## Introduction

Ultrafast electron diffraction (UED) methods1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 are powerful structural probes for gas-phase molecules due to the picometre-scale de Broglie wavelength of the electron, which permits to discern the location of a molecule’s atomic constituents. Laser-induced electron diffraction (LIED)7,9,16 is a strong-field variant of UED, which leverages the laser-controlled recollision of a single electron after tunnelling ionization1,5,8,11,13,14,16,17,18,19 of an attosecond electron wave packet (EWP)9,14,20, thus uniquely providing picometre and attosecond-to-femtosecond spatial-temporal imaging capabilities. Unlike any other diffraction method, the combination of mid-infrared LIED (MIR-LIED) with a reaction microscope permits the imaging of a single gas-phase molecule with one electron on a single-event basis9,10,14,21,22,23. These unique capabilities present many new opportunities, but they come with the caveat that the strong field itself triggers and modifies the molecular dynamics. In most cases, the influence of the strong field can be described very well with advanced quantum chemistry theory, thus allowing to extract the mechanisms of the processes under investigation despite the field-dressed dynamics. Examples of such treatment are the imaging of molecular bond breaking and deprotonation10, the photo-dissociation of carbonyl sulfide (OCS)19, or imaging skeletal deformation due to the Renner–Teller effect in carbon disulfide (CS2)22. We like to add that presence of the strong field can also be used to our advantage as intra-pulse pump–probe method in which the LIED field initiates and probes the dynamics. Variation of the pulse duration gives access to probing different dynamics, e.g. to distinguish between linear or bent stretching dynamics in OCS, or to probe the dynamics of the neutral molecule22. Notwithstanding these advances, extraction of the molecular structure from a measured electron interference pattern still requires some sort of pattern matching or minimization procedure, which can lead to molecular parameters with significant variances depending on the exact topography of extremal points in the solution space obtained from measured diffraction images. We note that this is a general challenge for all scattering-based methods, and not unique to LIED. All of these procedures require a degree of background subtraction, de-noising and fitting procedure and, thus, it is highly desirable to minimize any possible variance, which arises from the choice of methodology.

Here, we report an alternative scheme for retrieving the measured molecular structure from LIED experiments by exploiting its two-dimensional data. Firstly, the molecular structure is extracted from the measured MIT term in the laboratory frame to bypass the need to obtain the DCS in the laser frame. Secondly, two-dimensional DCS will be used whenever it is favourable, instead of limiting structural retrieval to one-dimensional DCS data as used in previous SD-LIED and FT-LIED methods. Thirdly, unlike SD-LIED, the molecular structure will be obtained by fitting not the whole MIT term of the DCS, but rather the critical zero crossing points (ZCPs) of the MIT term, making ZCP-LIED superior to SD-LIED as it is simpler and more efficient. We contrast our ZCP-LIED findings against FT-LIED using the same measured data to check if the bond lengths retrieved from a field-dressed carbonyl sulfide (OCS) molecule using both ZCP-LIED and FT-LIED methods are in agreement. We find that ZCP-LIED achieves a higher precision on bond lengths and it overcomes the difficulty of separating bond lengths that are nearly identical in FT-LIED/FABLES, where the radial distributions of bond lengths can overlap significantly. The paper starts with a description of the experiment on the molecular target OCS. Next, we extract the asymmetric top molecule’s structure with ZCP-LIED and contrast it against the well-established FT-LIED9,14,23/FABLES8,13 retrieval. Finally, we explain the molecule’s bent and asymmetrically stretched configuration with state-of-the-art quantum chemical and dynamical calculations.

## Results

### Experimental set-up

The set-up is shown in Fig. 1a, and we refer to refs. 10,20 for further details. Briefly, a gas mixture of 1% OCS seeded in helium was supersonically expanded through a 30-μm aperture, generating a molecular beam with an internal temperature of <90 K. The molecular beam was collimated using two skimmer stages and subsequently overlapped with a 96 fs (FWHM) 3.2 μm28,29 laser pulse generated by an optical parametric chirped-pulse amplifier (OPCPA) in the interaction region of the reaction microscope. The field strength of the 3.2 μm laser pulse corresponded to a Keldysh parameter of γ ≈ 0.3 and a ponderomotive energy of Up = 90 eV, which led to photo-ionization of electrons with maximal return and back-rescattered electron kinetic energies of Eret,max = 3.17 Up ≈ 285 eV and Eresc,max = 10 Up ≈ 900 eV, respectively. The resulting ions and electrons were guided towards separate detectors using homogeneous electric and magnetic extraction fields (19.1 Vcm−1 and 10.4 G, respectively). Both the electron and ion detectors consisted of chevron-stacked dual microchannel plates coupled with a quad delay-line anode set-up, enabling the detection of single molecular fragmentation events in full coincidence with 4π acceptance. LIED can be well-described through the framework of laser-driven electron recollision30,31,32,33 (see Supplementary Note 1 and Supplementary Fig. 1), as illustrated by the filmstrip inset in Fig. 1a. Here, a molecule is exposed to a strong laser field and an attosecond electron wave packet (EWP) is (i) emitted via tunnel ionization out of the target molecule; (ii) accelerated and returned back to the parent ion by the oscillating electric field of the laser pulse; and (iii) elastically recollides against the target ion where a momentum transfer leaves an imprint of the molecular structure within the EWP’s momentum distribution. Figure 1b shows the ion time-of-flight spectrum for all ions. Our analysis in this work will only consider electrons detected in coincidence with the OCS+ ion as part of our LIED imaging process to filter contributions from other processes such as multiple ionization or fragmentation. The two-dimensional momentum map of the transversal and longitudinal momenta (i.e. momenta emitted perpendicular and parallel, respectively, to the laser polarization direction—see Fig. S1) of electrons detected in coincidence with OCS+ is shown in Fig. 1c. In this map, electrons generated with a longitudinal momentum of >3 a.u. (<3 a.u.) correspond to LIED “rescattered” (“direct”) electrons detected with a kinetic energy of >2 Up (<2 Up) that have been returned by the laser field. The schematics shown in Fig. 1c correspond to the data analysis procedure for extracting the scattering differential cross sections in the laboratory frame (laser polarization frame) for the ZCP-LIED (FT-LIED) method as given by magenta (blue/red) lines and text.

### ZCP-LIED structural retrieval

Now, we turn to the ZCP-LIED analysis procedure for which we will show that the method allows retrieval of the entire molecular structure directly by identifying zero crossing point (ZCP) positions in the laboratory-frame photoelectron spectrum. First, we average out the local electron counts as a function of scattering momentum, p, and scattering angle, θ (see Fig. 1c). We obtain the laboratory differential cross-section (DCS) for 17 laboratory-frame scattering angles between 0.0 and 4.0° in steps of Δθ = 0.25° across the scattering momentum, p, range of 4.72–7.16 a.u. (see magenta line with origin (0,0) in Fig. 1c for one exemplary angular distribution that is subsequently shown in Fig. 2a). Including momenta lower than 4.72 a.u. is not considered desirable as the signal begins to rise quickly due to contributions from “direct” electrons and the IAM cannot be applied at small scattering energies, while for higher momenta above 7.16 a.u. the contrast in the modulated signal becomes very small due to an insufficient signal-to-noise ratio. To identify the ZCPs, we first smooth the laboratory DCS with a filter and then fit it with a third-order polynomial function using the least-squares method. The fitted polynomial corresponding to the background atomic IA signal is then subtracted from the laboratory DCS to obtain the laboratory molecular interference signal, IM. This procedure is repeated for each subsequent angle. The collection of laboratory molecular interference signal IM at varied θ is shown in Fig. 2a. The amplitude of the interference signal at smaller momenta is larger than that at larger momenta due to the reduced scattering probability with increasing electron scattering momentum. The roots of these molecular interference signals define the ZCPs. The indicated magenta arrows in Fig. 2a and vertical dashed lines in both Fig. 2a, b are the mean values of the four ZCPs at four momentum points (5.18, 5.84, 6.36 and 7.00 a.u.) obtained from the measured distributions extracted over the 17 laboratory scattering angles. Figure 2b provides an alternative view of the two-dimensional (2D) molecular interference signal. We represent positive (negative) values of IM as red (blue) squares. The boundaries where the IM changes sign are the locations of the ZCPs. To better visualize the variance, we use the mean value of the ZCP positions as the zero of the momentum axes for all four ZCP distributions. Due to a weaker molecular interference signal at relatively larger scattering momenta where signal-to-noise is lower, the ZCP positions at larger momenta possess a larger error in general. To retrieve the molecular structure, we apply the IAM to obtain the corresponding theoretical ZCPs for a variety of molecular structure parameters. We vary the bond lengths until the square of the difference between the experimental and theoretical ZCP positions is minimal. This optimization process is performed using a genetic algorithm. The optimal internuclear distances found are RCO = 1.18 Å, RCO = 1.70 Å and ROS = 2.74 Å.

Figure 3a displays the reconstructed ZCPs generated with the optimal bond lengths at the four momentum points (5.21, 5.82, 6.37, 6.97 a.u.), shown in magenta vertical solid lines, that best match the ZCPs of the experimental data over the 17 scattering angles according to the IAM. These magenta solid lines for the best-fit theoretically calculated ZCPs in Fig. 3a are at momentum positions nearly identical to those measured average ZCPs, shown as magenta vertical dashed lines in Fig. 2a–c (at 5.18, 5.84, 6.36 and 7.00 a.u.). Note that the magenta solid lines in Fig. 3a separate the positive and negative terms of the calculated IM with no dependence on the scattering angles, which is a general feature predicted by the IAM. Figure 3b shows the comparison of the experimental molecular interference signals (coloured lines) from 17 scattering angles with the reconstructed ones (black circled line) as a function of the electron momentum in the laboratory frame. We note that the positions of the maxima or minima are fairly independent of scattering angles as well reproduced to a fairly good degree.

despite the fact that these positions are not included in the optimization. We comment that the data used in the fitting covers the momentum from 4.7 to 7.2 a.u. only as in the experimental data analysis procedure described above. The lower limit was chosen at 4.7 a.u. as the IAM fails more as the electron momentum is further reduced. Thus, only four ZCP points are considered in the analysis which are sufficient to accurately retrieve the measured OCS+ molecular structure.

Finally, the distribution of experimental ZCP positions for different scattering angles (see Fig. 2c) can be used to estimate the error of the retrieved bond lengths. Since the molecular interference term is evaluated by averaging over a bandwidth of Δp = 0.1 a.u., Fig. 2c shows the number of angles where the ZCPs falls within 0.1 a.u. From this, we find that around half of the ZCPs fall below or above the magenta dashed line within 0.1 a.u. We can thus estimate the error of each ZCP, and, in turn, calculate the error of the retrieved bond lengths. We extract values of RCO = 1.18 ± 0.02 Å, RCS = 1.70 ± 0.02 Å and ROS = 2.74 ± 0.03 Å. These distances result in an O–C–S bond angle of ΦOCS = 144 ± 5°. The theoretical molecular interference signal (black circled line) that best reproduces the measured data (coloured lines), thus, corresponds to a significantly bent and asymmetrically stretched OCS+ structure, as shown in the lower right corner of Fig. 3b, which is drastically different from the equilibrium neutral ground state of OCS. Moreover, the corresponding theoretical ZCPs for equilibrium neutral ground state of OCS appear at four momentum points (grey vertical solid lines; 5.30, 6.03, 6.47, 6.84 a.u.) that are in poor agreement with the measured ZCPs (magenta vertical solid lines; 5.18, 5.84, 6.36, 7.00 a.u.).

### FT-LIED structural retrieval

Next, we contrast the results from ZCP-LIED against an FT-LIED analysis21 using the same experimental data as that used in the ZCP-LIED analysis. The experimentally extracted field-free DCS (blue dashed line) is extracted by integrating the electron counts along the return momentum pr for back-rescattering angles (see red arrow in Fig. 1c) in the laser polarization frame. Figure 4a shows the extracted DCS with its corresponding error-bars given by the longitudinal detected momentum (red horizontal) and a Poissonian distribution (blue vertical). The oscillations observed in the DCS above 50 eV arise from the coherent molecular interference signal, IM, as a result of the interference between the collision of the EWP and the OCS+ ion. In fact, our measured signal, IT, contains contributions from both IM and the incoherent sum of atomic scatterings, IA. The latter is independent of the molecular structure and therefore contributes a smooth background signal to our measured IT. This allows the extraction of IM with a simple subtraction of the empirical background IA obtained from a linear regression filter to the DCS (black starred line in Fig. 4a).

We subsequently enhance the oscillatory modulations in the IM by contrasting it against the IA through the molecular contrast factor, $${\mathrm{MCF}} = \frac{{I_{\mathrm{M}}}}{{I_{\mathrm{A}}}}$$, as shown in Fig. 4b. Here, the typical interference features are present (i.e. ZCPs, the local maxima and minima) over a wide momentum transfer range as seen earlier with ZCP-LIED. A simple Fourier transform of the MCF (Fig. 4b) results in Fourier amplitudes which are characteristic of the atomic separations within the measured system, as shown in Fig. 4c. The resulting radial distribution (black dashed line) exhibits three distinct peaks, with the area of uncertainty indicated by the shaded regions corresponding to the detected longitudinal error (red horizontal) and the Poissonian statistical error (blue vertical). The centre positions of these peaks are extracted with a least-squares fit of the measured distribution using the sum of three Gaussian distributions, which correspond to the C–O, C–S and O–S internuclear distances for which we measure values of RCO = 1.06 ± 0.06 Å, RCS = 1.87 ± 0.13 Å and ROS = 2.78 ± 0.10 Å. These distances result in an O–C–S bond angle of ΦOCS = 142 ± 22°. We find that these results are in excellent agreement with the results from the ZCP analysis and are summarized in Table 1 together with previously reported structural parameter values for field-free (FF) neutral OCS34,35 and OCS+ cation36.

### Quantum-classical calculations

Finally, we explain the retrieved bent molecular structure and contrast our findings against the linear OCS+ structure retrieved at 2.0 μm by Karamatskos et al.19. Therefore, we perform quantum chemical calculations on the CASSCF/aug-cc-pVTZ level of theory in combination with the classical surface hopping method to describe the dynamics of OCS in the presence of the 3.2 μm laser pulse used in our experiment. In the absence of a field, neutral OCS in its ground electronic state is linear (C∞v)34,37. The 11Δ ← 11Σ+ transition is dipole forbidden due to symmetry. However, field-dressing OCS on the rising edge of the 3.2 μm laser field results in a significantly bent molecule (see Fig. 5c) due to the Renner–Teller effect22,34,38. We find in Fig. 5, similar to our recent work in CS222 that in the bent geometry, a small population transfer to the 21A′ excited electronic state of the neutral OCS (lower bound of ~3%) occurs. As the ionization potential of this excited state is much lower than the 11Σ+ ground state (~5.2 eV lower), our LIED signal is dominated by neutral OCS in its 21A′ excited state.

Consequently, we expect, and measure, a significantly bent and asymmetrically stretched OCS+ structure in a 96 fs (FWHM; 9.0 optical cycles) 3.2 μm field. This is in stark contrast to the recent findings published by Karamatskos et al.19 who retrieve a linear OCS+ structure. This is accounted for by their much shorter 2.0 μm pulse duration of 38 fs (FWHM; 5.7 optical cycles) being too short in time to invoke the structural deformation of the neutral molecule during the rising edge of their pulse. We chose a longer pulse duration which allows the neutral molecule to undergo significant nuclear deformation on the rising edge of the laser pulse similar to those of CS2 and C6013,22. We note that our results are in excellent agreement with the work of Sanderson et al. in the Coulomb explosion of OCSn+ with a 55 fs (FWHM; 20.6 optical cycles) 0.8 μm laser pulse39. These authors assigned the recoil velocities from the (O3+, C3+, S4+) channels to a bent structure (ΦOCS = 140°) just prior to Coulomb explosion. Recent ab initio calculations at 0.8 μm also revealed that structural deformation in OCS occurs on a timescale longer than 13 fs40, with wave packet simulations reporting that a linear-to-bent transition occurs on a 50-fs timescale, reaching an OCS bond angle of ~135° 37.

## Discussion

While the ZCP-LIED method has been demonstrated for the structural retrieval of the OCS molecule only, we believe that the present work offers two important implications which deserve further discussion. The first one is presently limited to LIED experiments only. We have shown the advantage of extracting molecular structure using the full two-dimensional (energy and angle) electron scattering spectra in the laboratory frame, in contrast to existing methods that extract molecular structure in the laser frame. This avoids the tedious procedure of converting the 2D laboratory-frame electron spectra to the laser frame. Using the full 2D scattering spectra instead of 1D spectra improves the statistics of the retrieved results. Moreover, we have demonstrated that ZCP-LIED can retrieve the molecular structure with only the critical crossing points of the molecular interference term (MIT), avoiding the need to use the whole electron scattering spectra directly or the whole MIT. Since molecular bond lengths enter explicitly only in the phase of the interfering terms, by analysing only the ZCPs, the ZCP-LIED method relaxes the limitation of the Born approximation or the independent atom model (IAM) that are at the heart of electron diffraction theory. We further suggest that a combination of ZCP-LIED and FT-LIED methods to retrieve bond lengths from the same scattering data would improve the accuracy (as opposed to FT-LIED only) and avoid false results retrieved based on iterative methods. It should be noted that ZCP-LIED exhibits several advantages over FT-LIED. Firstly, ZCP-LIED retrieves internuclear distances with a higher precision, and it bypasses the challenge of separating internuclear distances that may be hard to distinguish in FT-LIED. Secondly, ZCP-LIED avoids the background polynomial fitting procedure that is used in FT-LIED. Thirdly, ZCP-LIED avoids converting between different frames of reference to retrieve structural information, as is required in FT-LIED and SD-LIED. Finally, ZCP-LIED is capable of structural retrieval at lower signal-to-noise as compared to FT-LIED, and the method is robust in that only a small number of ZCPs are required to accurately retrieve the molecular structure as opposed to requiring the whole MIT or scattering signal for successful retrieval. For example, while we used four ZCPs, in fact only two ZCPs were sufficient to distinguish between the linear (grey vertical lines) and the bent (magenta vertical lines) OCS+ structure (Fig. 3b). Moreover, we have tested the ZCP-LIED method on 10-atom molecules and found that the positions of ZCPs are independent of scattering angles. The accurate, reliable and simplified extraction of molecular structure from scattering data as demonstrated with the ZCP method is essential for the correct interpretation of diffraction imaging-based studies of chemical dynamics and the further evolution of dynamic imaging.

The second important implication from this study is whether the ZCP analysis method can be extended to ultrafast electron diffraction (UED) or even ultrafast X-ray diffraction (UXD). At present, conventional electron and X-ray diffraction methods utilize a monochromatic beam at high energy which can resolve molecular structure with few-picometre precision. For dynamical imaging, high spatial resolution of few-picometre is not needed given that the change in bond length is on the order of tens-to-hundreds of picometres. Would diffraction experiments with a broadband pulse at lower energies be a possible and more advantageous approach for diffraction-based dynamical imaging? Like LIED experiments, if the energy and angle of the scattered electrons can both be measured then two-dimensional differential cross-section data can be determined. This has the advantage that more data are collected in a single experiment. For high-energy beams, the two-dimensional data are redundant since the scattering cross sections depend only on momentum transfer, but similar to what we have demonstrated in this work, such data would provide additional complementary information on the retrieval process. In general, for time-resolved LIED/UED/UXD measurements, we believe that the ZCP analysis will prove to be an important and easy-to-use approach to track the transient geometric structure of a photoexcited molecule by simply tracking the time-resolved changes in the ZCPs of the transient molecular interference signal. Measuring the transient ZCPs will enable a direct retrieval of the transient molecular structure.

In summary, we demonstrate a simple and elegant retrieval of the molecular interference signal directly from the laboratory-frame photoelectron momentum distribution map. We achieve this by identifying the ZCPs of the molecular interference signal which provides a unique fingerprint of molecular structure. Using ZCP-LIED avoids complex retrieval algorithms, semi-classical analyses and ab initio calculations to retrieve structural information as is typically used in SD-LIED, UED and UXD. We confirm the ZCP-LIED results with comparison to those obtained using FT-LIED. For both retrievals, we find a significantly bent and asymmetrically stretched OCS+ structure. The observed bent OCS+ structure arises from a linear-to-bent transition assisted by the Renner–Teller effect in the presence of a strong laser field as confirmed by our state-of-the-art quantum-classical dynamical calculations. Avoiding strong deformation of the molecular structure on the rising edge of the 3.2 μm LIED probe pulse as seen in OCS (this work), CS222 and C6013 are important in capturing clean snapshots of transient structure in two-pulse pump–probe measurements of photo-induced chemical dynamics, which can be achieved by using 3.2 μm pulses with shorter pulse durations. We expect that ZCP-LIED will prove to be a powerful tool in the retrieval of larger, and more complex, molecular structures as well as transient structures where retrieval algorithms and ab initio calculations become increasingly challenging.

## Methods

### Experimental set-up

Our experimental set-up has been described in detail previously10,20, and briefly in the main text. The 20 W output of the OPCPA set-up, operated at a 160 kHz repetition rate, was focussed to a spot size of 6–7 μm which generated pulses with a peak intensity of ~9 × 1013 Wcm−2. The OPCPA system was seeded by the difference frequency of a dual colour fibre laser system to subsequently emit passively carrier-envelope phase (CEP) stable pulses41. The mid-infrared wavelength of 3.2 μm ensured that the target was strong-field ionized in the tunnelling regime. In all experiments, the laser’s polarization is aligned along the spectrometer axis, perpendicular to the jet.

### Theoretical calculations

The singlet electronic ground and excited state potential surfaces as well as the (transition) dipole moments have been calculated using the OpenMolcas program suite42 at the state-averaged CASSCF(12,10)/aug-cc-pVTZ level of theory43. The active orbitals were chosen from the full valence active space of OCS comprising 16 electrons in 12 orbitals, omitting the two lowest sigma orbitals with main contribution from the 1s atomic orbitals of the sulfur and oxygen atoms. The electronic structure calculations resulted in a minimum structure with an angle ΦOCS = 180° and bond lengths of RCS = 1.61 Å and RCO = 1.16 Å, respectively. Altogether, the first 12 configuration interaction (CI) roots have been calculated, without using symmetry.

In the dynamical calculations, we have employed the semi-classical programme package SHARC 2.044,45, which is interfaced with the OpenMolcas program suite42. For dynamics simulations, six CI roots have been calculated employing a smaller cc-pVDZ basis set. Within SHARC, the nuclear motion is treated classically, while the electronic states are calculated with the state-averaged CASSCF method specified above. Non-adiabatic effects, as well as field-induced population transfer is included in the SHARC code via jumps between electronic surfaces. Our mixed quantum-classical calculations on the strong-field driven dynamics of OCS in a 3.1 μm laser field show that upon interaction with the laser field, the molecule in the electronic ground state starts to bend by about 5–10°, accompanied by an asymmetric stretch vibration. Subsequently, near the maximum of the laser pulse, population transfer to the 21A′ excited state can occur, leading to a small population in the excited electronic state (lower bound of ~3%). Note that this represents a lower bound, as the transition dipole to the excited state is still very small at these nearly linear geometries (see Table 2) and thus is deficient; even small changes (e.g. in the size or choice of the basis set, or the choice of active orbitals) can easily vary the tiny transition dipole moments by a factor of 5–10. Table 2 summarizes the potential energies and oscillator strengths for the first four excited electronic states of neutral OCS. Our results agree with previous ab initio calculations40. Moreover, in order to minimize the computational effort, simulations assume instantaneous delta pulse excitation to the excited state at the beginning of the simulation. We used 50 initial conditions which were obtained from a Wigner distribution of the ground state. Trajectories were started for an ensemble of 20 initial conditions with the largest oscillator strength to the 21A′ state, which lead to a distribution of initial bent OCS angles between 171 and 178°. Subsequently, the trajectories are propagated using the classical equations of motion utilizing the ab initio potential surface of the 21A′ state. These trajectories are weighted by a Gaussian of width similar to that of our FT-LIED measured extracted Gaussian width (see Table 1) and shown in Fig. 5.