Capturing electron-driven chiral dynamics in UV-excited molecules

Chiral molecules, used in applications such as enantioselective photocatalysis1, circularly polarized light detection2 and emission3 and molecular switches4,5, exist in two geometrical configurations that are non-superimposable mirror images of each other. These so-called (R) and (S) enantiomers exhibit different physical and chemical properties when interacting with other chiral entities. Attosecond technology might enable influence over such interactions, given that it can probe and even direct electron motion within molecules on the intrinsic electronic timescale6 and thereby control reactivity7–9. Electron currents in photoexcited chiral molecules have indeed been predicted to enable enantiosensitive molecular orientation10, but electron-driven chiral dynamics in neutral molecules have not yet been demonstrated owing to the lack of ultrashort, non-ionizing and perturbative light pulses. Here we use time-resolved photoelectron circular dichroism (TR-PECD)11–15 with an unprecedented temporal resolution of 2.9 fs to map the coherent electronic motion initiated by ultraviolet (UV) excitation of neutral chiral molecules. We find that electronic beatings between Rydberg states lead to periodic modulations of the chiroptical response on the few-femtosecond timescale, showing a sign inversion in less than 10 fs. Calculations validate this and also confirm that the combination of the photoinduced chiral current with a circularly polarized probe pulse realizes an enantioselective filter of molecular orientations following photoionization. We anticipate that our approach will enable further investigations of ultrafast electron dynamics in chiral systems and reveal a route towards enantiosensitive charge-directed reactivity.

Chirality is a peculiar property that characterizes the majority of biochemical systems: a chiral molecule exists in two geometrical configurations that are non-superimposable mirror images of each other -defined as (R) and (S) enantiomers -exhibiting different physical and chemical properties when interacting with another chiral entity.This chiral recognition is central to many fields of applied sciences 14 , including enantio-selective catalysis 15 , drug engineering and biophysics 7 .Capturing the primary steps of chiral recognition and the mechanisms dictating the outcomes of a chiral interaction would thus have a significant impact in various fields dealing with chiral properties of matter.At the ultrafast electronic timescale, the opportunity to steer electrons responsible for chemical activity notably promises a way to control the outcome of enantio-sensitive phenomena through the concept of charge-directed reactivity [16][17][18] .Electron currents in photoexcited chiral molecules have indeed been identified as precursors for driving enantio-sensitive molecular orientation 13 .Nonetheless, the ability to track and manipulate electron-driven chiral interactions in neutral molecules is still pending.
In this context, the temporal resolution provided by attosecond technologies developed in the past twenty-two years gives access to some of the fastest electronic dynamics of matter on their natural timescale.Seminal pump-probe experiments using attosecond light pulses have revealed valence electron dynamics in atoms 19 , autoionization dynamics in molecules 20 , photoionization delays in solids 21,22 as well as electron-driven charge migration in ionized biomolecules 18,23 .In all these cases, the intrinsically high photon energy of the attosecond light sources inevitably leads to ionization of the target, restricting the measurements to ultrafast dynamics of cationic states.
Instead, investigating the light-induced electron dynamics of biochemically-relevant chiral molecules in their neutral states with high temporal resolution requires new experimental approaches, and important considerations must be taken into account.First, the pump pulse must have well-defined characteristics: (i) a photon energy below the ionization threshold, (ii) a broadband energy spectrum to trigger coherent electron motion among multiple electronic states and (iii) a time duration that provides a prompt excitation before any nuclear motion can take place, together with sufficient temporal resolution.Because of the low ionization potential of most molecular systems, laser pulses with such characteristics are confined to the spectral region covering the ultraviolet (UV) and vacuum-UV ranges, which also avoids triggering intricate high-order, strong-field multiphoton driven processes that do not exist in nature 24,25 .All these characteristics would provide the tools to harness temporal resolutions previously unattained in pump-probe spectroscopic techniques that are highly sensitive to chirality, such as time-resolved photoelectron circular dichroism (TR-PECD) [8][9][10][11][12] .
Here, we meet all the above requirements by using -for the first time -ultrashort UV pump pulses 26 in combination with circularly polarized near-infrared (NIR) probe pulses, to study coherent electronic dynamics in chiral neutral molecules with unprecedented temporal resolution.We apply the chiroptical method of TR-PECD to investigate electron-driven chiral interactions in neutral methyl-lactate (ML) C4H8O3 -a derivative of lactate, which has regained substantial interest due to its recently uncovered metabolic functions 27 .Fig. 1(a-b) shows an overview of the experimental approach.First, a linearly polarized UV pulse promptly launches a coherent electronic wavepacket just below the ionization threshold in the biorelevant molecule via a two-photon transition.Then, a time-delayed circularly polarized NIR probe triggers ionization from the transient wavepacket, providing an exceptional instrument response function of 2.90 ± 0.06 fs.For each pump-probe delay , the 2D-projected photoelectron angular distributions (PAD)  (") (, , ) are collected with a velocity map imaging spectrometer (VMIS), for both left (ℎ = +1) and right (ℎ = −1) circular polarizations of the probe pulse. and  stand for the kinetic energy and direction of ejection of the photoelectron in the (x,z) VMIS detection plan, respectively.The chiroptical response is characterized by a PECD image defined as the normalized difference (, , ) = 2 $ ("#) (%,',())$ (%#) (%,',() $ ("#) (%,',()*$ (%#) (%,',() , subsequently fitted using a pBasex inversion algorithm 8 .Snapshots of the measured (, , ) are presented in Fig. 1(c), discriminating between low ( ≤ 100 meV) and high ( >100 meV) energy electrons.Low energy electrons are preferentially emitted in the  = 180° backward hemisphere at  = 5 fs, and preferentially ejected forward at  = 11 fs.Their main direction of ejection reverses again at  = 17 fs.Higher energy electrons are more likely emitted forward for  ≥ 11 fs, with a magnitude that depends on .
The (, , ) images provide quantitative fingerprints of an ultrafast dynamics taking place on the few-femtosecond timescale.In order to further characterize the temporal evolution of the observed dynamics, we decompose the PAD images in series of Legendre polynomials,  (") (, , ) = ∑  + (") (, ) + (cos) , and calculate the multiphoton PECD 28 , defined as the normalized difference of electrons emitted in the forward and backward where  + (*/) (, ) = (see Methods). / (*/) (, ) refers to the isotropic part of the asymmetry in each hemisphere, whereas  1 (*/) (, ) encodes anisotropic features due to primary excitation 8 , leading to the angular shaping of the PECD illustrated in Fig. 1(c) - 3 (*/) (, ) has been found negligible in our measurements.Figure 2(a) shows  − (, ) while its  / (*/) (, ) component is displayed in (b).The results are shown for (S)-ML and a mirroring symmetric measurement in (R)-ML clearly confirms the chiral character of the Rydberg-induced dynamics, with minor discrepancies due to slightly lower enantiopurity and statistics (see Fig. S4 of the Supplementary Information, SI).We observe that  / (*/) (, ) closely matches the MP-PECD behavior of Fig. 2(a), indicating that the subtle anisotropic effects included in  1 (*/) (, ) play a minor role.The multiphoton MP-PECD can be partitioned into three kinetic energy ranges, as identified in Fig. 2(a).Between 25 and 100 meV, the photoelectron emission asymmetry strikingly reverses in ~ 7 fs (see Fig. 2(c)).A clear modulation of the asymmetry remains over few tens of fs, which is also observed at higher  between 100 and 300 meV (Fig. 2(d)) and 300 and 720 meV (Fig. 2(e)).These modulations are also visible in the time-resolved photoelectron yield  . (*/) (, ), albeit their contrast is considerably weaker (see Section 1.2 of the SI).This highlights the capabilities of TR-PECD, which relies on differential measurements, over conventional photoelectron spectroscopy.In the following, we aim at assigning the origin of the fast temporal modulation of the asymmetry, which could involve electronic and/or nuclear degrees of freedom.
We modeled the experiment including both the two-photon UV-excitation and the NIR photoionization steps as sequential perturbative processes, within the frozen-nuclei approximation.A detailed description of the theoretical model is provided in the Methods.
The electronic spectrum of ML and the two-photon excitation amplitudes are obtained via large-scale time-dependent density functional theory 29 .Ionization from the excited states is described using the continuum multiple scattering X approach 30,31 .
We present the results of our calculations in Fig.In our fixed-nuclei description, the electron coherences leading to oscillatory MP-PECD do not vanish and even lead to an overestimation of the MP-PECD amplitude at all delays .On the contrary, the oscillations observed in the experimental MP-PECD (Fig. 2(b-d)) become damped after about 40 fs.The time it takes for decoherence to occur in photoexcited 24,33 and photoionized [34][35][36][37][38][39][40][41] molecules is currently the topic of extensive investigations.Electronic wavepackets are subject to three main decoherence sources: (i) the decrease of the overlap between nuclear wavepackets evolving across different electronic states, (ii) the dephasing of the different wavepacket components, and (iii) the relaxation of electronic state populations induced by non-adiabatic couplings 36 .Chiral molecules provide an opportunity to increase the sensitivity of such studies due to the differential nature of PECD measurements.
In order to interpret the slow decoherence observed in our experiment, we investigate its dependence on the nuclear degrees of freedom.Describing the coupled electron and nuclear dynamics in an energy range where tens of electronic states lie is beyond state-of-the-art theoretical approaches.Therefore, we alternatively performed ab initio molecular dynamics calculations on the ground state of cationic ML to which all the HOMO Rydberg states involved in the pump-probe dynamics correlate upon ionization.Nuclear dynamics on the cationic and high-lying Rydberg surfaces are expected to be similar since the outer Rydberg electron does not significantly penetrate the molecular structure 42 .As detailed in Methods and Section 2.3 of the SI, two main classes of trajectories showed up, converging towards two main isomeric forms of the ML cation (see Fig. S10).Within each class of trajectories, the Rydberg state energies of neutral ML were found to remain approximately parallel to each other and to the ML cation along the reaction path (see Figure S11 of the SI).This favors the overlap of the nuclear wavepackets associated with different electronic Rydberg states over an extended time duration and thus minimizes the role of decoherence mechanisms (i) and (ii).This also strengthens the frozen-nuclei assumption in the description of electronic quantum beatings which are dictated by electronic energy differences and should therefore remain basically the same from  = 0 fs onwards.We assign the source of decoherence in the present investigation to non-adiabatic dynamics, not only between the states populated by the pump but also with the lower-lying states reached by internal conversion soon after the prompt excitation.This explains the decreasing amplitude of MP-PECD oscillations in Fig. 2(c-e) that is also compatible with the ~ 40 fs lifetime encoded in the time-dependent photoelectron yield (Fig. S4 of the SI).
We have seen that the oscillations of the chiroptical response for  = 250 meV results from the coherent superposition of two states, the 3d and 4p Rydberg states.We now investigate in more detail the role of these states in the chiroptical response.For a single molecular orientation  J , the excited electron wavepacket reads, at time  after the pump pulse vanishes, ΦL J , , N = ∑  7 ( J )Ψ 7 ()exp(− 7 /ℏ)  Ionization of the 3d and 4p state superposition leads, after averaging over the orientations  J , to the total photoelectron yield which can be decomposed similarly to (1): The computed yield is presented in Fig. 4 where the additional phase Δ arises from the interference of the state-selective continuum partial wave amplitudes building the asymmetry of the photoelectron yield (see SI).As usual, this interference is washed out at the level of the total photoelectron yield 30,43 .The temporal evolution of the unnormalized two-state MP-PECD is shown in Fig. 4 when expanding Φ on the (real) 3d and 4p bound eigenstates.The chirality of the molecule induces a curl in the generated electron current, whose rotation direction reverses periodically.Rotating electron currents are known to influence the ionization probability by circularly polarized light: the propensity rules 45 establish that 1-photon ionization is enhanced when the electrons rotate in the same direction as the electric field.Therefore, the molecules oriented such that their electronic current rotates in the same plane and direction as the ionizing laser pulse are preferentially ionized, see Fig. 4(d).Consequently, the produced molecular cations are selectively oriented along the probe polarization rotation axis, corresponding to the light propagation axis  h.
To quantify the degree of orientation of the photoionized molecules, we select an unitary vector  h B:D fixed to the internal C-C bond of the ML cation, as illustrated in the inset of Fig.
4(e) (see also Fig. S10) and calculate its averaged value over the probe-filtered molecular orientations in the laboratory frame as 13 where  h DE4 ( J ) is the  h B:D vector passively rotated in the laboratory frame and  (±/) ( J , , ) is the ionization rate associated with photoelectrons of energy .Importantly, the averaged orientation of the cations depends on  because the -dependence of the underlying photoionization yield is not the same for all orientations  J .The x and y components of 〈 h DE4 〉  G (±/) (, ) are found to be zero and only the z-component survives the averaging 13 , leading to where  8:+ is the angle between the internal C-C bond and the probe propagation  h axis (see inset of Fig. 4(e)).〈cos 8:+ 〉 9;:<< (±/) () involves chiral-sensitive products of 3d and 4p excitation and ionization amplitudes.The temporal evolution of 〈cos 8:+ 〉  G (*/) is illustrated in Fig. 4(e) for  = 250 meV.When ⟨cos 8:+ ⟩ M N (*/) (, ) < 0, the CO2CH3 moiety of the ML cations preferentially points forward with respect to  h while it rather points backward when 〈cos 8:+ 〉  G (*/) (, ) > 0. Such asymmetry could be detected by resolving the direction of fragmentation of molecular cations.Indeed, the relative numbers of molecules pointing forward and backward at time ,  * (±/) (, ) and  ) (±/) (, ), respectively, can be linked to 〈cos 8:+ 〉  G (±/) (, ) (see Methods).This ultrafast filtering of molecular orientation has consequences of paramount importance on subsequent reactive dynamics of ML cations.The prompt photoionization dictates the subsequent dissociation along the selected molecular orientation and a forward/backward fragment asymmetry (FBFA) naturally arises, which we define as The FBFA is illustrated in Fig. 4(f) for ℎ = +1 and  = 250 meV, reaching absolute values of ~30% while its temporal evolution is dictated by the behavior of the underlying electron current L J , , N.Similarly, to the MP-PECD, the FBFA switches sign for ℎ = −1 or when the other enantiomeric form of ML molecules is considered.Since the FBFA is created by the electron current, it vanishes in the case of incoherent population of excited states (Fig. 4(c)).
This shows that the chiral electronic coherence directly observed in our experiment via TR-PECD is crucial to achieve control over enantio-selective dynamics of the nuclei.
To conclude, the potential of time-resolved PECD to probe transient chirality had so far only been demonstrated experimentally for nuclear dynamics, internal conversion and photoionization delays in chiral molecules [8][9][10][11]46 . We ave taken an important step forward by resolving the coherent chiral electronic dynamics of a chiral molecule in the first instants following prompt excitation by an achiral few-fs UV pulse.The results demonstrate that TR-PECD can provide insights on the role of the primary electron dynamics in the light-induced chiral response of complex molecular systems such as chiral biomolecules and organometallic complexes.Offering a route to investigate the fundamental origin of chiral recognition that is ubiquitous in biological phenomena 47 , the possibility to control the photoelectron emission direction in the laboratory frame also offers the potential to engineer petahertz switching devices based on chiral interactions.Finally, we demonstrated that beyond its impact on the chiroptical properties of the system, the chiral currents generated in our experiment can be exploited for enantio-sensitive charge-directed reactivity leading to oriented fragmentation.
Steering the outcome of these photophysical and photochemical properties provides an important added-value to chirality at the molecular scale.

Experimental setup
The experiments were carried out with a 1 kHz titanium:sapphire laser (FemtoPower, Spectra-Physics), delivering 25-fs, 12-mJ pulses at 800 nm.5.6 mJ were used for spectral broadening in a 2.3-m long hollow-core fiber (few-cycle inc.) filled with a pressure gradient of helium gas.
The fiber setup seeds an all-vacuum Mach-Zehnder-like interferometer with 5-fs nearinfrared (NIR) pulses.One arm is used for the generation of the UV-pump pulse via thirdharmonic generation in a laser-machined glass cell filled with 7.2 bar of neon gas.A pair of silicon superpolished substrates (Gooch & Housego) is used at Brewster angle to attenuate the residual part of the NIR driving field by 3 orders of magnitude while reflecting ~16% of the UV radiation (50 nJ).In the second arm of the interferometer, the remaining part of the NIR beam is focused to the experimental region by a toroidal mirror (f = -900 mm) followed by a motorized zero-order quarter-waveplate (B.Halle) in order to control the helicity of the circularly polarized probe pulses (16 μJ), with an intensity of 5×10 12 Wcm −2 .The instrument response function of 2.90 ± 0.06 fs is obtained by a global fit of the non-resonant (gaussian) dynamics of all the ion masses acquired simultaneously with the photoelectron spectra (see SI). Liquid (S)-methyl-lactate (97% enantiomeric excess, Sigma-Aldrich) was evaporated and transported by diffusion to a velocity map imaging spectrometer to measure the photoelectron angular distribution as a function of the pump-probe time delay.To avoid condensation of the sample along the transportation line within the molecular source, a temperature gradient from 85°C to 95°C was applied.amplitude coefficients  + (*/) (, ), together with the resulting unnormalized  − (, ), are shown in Fig. S2 of the SI.Note that  3 (*/) (, ) is not included due to its negligible contribution to the total signal.The validity of the analysis protocol despite the lack of cylindrical symmetry induced by the anisotropy of excitation of the linearly polarized UVpump pulse has been demonstrated in 8,9 .

Computation of TR-PECD
At time  after the pump pulse vanishes, the electron wavepacket formed in a ML molecule whose orientation in the laboratory frame is characterized by  J reads .The trajectories are propagated using the Newton-X package 48,49 and the electronic potential  [ L•  ()•N is evaluated 'on the fly' using DFT at the CAM-B3LYP/6-311++G(d,p) level of theory [50][51][52] .Early stage ( ≤ 60 fs) and subsequent nuclear dynamics are discussed and illustrated in Sections 2.3 and 2.4.2 of the SI, respectively.Note that the sudden electron excitation and ionization processes mainly occur in classically allowed regions of the potential energy surfaces.The nuclei stay in these regions for short pump-probe delays of a few tens of femtoseconds, which is the timescale of interest in our work.The classical treatment of nuclear dynamics is thus appropriate.

3 .
photoionization by the probe pulse leads to the emission of photoelectrons with kinetic

Figure 4 (
a) shows, for one selected orientation  J , the coherent part L J , , N −  8+9:" L J , N of the electron density, oscillating back-and-forth along the molecular structure with a period  = 2ℏ/( 26 −  15 ) of 14.4 fs.

Fig. 1 :
Fig. 1: Light-induced chiral dynamics of methyl-lactate.(a) A few-femtosecond linearly polarized UV pulse excites an ensemble of randomly oriented chiral molecules, creating an electronic wavepacket of Rydberg states via 2-photon absorption.The dynamics is probed via 1-photon ionization by a time-delayed circularly polarized NIR pulse.The probing step leads to the ejection of photoelectrons along the light propagation axis defined along the z direction and the resulting angular distribution is recorded by a velocity map imaging spectrometer.(b) The red and blue structure shows the temporal evolution of the coherent electron density in the excited neutral molecule: the chiral evolution of the photoexcited Rydberg wavepacket leads to a reversal of the 3D photoelectron angular distribution at two distinct time delays t and t+Δt, captured by the measurements.(c) For each time delay, an image is recorded for both left and right circular polarization of the probe pulse.The differential image PECD(ϵ,θ,t) defined in the main text is shown for time delays of 5, 11,17 and 26 fs for photoelectrons with kinetic energies from 25 to 300 meV along the radial coordinate.The white circles identify the photoelectrons below 100 meV which experience an ultrafast reversal of their emission direction in the laboratory frame.

Fig. 2 :
Fig.2: Energy-resolved analysis.Temporal evolution of the unnormalized MP-PECD in (S)-methyl-lactate (a) and corresponding b1 coefficient (b).The white lines identifiy three different kinetic energy ranges of photoelectrons: 25-100 meV (c), 100-300 meV (d) and 300-720 meV (e).The standard error of the mean over 5 measurements is shown by the shaded areas.The solid blue lines show the fit of the oscillations from t = 0 fs (see the corresponding Fourier analysis in Fig.3c,e).The change of sign in (c) identifies a reversal of the photoelectron emission direction in the laboratory frame.

Fig. 3 :
Fig. 3: Modelling of the experiment.(a) Two-photon absorption (TPA) cross sections for the excited states stemming from almost pure HOMO excitation.The cross sections have been convoluted with the UV-pump intensity squared.The blue and green curves correspond to the spectral probe intensity, down-shifted in energy to elicit the transient Rydberg states leading to photoelectrons with energies  = 250 and  = 500 meV through ionization by one photon centered at frequency  = 1.75 eV.(b) Calculated MP-PECD for photoelectrons with  = 250 meV (green) compared to the experiment (blue).The calculations start at  = 10 fs corresponding to the end of the pump-probe overlap region (yellow area).(c) Corresponding power spectra from a Fourier analysis.The frequency axis is displayed for beatings of excited states with an energy spacing between 150 meV (27.6 fs period) and 500 meV (8.3 fs).The main peak from the computed MP-PECD evolution is at 291 meV (14.2 fs).The power spectrum of the experimental data shows a peak frequency at 280 meV (14.8 fs) and was evaluated up to  = 35 fs where the oscillations are damped.(d) Calculated MP-PECD for photoelectrons with  = 500 meV (green) compared to the experiment (blue).(e) Corresponding power spectra with a central component at 269 meV (15.4 fs) for the computed curve.The power spectrum of the experimental data is shown with a central frequency at ~329 meV (12.6 fs).

Fig. 4 :
Fig. 4: Electron-driven dynamics in the case of quantum beating monitored by 3d and 4p Rydberg states.(a) Temporal evolution of the coherent part of the electron density over one period of the quantum beating between the 3d and 4p states (see equation (1) of the text).(b) Photoelectron yield as a function of the pump-probe delay for  = 250 meV.The Rydberg quantum beating leads to an oscillatory behavior of the yield which is in phase with the variation of the electron density shown in (a), as expected from equations (1) and (2).(c) MP-PECD as a function of the pump-probe delay for  = 250 meV.The dichroism is delayed by  = 0.79 rad (1.8 fs) with respect to the variation of the electron density because of the interferences between the continuum partial wave amplitudes (see equation (3)).(d) Snapshots of the electronic current induced by the pump pulse, on a Rydberg sphere of 10 a.u.radius surrounding the molecule for two distinct orientations  0 !.The current is defined in the molecular frame (equation (4)) and ionization by the probe pulse is enhanced for the molecular orientation where (i) the current co-rotates with the circularly polarized probe field (red arrow) and (ii) the rotation axis of the current aligns with the light propagation vector.This happens here only for  0 " .(e) Active orientation of the produced cations along the light propagation axis  2 as a function of time for  = 250 meV.This orientation is defined as the mean value of   !#$ , where  !#$ is the angle between the internal C-C bond of the ML cation and , as shown in the inset (see equation(6)).(f) The resulting forward/backward fragment asymmetry along  2 in the reactive fragmentation of ML cations, following probe-induced ionization of the transient 3d-4p electron wavepacket leading to photoelectrons with energy  = 250 meV (see equation(7)).The insets illustrate the preferential directions of emission of CO2CH3 and CH3CHOH + fragments.

8 ,where Ψ 8 ( 8 where
ΦL J , , = ∑  8 L J NΨ 8 ()exp(−i 8 /ℏ) ) are excited states with energies  8 and two-photon absorption amplitudes from the ground state  8 L J N. These states, energies and transition amplitudes, have been obtained by large-scale TDDFT29 calculations, detailed in Section 2.1 of the SI.In the spectral region spanned by the pump pulse, most of the excited states have a Rydberg character and stem from the excitation of the ML HOMO (see Fig.S5of the SI).The absorption of one NIR photon of the probe pulse leads to the ejection of a photoelectron with wavevector ′ J in the molecular frame.The associated ionization dipole is Q ",B:D L J , N = y  8 L J Nz /)ORM ( 8 ) < Ψ Q ()) |  }. |Ψ 8 > exp(−i 8 /ℏ)  /)ORM ( 8 )is the spectral intensity of the probe pulse at frequency  8 =  Q0 /2 +  6 −  8 , with  6 the ML ionization potential,   } is the circular polarization of the probe pulse (ℎ = ±1) and Ψ Q ()) is the ingoing scattering state associated with the electron ejected in the continuum.Neither the scattering state nor the excited states explicitly depend on  since the calculations are made assuming that the nuclei remain frozen at their potential arising from the electrons within the ground state of the ML cation.The initial coordinates •  ( = 0)• and momenta •  ( = 0) =  7  ̇( = 0)• of the nuclei are randomly taken from a Wigner distribution of the harmonic vibrational ground state of neutral ML in its ground electronic state.The Wigner distribution is discretized in terms of 250 sets "•  ( = 0)•, •  ( = 0)•• Y-/,..,03.which are the starting points of 250 noninteracting trajectories "•  ()•, •  ()•• Y-/,..,03.