/abs/1305.1388 More Literature ● TDDFT: Concepts and Applications, by Carsten Ullrich (Oxford University Press 2012) ● Fundamentals of TDDFT (Springer, 2012) esp. Ch.4, “Introduction to TDDFT” by Gross and Maitra, available on my website, Ch. 8 and Ch. 24 on memory Recent Reviews on theory of TDDFT: