Photoelectron Diffraction Imaging of a Molecular Breakup Using an X-Ray Free-Electron Laser

A central motivation for the development of X-ray free-electron lasers (XFELs) has been the prospect of time-resolved single-molecule imaging with atomic resolution. Here we show that X-ray photoelectron diﬀraction - where a photoelectron emitted after X-ray absorption illuminates the molecular structure from within - can be used to image the increase of the internuclear distance during the X-ray-induced fragmentation of an O 2 molecule. By measuring the molecular-frame photoelectron emission patterns for a two-photon sequential K-shell ionization in coincidence with the fragment ions, and by sorting the data as a function of the measured kinetic energy release, we can resolve the elongation of the molecular bond by approximately 1.2 atomic units within the duration of the X-ray pulse. The experiment paves the road towards time-resolved pump-probe photoelectron diﬀraction imaging at high-repetition-rate XFELs.


I. INTRODUCTION
Following chemical reactions and structural changes in molecules in real time (and eventually steering/controlling these dynamics) has been a long-standing dream in physical chemistry. Accordingly, time-resolved single-molecule imaging has been one central aim triggering the development of X-ray free-electron lasers (XFELs). The intent of this new class of time-resolved imaging approaches is to investigate geometrical changes of a molecule with atomic resolution in real time, which is complementary to standard femtochemistry [1] approaches that follow the potential energy landscape of a molecule. Photoelectron diffraction is one of the envisioned techniques among others [2][3][4][5]. The idea of X-ray * Electronic address: demekhin@physik.uni-kassel.de † Electronic address: jahnke@atom.uni-frankfurt.de induced photoelectron diffraction imaging as an analytical tool is already well-established in surface sciences [6]. Almost two decades ago it has been pointed out that this technique can be used in the gas phase for deeply bound (i.e. localized) inner-shell electrons [7]. With the advent of XFELs, corresponding studies in the gas phase came into reach [8][9][10][11]. Detailed information can be retrieved when the electron angular emission distribution in the molecular frame of reference is obtained, which is the key to photoelectron diffraction imaging in the gas phase. Upon photoabsorption, an electron wave is launched from a well-defined location in the molecule, thus "illuminating the molecule from within" [7]. Its angular distribution is a manifestation of a complex interference pattern of this electron wave as it is multiplescattered by the molecular potential on its way out. Such distributions show very rich features depending on the molecular species under investigation, the initial orbital the electron is emitted from, the polarization, as well as the wavelength of the ionizing photons. In particu-lar, polarization-averaged molecular-frame electron angular distributions have been recently demonstrated to capture many molecular details [12], such as the internuclear distance at the instant of emission [13] or even the whole molecular structure as such [14]. In order to observe molecular-frame electron angular distributions, the orientation of the molecule at the instant of photoionization needs to be known. This is achieved by either actively aligning molecules in the gas phase via laser pulses [15] or by determining their orientation a posteriori from a coincident measurement of the ionic fragments, which are in many cases produced after inner-shell photoionization of a molecule [7,14,16]. The high repetition rate of the European XFEL finally opens the door to such coincidence experiments and we present a first successful implementation of this technique using a COLTRIMS-Reaction Microscope [17,18] at the European XFEL. Due to the high photon density in the focus of the XFEL beam, several photons can be absorbed sequentially by an oxygen molecule within a single light pulse. Absorption of the first photon and subsequent Auger decay triggers the break-up of the dication in a Coulomb explosion. During elongation of the molecular bond in the dissociation process, the molecule can absorb a second photon and emit another K-shell electron, which is then used for the photoelectron diffraction imaging in our experiment.
The photons have energies of approximately 670 eV. We were able to obtain internuclear distance-resolved photoelectron diffraction patterns, investigating the aforementioned naturally occurring fragmentation of the molecule upon K-shell ionization during the duration of the X-ray pulse of approximately 25 fs. Our experiment demonstrates that photoelectron diffraction imaging is finally possible using high repetition rate XFELs. In particular, it paves the road towards time-resolved pump-probe photoelectron diffraction imaging.

II. EXPERIMENT
In our experiment we employed soft X-ray pulses at the Small Quantum Systems (SQS) instrument at the European XFEL [19] to generate multiple core holes in O 2 molecules. A COLTRIMS Reaction Microscope [18] installed as a permanent end-station has been used for the coincident measurement of all generated molecular fragments. Details on the experimental setup and the X-ray properties can be found in Appendix A.
A scheme of the reaction triggered is shown in Fig. 1: A first photon is absorbed ionizing a O 2 -K-shell electron. Subsequently, an Auger decay causes the break-up of the molecule in a Coulomb explosion. During the dissociation of the molecule a second photon may ionize a further K-shell electron. This electron is used to illuminate the fragmenting molecule from within. By measuring the finally occurring singly and triply charged atomic ions and the second photoelectron in coincidence, we were able to perform internuclear distance-resolved photoelectron  As the absorption of the second photon occurs at random delay times during the light pulse, the information on the delay between the two photon absorption events needs to be extracted from other observables in our coincidence measurement. In the example of H 2 , it is possible to obtain information on the internuclear distances from the kinetic energy of the ionic fragments after a Coulomb explosion, as this energy depends only on the initial internuclear distance of the two protons [20]. However, in the present study on oxygen, the relation between interatomic distance and kinetic energy release (KER) is not straightforward: Two sequential inner-shell photoabsorptions with subsequent Auger decay yield a total charge state of four, thus involving four excited-state potential energy curves. A simplified sketch of the involved charge states is shown in Fig. 2. This sketch neglects the comparably complex electronic structure of O 2 , and for example, in case of a break-up into O + /O + after core-ionization the KER-spectrum exhibits already several distinct features that have been attributed to a multitude of different decay pathways towards several final O + /O + states [21,22]. Thus, a direct extraction of the internuclear distance from the measured KER of individual quadruple ionization events is expected to be futile. We therefore pursue a different path for our analysis and focus first on polarization-averaged molecular-frame photoelectron angular distributions obtained from a full theoretical modeling in order to compare these to the ones obtained from our experiment.

III. THEORY
For linearly polarized ionizing radiation, the angular emission distribution of photoelectrons in the frame of a diatomic molecule is given (within the dipole approximation) by: Here, β is the angle between the molecular axis and the direction of the polarization of the ionizing radiation (the remaining orientation Euler angles α and γ are set to zero), θ and φ are the photoelectron emission angles defined with respect to the molecular axes, Y lm are spherical harmonics, and D 1 k0 are the Wigner's rotation matrices. Electron dynamics of the considered process is imprinted in the amplitudes A lmk for the emission of the partial electron continuum waves of energy with the angular momentum quantum numbers l and m through the absorption of one photon of polarization k = 0 ± 1, as defined in the frame of the molecule. Within our theoretical model, the photoionization transition amplitudes A lmk were computed by using the stationary Single Center (SC) method and code [23][24][25], which provide an accurate theoretical description of the angle-resolved photoemission spectra of molecules. The calculations were performed in the Relaxed-Core Hartree-Fock approximation enabling relaxation of the molecular orbitals in the field of the core vacancy. The SC expansion of all occupied orbitals of O 2 with respect to the geometrical center of the molecule was restricted to partial harmonics with l ≤ 99, and for the photoelectron in the continuum by l ≤ 49 (note that in diatomic molecules, the projection m of the angular momentum l is a well-defined quantum number).
For the first photoelectron, the neutral O 2 molecule in the initial photoionization step was considered to be in its ground electronic state at the respective equilibrium internuclear distance. Calculations were performed for a photoelectron kinetic energy of = 127 eV and averaged over the two contributions from the degenerate initial orbitals 1σ g/u (or 1s R/L ). For the second photoelectron, the doubly ionized O 2+ 2 molecule after the initial photoionization step was considered to be in the ground electronic state of the dication. These calculations were performed for a kinetic energy of = 93 eV and several internuclear distances, as indicated in Fig. 3. Only the contributions from the initial orbital 1s R , which is localized at the oxygen atom being triply charged at the end, was considered. In both cases, angular emission distributions were computed in the orientation interval β ∈ [0, 180] in steps of ∆β = 1 • and then averaged over these orientations of the polarization axis of the light with respect to the molecular axis.

IV. RESULTS AND DISCUSSION
We can distinguish the first photoelectron stemming from the pump and the second originating from the probe step from the measured electron energy. At the photon energy employed in our experiment (hν = 670 eV), the first photoelectron is emitted at an electron energy of 127 eV, the second within a range of 80 eV < E e < 105 eV depending on the emission time. As a check of the validity of our modeling, we examine the results obtained for the first K-shell electron (Fig. 4). The angular distribution of this photoelectron -emitted in the pump step -is fully symmetric with respect to the O + and O 3+ sides of the fragmentation. This highlights that the symmetry of the molecule is not yet broken during this first ionization step. The modeled molecularframe electron angular distribution corresponds nicely to the measure one. The theoretical predictions of the angular distributions of the photoelectrons from the probe step (i.e. the second photoionization) are depicted in Fig. 3. They are highly asymmetric. This is a direct consequence of the fact that the second photoionization is assumed to occur at the K-orbital localized at the right hand Oxygen ion (labeled "3" in the inset). The main lobe is directed towards the singly charged oxygen ion, showing the ability of photoelectron diffraction to image the neighboring atom. Inspecting the calculated distributions in Fig. 3 reveals a main trend, which is the occurrence of additional lobes and minima as the internuclear distance increases. For example, the maximum along the molecular axis on the right of Fig. 3(a) (labeled as m3) rotates towards larger angles in Figs. 3(b,c) as a minimum occurs along the molecular axis, which finally turns into a maximum m4 in Fig. 3(d). Such behavior is well known from interference-based effects (e.g. Bragg-diffraction, double slit scattering), where an increase of the distance between the interfering virtual sources of the waves leads to more narrowly spaced interference fringes. For the homonuclear molecule O 2 , this behavior is expected as well and occurs as a direct consequence of the electron scattering process: Even though the electron emission in question can be considered to originate from the eventually triply charged ion, the electron is scattered by the other ion and the interference of direct and scattered pathways forms the observed body-fixed electron emission pattern. For illustration, we show in Fig. 5 a sequence of corresponding angular distributions as obtained from interfering a direct and a (phase-shifted) scattered wave with wavelength λ. Every time the distance between the emitter E and the scatterer S is increased by λ/2 a new maximum emerges on the right and one further increase by λ/4 yields a new minimum. The range of internuclear distances covered by Fig. 3 corresponds to the distances that the two point charges (m = 16, q = 1) can reach within the duration of the XFEL pulse of approximately 25 fs, after being initialized at rest at the O 2 mean inter- shortest distances no diffraction lobes are visible. b) As the distance increases a first maximum m1 forms towards the emitter side. c) Further increase yields the first minimum in direction of E and the maximum m1 moves towards the main lobe in direction of S. As the distance is increased a new lobe m2 appears (d) and moves towards S, as well (e).
nuclear distance and then driven solely by their Coulomb repulsion.
Now, we investigate the experimental, polarizationaveraged molecular-frame electron angular distributions for different KER-values. We find within a large range of KER (28 eV< KER < 36 eV) a good qualitative agreement with the theoretical predictions. A movie of a sweep through this region can be found in the online supplementary materials section, and several examples of the molecular-frame electron angular distributions from within this range are depicted in Fig. 6. The results of our theoretical modeling are overlayed as a red line. It is not surprising that the agreement is only qualitative: The calculations for this second photoelectron have been performed only for fixed (i.e. single) internuclear distances and for a fixed electron energy of 93 eV (which is the mean energy of this electron). Moreover, only the ionization of the dicationic O 2+ 2 ground state by the second photon has been taken into account. In the experiment, as argued before, a fixed KER does most likely not correspond to a distinct internuclear distance and, in addition, the experimental data have been integrated over the electron energy range expected for the second K-shell electron, which has been emitted after the first Auger decay (i.e. 80 eV< Ee < 105 eV). Moreover, the experimental data include contributions from all final Auger states being ionized by the second photon, and, furthermore, the probability that the third charge created in the process remains at the Oxygen ion where the electron was ejected from is not 100%. Despite these conceptual flaws, the experimental distributions resemble those obtained from our theory and indicate that a similar range of internuclear distances is covered here as in the calculations (Fig.  3). This finding prompts us to employ a simplified classical model of the transitions and potential energy curves involved in the overall ionization pathway, which is described in appendix B. This model provides values for the assumed internuclear distances computed from the corresponding KER-values. The values are stated below each panel in Fig. 6, and the theoretical distribution overlaid for comparison was selected accordingly. Thus, Fig. 6 a), d), and e) are compared to Fig. 3 a), c), and d), respectively. The internuclear distance of Fig. 3 b) is somewhat in between that belonging to the KER values employed in Fig. 6 b) and c) and, accordingly, the distribution shown in Fig. 3 b) has been employed for comparison to both measured angular distributions. Taking the simplicity of the model and the complexity of the O 2 electronic structure into account, the level of agreement is surprisingly good, even though the measured angular distributions obtained for largest internuclear distances are more washed out. Furthermore, the measured angular distribution shown in Fig. 6 d) suggests contributions from slightly larger internuclear distances as-instead of the predicted minimum-a further lobe seems to occur in the direction of the emitting, finally triply charged O 3+ ion.

V. CONCLUSION
Our work demonstrates that molecular-frame photoelectron diffraction imaging is a viable route for analyzing molecular dynamics while employing high repetition rate XFELs. Even without direct access to real-time information, the evolution of the dissociation process could be followed via scanning the KER. Remarkably, this was possible despite the multi-step ionization process and revealed a first molecular movie of a sequential core-hole ionization. Upcoming two-pulse pump-probe schemes are expected to open a new realm of time-resolved studies of chemistry on the most fundamental level. With such two-pulse schemes the need of extracting geometrical information on the molecule and thus the timing of the process from ion properties, as in our case, vanishes, allowing for performing direct time-resolved photoelectron diffraction imaging. Additionally, with the future availability of femtosecond lasers synchronized to the X-ray pulses molecular dynamics can be triggered, and photoelectron diffraction imaging can be used as a powerful probe step in time-resolved experiments on molecular dynamics.  Fig. 3 are overlayed as red lines (see text for details). The internuclear distances R and the absorption time t of the second photon have been obtained using a simple Coulomb explosion model (see Methods section). The ion, which is triply charged in the final state, is located on the right.

Acknowledgments
We acknowledge the support by the European XFEL staff. We are indebted to A. Bräuning-Demian for timely support during the beamtime preparations. This work has been supported by the Bundesministerium für Bildung und Forschung (BMBF), grants 05K13RF4 and 05K16RF1. The

VI. APPENDIX A: EXPERIMENTAL DETAILS AND XFEL PROPERTIES
The experiment has been performed at the Small Quantum Systems (SQS) instrument of the European X-Ray Free-Electron Laser in Hamburg (Germany). A Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS) reaction microscope [17,18] has been used to perform ion-ion-electron coincidence measurements of the charged fragments of O 2 molecules after irradiation with femtosecond X-ray pulses. In brief, a supersonic jet of O 2 molecules has been intersected at right angle with the ionizing light. Charged fragments have been guided by weak electric (43.5 V/cm) and magnetic (26 G) fields towards two time-and position-sensitive micro-channel plate detectors with an active area of 120 mm. The ion arm had a length of 25.5 cm; the electron arm consisted of a single acceleration region of 60.8 cm length. From the flight times and the positions of impact the initial vector momentum of each particle can be deduced. Accordingly, the emission directions of the ionic fragments of the molecule are measured in coincidence with the photoelectron emission direction. This provides all information needed to obtain molecular-frame angular distributions, assuming that the axial recoil approximation holds (i.e. the diatomic molecule does not significantly rotate prior to fragmentation). The data presented in this manuscript has been obtained during a total acquisition time of approximately 17 h.
We employed X-ray pulses with initial pulse energies of 2.4 mJ, which were attenuated to 30 ± 5 µJ by using a 15 m long gas absorber filled with nitrogen gas. This results in 14 ± 2 µJ on target based on the calculated beamline transmission of 0.46 at a photon energy of 670 eV. The focus size has been determined to be ∼ 0.9 × 1.6 µm 2 at a photon energy of 1 keV using a wavefront sensor. The EuXFEL operated at a base repetition rate of 10 Hz, providing bursts of X-ray pulses with a spacing of 1.1 MHz, out of which we used every 6 th pulse for our experiment, i.e. the X-ray pulses had a spacing of 5317 ns. We received 42 bunches per burst, i.e., 420 X-ray pulses per second. The pulse duration was not measured but is calculated to be ∼25 fs based on the electron bunch charge of 250 pC.

VII. APPENDIX B: SIMPLIFIED DECAY AND COULOMB EXPLOSION MODEL
As outlined in the main article, the complexity of the electronic structure and the overall decay path makes a direct determination of the internuclear distance from the measured kinetic energy release of the ions difficult. We employed a simplified model of the ionization and decay process in order to estimate the internuclear distance at the instant of the emission of the second K-shell photoelectron for the movie (see online supplementary material) and the values obtained fit surprisingly well to the observed number of interference fringes and seem to confirm the assumed range of internuclear distances covered in the experiment. A scheme of the involved potential energy curves is shown in Fig. 2. In our experiment, we concentrate on a break-up of the molecule into O + /O 3+ , which is reached by the following overall decay pathway: A Coulomb explosion is triggered, as an Auger decay occurs after the removal of the first K-shell photoelectron from the molecule. The system propagates along the repulsive O + /O + potential energy curve, which is approximated to be 1/R (in atomic units) in our modeling. After an unknown delay (within the X-ray pulse duration of 25 fs), a second photoabsorption can take place, and the second K-shell electron is ionized at the internuclear distance R 2 as the molecule continues its fragmentation along the O + /O 3+ potential energy curve (approximated by 3/R) after the emission of a second Auger electron (assumed as instantaneous). The overall kinetic energy gained within this dissociation path consists of the two contributions KER A and KER B as shown in Fig. 2. Given, in addition, the O + /O 3+ potential energy curve, R 2 can be extracted from the measured total KER = KER A + KER B .
In order to validate that the range of internuclear distances observed in our experiment is within reach during the duration of the XFEL pulse, we simulated a Coulomb explosion of two point charges with q = 1 that have been placed at the molecule's mean internuclear distance of R = 2.28 a.u. without initial kinetic energy. These point charges reach internuclear distances of up to R ∼ 6.5 a.u. within the XFEL pulse duration of approximately 25 fs. This internuclear distance, thus, should be considered as a maximum value for R 2 , assuming an instantaneous Auger decay of the O + 2 ionic state. However, considering the mean decay time of this state, the range of reachable internuclear distances reduces to approximately R ∼ 3.8 a.u., which is still well within the range observed in our experiment. The time information provided in the movie (supplemental material) and Figs. 3 and 6 have been retrieved using this dissociation model along the O + /O + potential energy curve, adding the mean lifetime of the decaying O + 2 state. A possible reason why our modeling of the charge-up and decay of the molecule yields reasonably good agreement between theory and experiment in Fig. 6 can be found in the shape of the kinetic energy release distribution. Fig. 7 depicts the measured KER for a) the O + /O + channel and b) the breakup of the molecule into O + /O 3+ . The distribution shown in Fig. 7 a) exhibits several distinct peaks resulting from intermediate O + 2 states which have potential energy curves which are to some extent not repulsive. A full assignment and disentangling of these can be found in [21]. The distribution of the quadruply charged final state shown in b) is different. It is in principle featureless and consists mainly of one single broad peak. In line with the short length of the X-ray pulse, this suggest a scenario where high charge states are generated very rapidly, and the final kinetic energy of the ions is indeed dominated to large extent by the Coulomb explosion along steeply repulsive potential energy curves. The KER region used to generate Fig. 6 is marked in red. The molecular frame angular distributions outside of that red region resemble those shown in Fig. 6 without showing a systematic evolution with respect to the KER.