Nonlinear dynamics of attosecond charge migration in model carbon chains
Francois Mauger, Aderonke Folorunso, Kyle Hamer, Cristel Chandre, Mette Gaarde, Kenneth Lopata, Kenneth Schafer
NNonlinear dynamics of attosecond charge migration in model carbon chains
Fran¸cois Mauger , Aderonke S. Folorunso , Kyle A. Hamer , CristelChandre , Mette B. Gaarde , Kenneth Lopata , , and Kenneth J. Schafer Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803, USA CNRS, Aix Marseille Univ, Centrale Marseille, I2M, Marseille, France Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803 (Dated: July 17, 2020)We investigate attosecond charge migration (CM) in model carbon chains by employing tools fromnonlinear dynamics. We show that mean-field interactions drive synchronization of the electronmotion and give rise to a variety of CM dynamics. Notably, we find that a molecule can supportseveral current-like CM modes, depending on the initial conditions, with periods varying by severalhundred attoseconds. We also show that functionalization and hybridization can play an importantrole in synchronizing CM dynamics. Our results pave the way to understanding the mechanismsthat regulates CM dynamics in complex molecules using nonlinear collective effects.
Electron and hole motions in matter are central tophysical and chemical processes. For instance, they caninduce chemical reactions, photosynthesis and photo-voltaics, and charge transfer [1, 2]. At the natural lengthand time scales of electron motion in molecules – theAngstrom and the attosecond – these coherent ultrafastdynamics are often called charge migration (CM) [3–8]. In this context, CM is understood as the purely-electronic-driven dynamics that unfolds before nucleihave time to move. It can be the precursor for manyof the down-stream processes mentioned above [2, 9–11],and therefore a means of understanding and ultimatelysteering them. The study of molecular CM is currently anactive field of research [12–15], however, it is a formidableendeavor. Experimentally it requires the development ofmulti-dimensional coherent probes with attosecond reso-lution [2]. Theoretically, it necessitates models with mul-tiple interacting electrons, even when nuclear motion isignored.Systematic studies of the mechanisms underlying, andresponsible for regulating, CM can appear beyond reachsince they involve the analysis of systems with a largenumber of coupled degrees of freedom. One approach tosuch large-scale problems is provided by nonlinear dy-namics, which has developed general-purpose methodsfor tackling and understanding the structure of high-dimensional phase spaces. For instance, nonlinear dy-namical analyses have been instrumental in many ar-eas of atomic, molecular and optical science, includingunderstanding the bunching mechanism responsible forlasing in free-electron lasers [16], in transition-state the-ory of chemical reactions [17–19], and for strong-fieldphysics [20–23].In this Letter, we study the field-free dynamics of a lo-calized hole suddenly introduced into a one-dimensionalcarbon chain, as a paradigm for CM following ionization.Figure 1 introduces an example of CM within our model.Strikingly, the foreground colormap in the figure revealsa nearly smooth, current-like migration of the hole, which
FIG. 1: (color online) Charge-migration dynamics in (C ) following a sudden ionization perturbation localized on theleft end of the chain. The graphs at the top show the electrondensities of the end C center alone, in the neutral groundstate, and in the cation immediately after the ionization per-turbation. The foreground colormap corresponds to the sub-sequent electron-hole-density dynamics, defined as the differ-ence between the neutral ground-state and time-dependentcation densities. moves through the entire chain in about 1.7 fs. The ap-parent sustained coherence of a particle-like hole, whichremains localized both in space and time, naturally raisesthe questions: (i) How common is this? (ii) Are thereother types of coherent dynamics? If so, what are they?(iii) What are the physical mechanisms that organize andregulate those CM motions? In what follows, we an-swer these questions by employing tools from nonlineardynamics. In doing so, we highlight the central role of dynamical electron-coupling and synchronization as theengine for CM dynamics in our models, as an alternativeto, e.g. , few-orbital beating mechanisms that have pre-viously been employed [6, 9, 12]. We also demonstratethe importance of functionalization and hybridization insynchronizing the dynamics, which would open the wayfor chemically controlling CM in the future. a r X i v : . [ phy s i c s . a t m - c l u s ] J un We focus on generic mechanisms that regulate CMin molecules with multiple interacting electrons. To doso, we consider dimensionally- and functionally-reducedmodels that reproduce the key elements of the coupleddynamics. More specifically, we systematically considerone-dimensional carbon chains, denoted (C ) n , with n the number of pairs of “C” centers. These chains are theone-dimensional chemical analogs to three-dimensionalalkenes, without the hydrogen centers. We functional-ize the chain by adding an electron-heavy “X” center atone end of the molecule, meant to emulate a halogengroup. Building on previous successes of time-dependentdensity-functional theory (TDDFT) in modeling CM invarious molecules [12, 13], we likewise consider a mean-field approach. Specifically, we use an average-densityself-interaction corrected [24] time-dependent Hartreepotential, i.e. , TDDFT without an exchange or corre-lation potential. This allows us to unambiguously de-termine the role of dynamical mean-field interactions inCM. We describe the electronic structure and dynam-ics with an orthonormal family of non-interacting one-electron Kohn-Sham (KS) orbitals { φ k ( x ; t ) } k [25] suchthat, in atomic units, i∂ t φ k = ˆ H eff [ ρ ] φ k with ρ = (cid:88) k ≥ n k | φ k | (1)where the one-body electron density ρ is an explicit non-linear variable in the mean-field Hamiltonian ˆ H eff . Here0 ≤ n k ≤ ) n molecules’parameters and the mean-field model can be found in thesupplemental material section I [26].We begin by studying the CM dynamics in a (C ) n chain. The bare chain, without functionalization, ishighly symmetric and therefore an impractical systemfor generating a localized ionization-hole in a realisticexperimental scenario. However, since the chain is thebackbone for the migration dynamics we will consider, itis appropriate to study its dynamics first. Below we willdiscuss the effects of adding an X center to one end of thechain, as a handle for introducing a localized ionizationhole or for steering the CM dynamics.In Fig. 1 we show an example of CM dynamics in (C ) .We create the initial ionization perturbation by suddenlyremoving a full electron localized on the left end of thechain – see the curves at the top of Fig. 1. As mentionedabove, the subsequent field-free evolution, shown on theforeground colormap, reveals an alternating current-like motion of the hole with a 1.7-fs half period. We have ob-served similar current-like migration motions when vary-ing the length of the chain (not shown), demonstratingits robustness across the (C ) n family. We model ion-ization from the highest-occupied KS channel, in whichwe construct the initial localized hole with a linear com-bination of a few occupied and unoccupied orbitals ofthe corresponding cation. Note that this linear combina-tion solely reflects the spatial configuration of the elec-tron density right after ionization and does not infer anyorbital-beating process in the subsequent dynamics. Ad-ditional details about the method we use to create the ini-tially localized hole and how we integrate the subsequentdynamics of Eq. (1) can be found in the supplementalmaterial section II [26].Aside from the computational cost of integrating themean-field system of Eqs. (1), one of the main obstaclesto systematic studies of CM in molecules is the very largedimension of phase space associated with the many ac-tive electrons in the target. These analyses are furthercomplicated by the structure of the TDDFT formalism:The KS-orbitals dynamical variables { φ k } k represent fic-titious electrons interacting via the mean field. As such,they hold very limited physical or chemical meaning fordynamics far from equilibrium, including CM (see dis-cussion below). The physical one-body density ρ , onthe other hand, hides the organization of the dynam-ics by incoherently adding the KS-orbital contributions.In the following, we reveal this organization using fre-quency map analysis (FMA) and explain the mechanismsthat regulate coupled multi-electron motions like the al-ternating current-like migration of Fig. 1.Generally speaking, FMA follows the main frequencycomponents associated with a dynamical process as afunction of a continuously-varied initial condition [27,28]. In doing so, it can discriminate chaotic regions ofphase space – with strong sensitivity to the initial condi-tion – from regular ones. We show an example of FMAfor (C ) in Fig. 2. Here we continuously vary the de-gree of localization of the initial one-electron-hole labeledby ε on Fig. 2: ε = 0 corresponds to the delocalizedground state of the cation while ε = 1 corresponds tothe initial hole being fully localized on one end of thechain (similar to the top curves of Fig. 1). In order tohighlight migration motions through the entire chain, thecomplex-valued signal we use in the FMA tracks the time-dependent electron density around one or the other endof the molecule in its real or imaginary part. Furtherdetails about the FMA and its implementation can befound in the supplemental material section III [26].In the region of phase space around the ground state ofthe cation, roughly 0 ≤ ε (cid:46) . not evolve in time in Eq. (1). In the interme- FIG. 2: (color online) Frequency-map analysis of the charge-migration dynamic in (C ) . The horizontal axis labels thedegree of localization for the initial ionization perturbation: ε = 0 corresponds to the ground state of the cation while ε = 1is associated with a fully localized hole on one end of the chain.The frequency colormap is obtained with a windowed-Fouriertransform, with 100-fs total duration, of a signal tracking mi-gration motions through the entire chain (see text). Dashedhorizontal lines label the frequency components in the linearapproximation ˆ H eff [ ρ ( t )] ≈ ˆ H eff [ ρ ε =0 ] in Eq. (1), with thetransparency encoding the intensity of those frequency lines. diate region 0 . (cid:46) ε (cid:46) . . (cid:46) ε ≤
1, we observe a succession of plateaus, between1 and 1 . ε = 0 . dynamical stability – plateaus like theseare signatures of stable regions of phase space, like theislands surrounding elliptic periodic orbits [27]. In turn,this explains the sustained particle-like hole migrationobserved in Fig. 1 and discussed above. (ii) parametricstability – within each plateau of the FMA of Fig. 2, irre-spective of the details for the initial ionization perturba- tion, all current-like motions have the same period. Thisis essential for experimental applications as it provides arobustness of the migration dynamics against uncertain-ties in the way the hole might be created. (iii) multiplemigration modes – the multiplicity of plateaus shows thatthe same molecule can support several modes of alternat-ing current-like migration, each with a different periodvarying with as much as 700 as in the FMA of Fig. 2 –see right margin. When varying the length of the chain,we systematically observe similar multiple plateaus, eachassociated with a current-like CM, therefore demonstrat-ing their robustness across the (C ) n family.As a first attempt to identify the mechanisms respon-sible for the migration dynamics in (C ) n chains, weconsider a linearization of the mean-field Hamiltonianof Eq. (1) by fixing ˆ H eff [ ρ ( t )] ≈ ˆ H eff [ ρ ε =0 ] , with ρ ε =0 the time-independent ground-state density of the cation.The linearization reduces the KS-orbital dynamics to thebeating between the ground-state molecular orbitals ofthe cation. In Fig. 2, we plot the frequency componentsof the FMA, for the linear approximation, with dashedhorizontal lines. While, at first glance, the plateaus inthe FMA of the full time-dependent mean-field system(colormap) seem to gravitate around some of the linear-approximation frequencies, a closer comparison betweenthe two reveals qualitatively different dynamical struc-tures: over the entire range of phase space we investi-gate, nowhere does the linear approximation manage toaccurately predict the electron/hole dynamics. Instead,we have found that the orbital-beating picture associatedwith the linear approximation is only valid in the neigh-borhood of the ground state of the cation, where there isbarely any electron dynamics – see supplemental materialsection III. B [26]. Our analysis therefore highlights thecentral role played by time-dependent mean-field corre-lations between electrons in regulating the observed CM,since these correlations are removed in the linear approx-imation, which differs essentially from the full analysis.Globally, the FMA of Fig. 2 shows that, in the partof phase space we investigate here, the density dy-namics associated with Eq. (1) has at most a hand-ful of well-defined contributing fundamental frequen-cies. This fully correlated dynamics also has notice-ably fewer frequency components than its linear ap-proximation (compare dashed horizontal lines and thecolormap), which seems consistent with observations infull-dimensional simulations performed in various organicmolecules [6, 12]. Both findings are rather unexpected fora dynamical system with many coupled degrees of free-dom. Instead, our analysis suggests that CM motions in(C ) n are regulated by synchronization-type dynamics.Here, synchronization refers to the self-organization andcoalescence of several degrees of freedom around an ef-fective dynamics in reduced dimension [29]. Synchroniza-tion phenomena have been identified throughout physics,engineering and biology [30–32]. FIG. 3: (color online) Evolution of the contribution to the to-tal electron density from the ionized KS channel. Panels (a,b)correspond to the (C ) -chain model, with initial ionizationlocalization ε = 0 . ε = 0 .
95, respectively. Panel (c)corresponds to the functionalized X – (C ) system. Detailsabout the molecular models and parameters can be found inthe supplemental material section I. B [26]. The synchronized CM dynamics is most clearly appar-ent in the contribution to the total electron density fromthe ionized KS channel. To illustrate this, in Fig. 3 (a),we show the density in the intermediate part of phase-space, with ε = 0 .
2. Because of the two irrational fun-damental frequency components in the dynamics – seediscussion above – the density exhibits a succession ofoscillations around each C center, with no apparentglobal period or pattern. Accordingly, we observe a sim-ilar aperiodic dynamics in the hole density (not shown).Looking back at Fig. 2, we see that these two funda-mental frequencies ν ≈ . ν ≈ . ε ). This shift is drivenby the nonlinear electronic coupling in Eq. (1). Eventu-ally, around ε ≈ .
7, the higher fundamental frequencymerges with the first harmonic of the other component ν = 2 × ν ≈ . ) chain on its own is limited by the impracticality of inducing a localized hole at one end of a symmetrictarget. To circumvent this issue, we can break the chainsymmetry by attaching a suitable functional group atone end of the chain, and leverage this to generate theinitially-localized electron-hole, for instance using strong-field ionization [7]. To model functionalization, we addan electron-heavy “X” atomic center to the (C ) n chain,meant to emulate a halogen – see supplemental materialsection I. B [26] for details about the X – (C ) model andparameters.In Fig. 3 (c), we show a successful example of X – (C ) functionalization where the X center is used to createthe initial localized electron perturbation in the ionizedKS channel, which then migrates back and forth throughthe entire molecule. Here as well, we have confirmedthat the total hole density in the molecule follows thesame current-like motion as that of the density in thatKS channel. Importantly, with varying the parameters ofthe X center, e.g. , as would be obtained by using differ-ent functional groups, we have found that not all man-age to transmit their initial X-localized hole in to thechain, as in Fig. 3 (c). For that to happen, the X-atomicand (C ) -molecular orbitals need to hybridize in formingthe X – (C ) compound. Intuitively, such delocalized hy-bridized orbitals provide a bridge for the electron densityto move between the two parts of the molecule.Through the phase-space analysis of X – (C ) n systems,we have identified an additional potential use of function-alization for controlling CM. Specifically, we comparedthe dynamics when starting the localized ionization-perturbation on the chain side of the molecule and us-ing various X effective potentials. Even when the migra-tion remains confined to the (C ) n part of the molecule,the attached X function can alter the properties of thechain, including its ability to support current-like migra-tions. We provide an example of this point by compar-ing FMA and sample CM dynamics in the supplementalmaterial Fig. S3 [26]. Clearly, this demonstrates the po-tential for chemically tailoring CM in molecules.In conclusion, using nonlinear dynamical tools, we haveperformed systematic analyses of charge migration incarbon-chain-like molecules. Our analyses reveal thatthe same molecule can support several types of ultra-fast electron motions, including regular aperiodic andquasi-periodic dynamics, each identified with their char-acteristic signatures in their frequency maps. Drivenby nonlinear coupling between the time-dependent mul-tiple interacting electrons, we have identified apparentsynchronization-like mechanisms that ultimately lead tocurrent-like CM modes in the molecule. Notably, qual-itatively similar current-like modes with periods vary-ing by several hundred attoseconds can be found in thesame compound. Finally, we have shown the potentialfor chemical control of migration motions with molecularfunctionalization, both in creating the initially localizedelectron hole and for its subsequent time evolution.In this Letter, we elected for simplified molecular mod-els to demonstrate our analysis tools and highlight themechanisms that regulate CM motions in them. Criti-cally, these emerge as a result of the dynamical mean-field interaction alone. It suggests that synchronizationeffects are sufficient to organize the electrons into CMmodes, even without explicit exchange or correlation in-teractions. 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