Abstract
The manybody quantum nature of molecules determines their static and dynamic properties, but remains the main obstacle in their accurate description. Ultrashort extreme ultraviolet pulses offer a means to reveal molecular dynamics at ultrashort timescales. Here, we report the use of timeresolved electronmomentum imaging combined with extreme ultraviolet attosecond pulses to study highly excited organic molecules. We measure relaxation timescales that increase with the state energy. Highlevel quantum calculations show these dynamics are intrinsic to the timedependent manybody molecular wavefunction, in which multielectronic and nonBorn−Oppenheimer effects are fully entangled. Hints of coherent vibronic dynamics, which persist despite the molecular complexity and highenergy excitation, are also observed. These results offer opportunities to understand the molecular dynamics of highly excited species involved in radiation damage and astrochemistry, and the role of quantum mechanical effects in these contexts.
Introduction
Thanks to the development of ultrafast spectroscopic techniques, the investigation of the dynamics of fewbody atomic or molecular systems has led to the observation of mechanisms down to the attosecond timescale^{1,2}. Such experiments have provided information on how nature functions at the quantum level^{3,4} through the study of resonances^{5,6,7}, quantum coherences^{8,9,10}, and energy flow^{11,12,13}. However, manybody quantum systems remain a major challenge^{14} both for experiment and theory. The main difficulty in describing manybody systems is that the two conceptual pictures for structure and dynamics often break down. Indeed, the orbital picture describing the electronic molecular states as single electron configurations distributed in molecular orbitals becomes less adequate for highly excited states^{15,16}. It can also be the driving force for nontrivial dynamics, such as ultrafast correlationdriven hole migration^{17,18}. At the same time, with an increase of the density of electronic states, the Born−Oppenheimer approximation, which decouples the electronic and nuclear wavefunctions, may fail leading to the appearance of nonadiabatic relaxation through conical intersections (CIs)^{19}. In this context, ultrashort extreme ultraviolet (XUV) pulses are a useful tool to reveal the dynamics of manybody systems, because they combine highenergy photons and short duration times. Indeed, XUV excitation may populate a multitude of nonBorn−Oppenheimer states with strong multielectronic character, leading to a truly manybody molecular wavefunction whose time evolution is driven by the complex entanglement between the large number of electronic and nuclear degrees of freedom.
Here, we investigate naphthalene (Naph, C_{10}H_{8}) and adamantane (Ada, C_{10}H_{16}) using an XUVpump−IRprobe scheme (Fig. 1a). The XUV pulse ionizes the neutral molecules and populates excited cationic states, with energies near the doubleionization level (IP2 = 21.5 eV) (Fig. 1b). The dynamics of these excited cations are probed with a second ionization, performed using a delayed IR pulse (800 nm, 1.55 eV). Hence, the timeresolved photoelectron spectrum (TRPES) probes the dynamics of the manybody quantum cationic states (Fig. 1c). These dynamics can only be explained when both strong multielectronic^{20} and nonBorn−Oppenheimer^{21} effects are taken into account.
Results
Angle resolved photoelectron XUV−IR pump−probe spectroscopy
The experiment was performed on an XUV beamline consisting of a compact XUVIR interferometer based on an XUV attosecond pulse train with an envelope of sub30 fs and fs IR beams, coupled to a velocity map imaging (VMI) spectrometer (see Methods). The XUV pulse was synthesized via high harmonic generation (HHG). The experimental results were obtained with an HH spectrum containing harmonics between 17 and 35 eV, centered around 26 eV.
We recorded a statistical set of twocolor photoelectron spectra as a function of the XUVpump–IRprobe delay. The photoelectron spectrum was obtained after angular integration of the VMI image and the twocolor signal defined as the difference between the signal measured when the two pulses are present and the signal measured with XUV only. By doing so, we subtract the contribution of electrons originating from the single and double ionization of the molecule by one XUV photon. Above an electron kinetic energy (E_{kin}) of 1 eV, the signal is symmetric in time and stems from the XUV ionization of the neutral Naph assisted by the IR near t = 0 (not shown). Below 1 eV, the signal is asymmetric in time and corresponds to an XUVpump–IRprobe scheme. We thus focus our analysis on those photoelectrons below 1 eV that are produced upon IR ionization of the cationic Naph.
The timedependent twocolor TRPES is plotted in Fig. 2a. In this map we distinguish three features, corresponding to three maxima in the spectrum: around 0.4, 0.64 and 0.88 eV, respectively. By fitting the signal at these energies integrated between ±0.075 eV (Methods) the extracted timeconstants τ_{decay} are 24 ± 5, 33 ± 6, and 46 ± 7 fs, respectively (Fig. 2b). This leads to the main observation of this work, which is that the timeconstant increases with the photoelectron energy indicating that the closer the cationic states are to the doubleionization threshold, the longer the dynamics are. We performed the same experiment on Ada that is a similar size carbonbased molecule but with a nonaromatic, 3D structure. Here again we observed that the decay timescale increases with the photoelectron energy (Supplementary Fig. 6) which demonstrates that this effect is not specific to Naph. We have also changed the XUV spectrum and IR intensity and no significant variation of the timescale was observed (Fig. 2d, f).
What kind of dynamics does this reflect? As shown in Fig. 3a, b, within the monoelectronic and Born−Oppenheimer approximation, one would expect three completely decoupled cationic states (corresponding to the ionization out of three innervalence orbitals) and no ultrafast dynamics. If nonBorn−Oppenheimer effects are present (see Fig. 3c), population may be transferred between the electronic states due to the nonadiabatic couplings. In this situation it is expected that higher energy states are depopulated and the population is transferred to lower energy states. In the case considered here, both multielectronic states and nonBorn−Oppenheimer effects are present. Due to the multielectronic effects, when approaching the doubleionization threshold, the density of states may increase dramatically (see Fig. 3e). In principle, when states get closer, the nonadiabatic effects get stronger, suggesting that with the increase of the state energy, the relaxation time should become shorter. The nonadiabatic relaxation should, however, occur through a large number of CIs, making the dynamics of many strongly entangled electrons and nuclei highly nontrivial (Fig. 3f). In the following, we will show that this leads to the specific behavior observed in our experiment in which the population is trapped for a much longer period of time than expected. We note that in previous experiments, only timeresolved ionic yields could be measured, allowing to trace either simple dynamics or averaged relaxation processes^{22,23}. Our present approach gives access to the relaxation of energyresolved density of states and therefore allows for a much more refined investigation which requires a complex theoretical description that takes into account a large number of strongly coupled states as described below.
Signature of coupled multielectronic and nonBorn−Oppenheimer dynamics
To simulate the manybody quantum dynamics, we constructed a vibroniccoupling model based on 23 electronic states and 25 active vibrational modes (Methods). The cationic eigenstates have been computed with the nonDyson ADC(3) Green’s function method^{24}. The method takes into account the multielectronic effects up to third order of perturbation theory. All cationic states with a binding energy E_{i} lying below the XUV photon energy can be populated. However, in the experiment, the weak IR pulse ionizes only states lying just below the doubleionization potential. The cationic eigenstates in this energy range are shown in Fig. 2e. A comparison of the computed cationic spectrum and the experimental TRPES map shows that the three features observed experimentally around 0.4, 0.65, and 0.9 eV can be directly related to three groups of states that stem from the ionization out of three innervalence orbitals: 6a_{g}, 5b_{1u}, and 4b_{2u}. The measured timeconstants correspond, therefore, to the timescale of the nonadiabatic relaxation dynamics of these groups of states. This dynamics was obtained by simultaneously propagating the nuclear wavepackets on all nonadiabatically coupled multielectronic states using the MCDTH method^{25,26}, taking into account the initial population of the states upon the XUV ionization.
Looking only at the most populated states within each of the three groups, computed to be at E_{i} = 19.54, 19.74, and 20.10 eV (Fig. 2e), we obtain relaxation times of 8.1, 8.2, and 5.5 fs, respectively. These timeconstants not only differ significantly from the experimentally observed ones, but show also a different trend: lower in energy states relax slower, as expected for the monoelectronic/nonBorn−Oppenheimer case (see Fig. 3d). In fact, although the respective populations leak out very fast from these three states, they are transferred to very closelying states, which cannot be resolved, but where those populations can be trapped for longer. We can account for this by integrating the population evolution of all symmetryrelated shakeup states lying within 150 meV below the three most populated ones, yielding the following timeconstants: 15 fs around 0.4 eV, 21 fs around 0.64 eV, and 35 fs around 0.88 eV (Fig. 2c).
These results are not only much closer but also show the trend observed experimentally. As a result, the physical picture seems to be clear. The correlation effects become stronger going deeper in the valence shell. Therefore, removing a deeper electron results in populating an increasing number of states. This leads to a complex shakeup zone with a large number of CIs. Due to the high density of electronic states and because of the ultrashort lifetime of these states, the individual electronic states cease to be an adequate description of the transient molecular state. The number of CIs through which the wavepacket has to go increases with the binding energy which leads to an increase of the relaxation time as observed in the experiment. This suggests that the observed dynamics is intrinsic to the strongly nonadiabatically coupled multielectronic states. We note that increasing the number of states considered in the model will lead to even more accurate description of the manybody molecular wavefunction and will bring the relaxation times even closer to the experimental values. A better description of the shakeup zone can be done, for example, by increasing the basis sets used in the ADC calculations, which however will face a prohibitive computational time.
XUVinduced vibronic coherences
Let us now turn to our second important observation. We note that the calculations predict smooth oscillations in the population decay around 100 fs, which is illustrated in Fig. 4a with the timedependent population of the main state mapped to 0.4 eV. Comparing to the experimental data, and taking into account the crosscorrelation between the XUV and IR pulses, we see a similar recurrent signal (Fig. 4b, c). The analysis shows that these recurrences correspond to coherent vibrational dynamics of the lowfrequency a_{g} mode at 514.3 cm^{−1} (inset of Fig. 4a) that couples two electronic states, one lying within (2a_{g}) and one lying below (1a_{g}) the energy region probed experimentally, making the wavepacket experimentally accessible only during the time it evolves on 2a_{g} surface. This mode is of particular importance, as it is the slowest totally symmetric mode of the molecule and thus easy to excite. Moreover, many states in this energy region share the same symmetry and are therefore coupled by this particular mode. We conclude that the observed dynamics correspond to cationic excited molecules that coherently vibrate. We note that the effect is observed only at the lowest electron energy (0.4 eV), which corresponds to the ionization out of 6a_{g} orbital. Neither the calculations, nor the experiment show oscillations in the dynamics triggered by the ionization of the other two orbitals 5b_{1u} (mapped to 0.64 eV signal) and 4b_{2u} (mapped to 0.88 eV). We also note that this effect is not observed in the case of Ada, which we attribute to the fact that slow vibrational mode are not active for Ada due to its compact structure.
While XUVinduced coherent dynamics have been observed in small molecules^{27}, its existence in large polyatomic systems like Naph has never been proven to date. Lightinduced vibrational coherences is known for low excited molecular states^{28}; it is striking that such coherence can also survive the molecular complexity in highly excited states.
In our experiment, relaxation dynamics with timescales that increase with the state energy has been observed in XUVexcited Naph and Ada molecules. This behavior is intrinsic to the manybody quantum nature of the molecule, i.e. their fully entangled multielectronic and nonBorn−Oppenheimer character. These results open new avenues in controlling molecules using the entangled quantum properties of their constituting particles. We have also observed XUVinduced vibronic coherence in Naph. The observation of such quantum effect at low excitation energy has led to suggesting that quantum coherence might play an important role in liferelated phenomena such as photosynthesis^{29,30,31}. Similarly, our observation raises the question of the role of the ultrafast quantum manybody dynamics initiated by the illumination of molecules with energetic radiation in space^{32} which predetermines the chemical reactivity of the irradiated species.
Methods
Experimental setup
The setup (Supplementary Fig. 1) consists of a Mach−Zender interferometer coupled with a VMI spectrometer. The output of a commercial amplified laser system producing pulses of 2 mJ, centered around 810 nm, 5 kHz repetition rate and 25 fs pulse duration, is split into two arms. One part is focused, with a 30 cm lens, inside a windowless gas cell filled with rare gas, 2 mm thickness, where the fundamental frequency is upconverted by means of highorder harmonic generation (HHG). The pressure inside the gas cell is maintained around 10 mbars. The XUV frequency comb generated under such conditions is composed of odd harmonics from the 7th (H7) up to the 21st (H21). The HH spectrum can be tuned by adjusting the phasematching conditions and by choosing an adequate rare gas (xenon, krypton or argon). The contribution of the XUV radiation that is due to the long electron trajectories in the HHG process is filtered out using a motorized iris just after the cell and then the copropagating IR is removed in two stages. First, a beam splitter with Nb_{2}O_{5} outermost layer reflects mainly the XUV part. Second, the XUV beam passes through a 200 nm Al filter that transmits light only above 17 eV. The XUV beam is focused by a 30 cm toroidal mirror. A goldcoated grating mounted on a rotational stage can be inserted to measure the XUV spectra at the first order of diffraction. Typical spectra optimized to variable order (around H13 up to H25 is shown in Supplementary Fig. 2). The XUV beam then passes through a 45° drilled mirror (3 mm hole diameter), and the other arm of the interferometer is time delayed and recombined with the XUV beam by a reflection at the vicinity of the hole. Both beams are focused in the interaction region of the VMI spectrometer. The energy of the second arm of the interferometer is controlled by a half–wave plate combined with a thin film polarizer (vertical polarization). Then, the beam is reflected by flat mirrors. Each mirror is placed on a manual translational stage, in order to reach an approximate temporal overlap between the pump and the probe pulses. The beam then passes through a refractive delay line, consisting of a pair of parallel wedges, which allows to introduce a time delay proportional to the displacement of the wedges by a linear actuator. Finally, the beam passes through a 75 cm lens and is reflected at the vicinity of the hole of the 45° drilled mirror.
The VMI spectrometer follows the standard design introduced by Eppink et al.^{33}. Electron trajectories are focused using a static electrostatic lens and detected by a double MicroChannel Plate (MCP) and a phosphor screen assembly imaged on a CCD camera. The two molecules studied in our experiment, Naphthalene and Adamantane (Supplementary Fig. 3), have a sufficient partial pressure at room temperature to be directly injected into the vacuum chamber through a capillary. The capillary is connected to the repeller electrode of the VMI and the molecules are ejected towards the detector and cross the XUVIR beams.
Data analysis
The VMI images are Abel inverted and angularly integrated to obtain the kinetic energy spectrum. Timeresolved electron kinetic energy spectrum has been recorded and the twocolor signal has been extracted by subtracting the contributions from the XUV and NIR pulse at each delay. The timedependent signal obtained at each electron kinetic energy is integrated over a ΔE = 150 meV energy range, leading to a spectrum that is used to extract the decay times as described in the following.
The measured signal can be schematically described as follows: excited states are produced at t = t_{0}, part of the initial population relaxes following an exponential decay τ_{decay} while a fraction of the initial population remains constant. The temporal resolution is determined by the crosscorrelation between the XUV and IR pulses. This crosscorrelation of duration τ_{crossco} is represented by a Gaussian function. Therefore, we use the following formula to extract the decay lifetime from the electron signal obtained at a given kinetic energy:
where τ_{crossco} and t_{0} are fixed parameters (that can be extracted from the high KE_{e} features) and A_{decay}, τ_{decay}, and A_{step} are free parameters. θ(t – t_{0}) is the Heaviside step function.
Experimental results for naphthalene
The photoelectron signal arises from the photoionization of the XUV excited cations. This assumption is also validated by the comparison between the timedependent dication signal obtained by measuring the variation of the ion yield, and the timedependent lowenergy electron signal that we integrate over energy. In the first case, the fitting procedure leads to a time scale of 36 ± 4 fs and in the second case we obtain 40 ± 7 fs (Supplementary Fig. 4).
The error bar on the extracted decay timeconstant is defined as the error to the fit. We have also measured decay time constant defined as the central value of a statistical set of independent measurements. The associated error bar is then given by the dispersion of the experimental results. We have included in the statistics the measurements performed at different probe intensities and with different XUV spectra, since no significant changes were observed in the dynamics while changing such experimental parameters. These results are presented in Supplementary Fig. 5.
Comparison between naphthalene and adamantane
Similar measurements have been performed for adamantane (Supplementary Fig. 3b). The results of the measurement are presented in Supplementary Fig. 6. They show a very similar trend as observed in the case of naphthalene: the extracted time scale increases with the electron kinetic energy from 25 ± 3 to 80 ± 20 fs.
Theory
The ionization spectra of the naphthalene molecule have been calculated with the third order nonDyson Algebraic Diagrammatic Construction scheme (ndADC(3))^{34,35} for representing the oneparticle Green’s function. The spectrum at the ground state equilibrium geometry is shown in Supplementary Fig. 7. Each line in the spectrum corresponds to a cationic eigenstate of the molecule. The intensity of each line is given by the weight of the one hole (1h) contributions to the corresponding cationic state. The missing to one part of the line thus reflects the multielectronic contribution to the corresponding state, which at third order means all twoholeoneparticle (2h1p) contributions. In other words, even though only the 1h part of the cationic state is detected experimentally, its multielectronic nature is encoded through its initial population (i.e. the intensity of the line) and its energy position. We see that within the energy range accessible in the present experiment (close to the doubleionization threshold (DIT), between approximately 20 and 19 eV), the majority of the states come from the ionization out of only three orbitals. These orbitals are 6a_{g}, 5b_{1u}, and 4b_{2u}. The fact that only three symmetries out of eight (the naphthalene molecule belongs to D_{2h} symmetry point group) are present will limit the number of relevant vibrational modes. The symmetry selection rules^{36} reduce the problem to the following coupling scheme: Apart from the totally symmetric a_{g} mode, the states of interest can be coupled by the nonsymmetric b_{1u} mode (coupling the states of a_{g} and b_{1u} symmetry), b_{2u} mode (coupling a_{g} and b_{2u}) and b_{3g} mode (coupling b_{1u} and b_{2u}). The vibrational modes have been obtained at the MP2 level using the GAUSSIAN program package^{37}. The ionization spectra of the naphthalene have been computed along all the vibrational modes that belong to the four symmetries a_{g}, b_{1u}, b_{2u}, and b_{3g}, up to 2 Q_{i} with step of 0.25 Q_{i}, where Q_{i} is the dimensionless coordinate associated with the normal mode i. All calculations have been performed using ccpVDZ basis set^{38}.
Vibroniccoupling Hamiltonian
In order to simulate the coupled nuclear and electronic dynamics triggered by the innervalence ionization of naphthalene, we have built a vibroniccoupling Hamiltonian^{39} for 23 electronic states present in the energy range of interest, and 25 relevant vibrational modes. The included states, 5 states of a_{g} symmetry, 11 of b_{1u} symmetry, and 7 of b_{2u} symmetry, are listed in Supplementary Table 1 and consist of all ionic states in the range 19.02–20.21 eV, which can be unambiguously identified and followed along the whole distortion range of the vibrational modes. The vibrational modes that can couple the states of interest belong to a_{g}, b_{1u}, b_{2u}, and b_{3g} irreducible representations and are listed in Supplementary Table 2. The vibroniccoupling Hamiltonian can be written as:
where τ_{N} and ν_{0} denote the kinetic and the potential energy of the neutral unperturbed reference ground state, respectively. Using a harmonic approximation for the vibrational modes, τ_{N} and ν_{0} can be written as:
with ω_{i} being the frequency of mode i. The matrix W in Eq. (2) contains the diabatic cationic states, and the couplings between them. Using the standard for the vibroniccoupling theory Taylor expansion of the matrix elements, W can be written as:
In the above expressions, E_{j} is the vertical ionization energy of state j, κ_{i}^{j} and γ_{i}^{j} are the linear and quadratic coupling parameters of state j for normal mode i, respectively, and λ_{i}^{jk} is the linear coupling parameter between states j and k by the normal mode i. These quantities are obtained directly from the ab initio ndADC(3) calculations through a leastsquare fit procedure performed by the VCHam module, included in the Heidelberg MultiConfiguration TimeDependent Hartree (MCTDH) program package (http://www.pci.uniheidelberg.de/tc/usr/mctdh/doc/). An example of the result of the fitting procedure is shown in Figure S8. Our analysis shows that the highfrequency CH vibrational modes lead to a very weak coupling between states. That is why we have neglected the 8 modes of this type, leading to a vibroniccoupling Hamiltonian based on 25 modes (presented in Supplementary Table 2).
MCTDH calculations
The vibroniccoupling Hamiltonian was used to propagate nuclear wavepackets on the coupled manifold of electronic states via the MCTDH methods^{40}. MCTDH is a powerful gridbased method for numerical integration of the timedependent Schrödinger equation, particularly suitable for treating multidimensional problems^{41}. The Heidelberg MCTDH package has been used for this calculation.
We have performed three independent propagations describing the dynamics triggered by ionization out of 6a_{g}, 5b_{1u}, and 4b_{2u} orbitals. The initial population has been distributed between 5, 11, and 7 states, respectively, according to the spectral intensity of the corresponding ionic state at equilibrium geometry (Supplementary Fig. 7). These wavepackets have been then propagated for 200 fs taking into account all coupled 23 states. The results are presented in Supplementary Fig. 8. On the left set of panels, we report the evolution of the populations of the states located within an energy window of 150 meV below the initially most populated state (i.e., having the largest intensity at equilibrium geometry) in each group. The energy window chosen reflects the experimental resolution. We see that although the initial population is transferred very fast to lower lying states (timeconstants of ~ 8.1, 8.2, and 5.5 fs are obtained for the relaxation of a_{g}, b_{1u}, and b_{2u} states, respectively), part of the population is transferred and somewhat trapped in closelying states. As the latter states lie within the experimental resolution, for simulating the experimental observation we need to sum all populations that lie within the energyresolution window. The result of this procedure is shown in the right set of panels of Supplementary Fig. 8.
We thus obtain the following relaxation timescales: 15 fs for the a_{g} states, 21 fs for the b_{1u} states, and 35 fs for the b_{2u} states. Although the computed timeconstants are about 30% smaller than those extracted from the experiment, they correctly reproduce the general trend, namely the states closer to the DIT relax slower. The explanation for this somewhat counterintuitive result can be well understood with the increasing density of states when approaching the DIT. In dense spectral regions, the wavepackets have to go through a larger number of CIs in their relaxation path, compared to more sparse spectral regions, thus resulting in a slower relaxation time. This also explains the shorter relaxation times obtained in our theoretical modeling. Although quite advanced, our model only includes a fraction of all the states in the energy range of interest.
Code availability
The ADC and MCTDH custom codes are available upon request with no restrictions.
Data availability
All other relevant data supporting the key findings of this study are available within the article and its Supplementary Information files or from the corresponding authors upon request.
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Acknowledgements
The research has been supported by CNRS, ANR16CE300012 “Circé” programme Blanc, Fédération de physique MarieAmpère, Région Aquitaine (Caracatto 20131603008). V.D. acknowledges the financial support of DFG through QUTIF priority programme. A.I.K. thanks US ARO for financial support under grant No W911NF1410383. We thank Horst Köppel for fruitful discussions.
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F.L. and A.I.K. led and designed the research program and wrote the manuscript. A.I.K. and V.D developed and performed the calculations. A.M., V.L., G.K., and M.H performed the measurements and analyzed the experimental data. E.C. and L.Q developed an early version of the XUV beamline. F.C. and C.J. have contributed to the discussion of the results. All authors have contributed to the final version of the manuscript.
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Marciniak, A., Despré, V., Loriot, V. et al. Electron correlation driven nonadiabatic relaxation in molecules excited by an ultrashort extreme ultraviolet pulse. Nat Commun 10, 337 (2019). https://doi.org/10.1038/s41467018081318
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DOI: https://doi.org/10.1038/s41467018081318
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