Introduction

In 1979, Kanter et al. introduced the Coulomb explosion imaging technique (CEI)1. In this technique, the molecule under investigation is ionized rapidly, multiple times, on a femtosecond time scale and then breaks up, due to the Coulomb repulsion between the positively charged ionic fragments. By measuring the momentum of all the ions in coincidence, information about the initial molecular structure is theoretically available. Experimentally, the Coulomb explosion occurs in a momentum imaging apparatus that consists of an electric field that allows all ionic fragments to be captured in coincidence over 4π. The momentum imaging apparatus ends in a Position Sensitive Detector (PSD) to record the position, as well as the time information. From this information, we can obtain the momentum of the ions which are directly related to the parent molecule structure before ionization2. There are many ways to ionize the parent molecule: the foil-induced ionization technique2,3, highly charged ion impact4,5, electron impact6,7, X-ray and XUV pulses from Free Electron Lasers (FEL)8,9,10,11,12,13, synchrotron radiation14,15,16,17,18, and table-top femtosecond laser pulses in the near IR region19,20,21,22,23,24. The two laser ionization processes involve very different physics, with FELs ionizing directly or in the multiphoton regimes, according to the Keldysh parameter25 and giving rise to the inner shell and Auger decay ionization, while tabletop IR lasers ionize in the strong field tunneling regime, ionizing valence electrons.

Femtosecond laser-triggered CEI has been used to image the structure and dynamic evolution of molecules and clusters of diatomic26,27, triatomic21,22,23,28,29,30,31,32,33,34, and polyatomic molecules35,36,37,38,39. It has also been proven to be capable of imaging statistically occurring signals of dissociation and roaming dynamics40,41,42. A persistent challenge in CEI involves establishing a connection between measured momentum values and the precise geometric structure of molecules at the initial stage in the ground state. This arises in both inner shell and valence ionization because charge changes and migrates on timescales which are of the order of the nuclear motion of the molecule, and so significant geometric change can have occurred during the stepwise ionization process before the Coulomb explosion is complete. To address this challenge, it is helpful to distinguish forward and inversion approaches. Firstly, in an inversion approach, the aim is to deduce the molecule’s structure, by comparing experimental momentum measured from a coincidence event with the momentum resulting from a simulation of the Coulomb explosion process for a randomly chosen geometry32,43,44, with no preconception of the correct geometry is used before comparison between simulation result and experiment are compared, and geometries are generated repeatedly until an agreement is found. This approach lacks the possibility of providing information about how the molecule dynamically interacts in the presence of the laser electric field (E field) inducing the Coulomb explosion. In a previous study32, we followed the Coulomb explosion of CO2 up to 6+ and reconstructed the geometry of the molecule right before dissociation using a simplex reconstruction method44, by comparing the final experimental momenta with those resulting from the Coulomb explosion of guessed initial geometries assuming a simple point-charge ion model, and instantaneous multiple ionization to the final Coulombic state. In this inversion process, once agreement is found between experimental momentum and the momentum of the guessed structure, the geometry of the molecule right before dissociation is considered to have been found. The inversion process is an ill-defined problem, but results have shown that, for the shortest pulses available, there is some quantitative agreement between the inversion results and the known geometry distributions at least in terms of average bond length and average bend angle, but distributions have been typically substantially wider than expected from equilibrium geometry distributions4,45. This is explained by molecular dynamics on intermediate ionic states as demonstrated for D2 when doubly ionized with few-cycle pulses24.

A second powerful approach is to deduce the initial spatial nuclear distribution, by modeling the molecular wave function first and then performing a “forward” simulation of the multiple ionization pathway and comparing the simulated momentum distribution with the experimental measurement, in order to judge the accuracy of the geometry determination. In the forward approach, we can simulate the molecule’s evolution in the presence of the E field for high-field experiments, or during charge redistribution for XUV-initiated ionization. This allows us to obtain comprehensive information about the molecule’s journey from the ground state to its final ionization state and the progression of the fragments to the detector, where we measure the momentum vectors. We can then compare these simulated results with experimental data, bridging the gaps left by the inversion approach and gaining insights into previously unexplored aspects of the process.

Recently this forward approach has shown promise when it comes to revealing quantitative properties of molecules significantly larger than triatomic, either simulating the physics of XUV multiple ionization, when coupled with experimental results from an FEL11 or modeling the influence of vibrational zero-point motion probed by a tabletop laser12. In these calculations, a ground state initial geometry or distribution is simulated and allowed to either ionize stepwise, during a simulated laser pulse, or in a single step. The final momentum vectors of the simulated fragments are then correlated in a correlated momentum plot to compare with the momentum patterns created by the experimentally generated ionic fragments. Only partial correlation between the experimental fragments has been necessary, in order to observe a mapping with the simulation. Similarly, a forward process can give qualitative agreement about momentum angular distributions from IR-initiated CEI for large molecules from a simple single-step multiple ionization simulation11. Here we employ another forward simulation approach, to model the stepwise ionization process resulting in OCS6+ in the (2,2,2) channel, due to ionization by 7 fs laser pulses at 800 nm, in an effort to precisely map the ground state vibrational bend distribution and equilibrium bond lengths, on to the final pattern of experimentally observed momentum and energy release. We select OCS because of its asymmetric bond lengths, and its cation has been the subject of precise imaging studies using laser-induced electron diffraction (LIED)46.

We attempt to model the stepwise ionization process which happens in the laser pulse; the calculation is computationally expensive and so rather than using a Monte Carlo approach to sample the ground state geometries randomly, we choose six initial bend geometries in 5-degree steps from 155° to 180°. These geometries significantly sample the ground state bend distribution and so by using time-dependent density functional theory (TDDFT)47 to follow the ionization and dissociation process in the laser pulse, we can predict the final measured momentum, and then weight the result according to the initial geometry probability. Because the ionization and dissociation processes also depend on the angle between the laser polarization and the molecular axis, we also sample eight different angles between the laser polarization and the molecular axis (α) for each geometry. The TDDFT approach has been a successful method for studying ultrafast and strong-field processes in gas phase systems, such as strong-field ionization32,48, linear and nonlinear spectroscopies49,50,51, and charge migration52,53. It can reproduce and describe experimental observations, including the photoelectron-momentum distribution from strongly aligned molecules48, multi-orbital effects observed in ionization and dissociation of polar molecules54, angle-dependent strong-field ionization of molecules55,56, resonances in the tunneling spectra of different charge states of molecules50, and time-resolved X-ray absorption spectroscopy of the dissociation reactions of molecular cations57.

We use TDDFT to describe the many-body system and its dynamics during the stepwise ionization process in the presence of the laser field, incorporating fully quantum charge exchange; once the laser pulse is complete, we allow the ions to behave as classical particles as they move to their asymptotic momentum, where we compare their trajectories with our experimental results. Previously, TDDFT was used for example, to give insight into the presence of Charge Resonance Enhanced Ionization (CREI) for a CO2 molecule under controlled conditions32, and to distinguish the polarization dependence of ionization in acetylene35 for 266 nm pulses; but here we use it to simulate many experiments each taking place with different starting points within the molecular ground state bend distribution to show that TDDFT also offers the possibility of mapping molecular break-up in a few-cycle laser pulse with detailed accuracy.

Methods

Experimental technique

CEI was carried out at the ALLS (advanced Laser Light Source) laboratory, using a momentum imaging apparatus58 consisting of a time and position-sensitive detection system incorporating a delay line PSD (Roentdek). Pulses from a kHz, 35 fs Ti:Sapphire laser, centered at 800 nm (Red Dragon, KMLabs), were compressed to 7 fs, using self-phase modulation (SPM) in a hollow-core fiber filled with Argon. A half-wave plate combined with two germanium plates (at Brewster angle) was used to control the energy that entered the CEI apparatus. A supersonic gas jet of OCS molecules, with 99.99% of molecules estimated to be in the ground vibrational state, was intersected by the focus of a 10 cm parabolic mirror, generating intensity of the order of 1015 W cm−2 at its center. The ionic fragments produced were accelerated to the detector by a uniform electric field. The momentum imaging apparatus is triggered by the fs laser pulse and so by measuring the time of flight (TOF) and the positions of impact for each fragment ion on the PSD, we can obtain its initial momentum vector. By detecting each ionic fragment produced, from one laser shot, in coincidence and using conservation of momentum, the momentum imaging apparatus can record the three positive ions produced during a Coulomb explosion event.

Theoretical method

The laser-molecule interaction has been simulated within the framework of TDDFT by employing the OCTOPUS package59. Based on this approach, the time-dependent Kohn–Sham (TDKS) orbital \({\psi }_{i}({{\bf{r}}},t)\) is obtained using the TDKS equation as follows:

$$i\frac{\partial }{\partial t}{\psi }_{i}({{\bf{r}}},t)=\left[-\frac{1}{2}{\nabla }^{2}+{{{\rm{V}}}}_{{{\rm{H}}}}\left({{\bf{r}}},t\right)+{{{\rm{V}}}}_{{{\rm{xc}}}}\left({{\bf{r}}},t\right)+{{{\rm{V}}}}_{{{\rm{ext}}}}({{\bf{r}}},{{\bf{R}}},t)\right]{\psi }_{i}({{\bf{r}}},t).$$
(1)

In Eq. 1, VH is the Hartree potential, Vxc is the exchange-correlation potential, and Vext is the external potential. Additionally, r and R = {\({{{{\bf{R}}}}_{1},\ldots ,{{\bf{R}}}}_{{N}_{i}}\)} represent the position vectors of electrons and ions, and Ni is the number of ions. The external potential can be written as:

$${{{\rm{V}}}}_{{{\rm{ext}}}}({{\bf{r}}},{{\bf{R}}},t)={{{\rm{V}}}}_{{{\rm{ne}}}}({{\bf{r}}},{{\bf{R}}},t)-{{\bf{E}}}(t).{{\bf{r}}},$$
(2)

where Vne is the Coulomb potential between the set of ions and the electron, and E is the electric field of the laser. Norm-conserving pseudopotentials for the interactions between ions and valence electrons (Vne), and generalized gradient approximation (GGA) for the exchange-correlation potential (Vxc) based on a Perdew–Burke–Ernzerhof functional60 have been applied. The motion of nuclei has been treated by the Ehrenfest theorem61 under the influence of the ion-electron potential, the internuclear Coulomb repulsion, and the interaction of the nuclei with the external laser field, as follows:

$${M}_{{{\rm{j}}}}{\ddot{{{\bf{R}}}}}_{{{\rm{j}}}}=-\int n\left({{\bf{r}}},t\right)\frac{\partial {{{\rm{V}}}}_{{{\rm{ne}}}}}{\partial {{{\bf{R}}}}_{{{\rm{j}}}}}{{\rm{dr}}}-\frac{\partial {{{\rm{V}}}}_{{{\rm{nn}}}}}{\partial {{{\bf{R}}}}_{{{\rm{j}}}}}+{Z}_{{{\rm{j}}}}{{\bf{E}}}\left(t\right),$$
(3)

where Vnn is the Coulomb repulsion between the nuclei, and Mj and Zj are the mass and core charge of the nuclei, respectively. The time-dependent electronic density n(r, t) for a spin-compensated system containing Ne electrons is given by:

$$n\left({{\bf{r}}},t\right)=2\mathop{\sum }_{i=1}^{\frac{{N}_{{{\rm{e}}}}}{2}}{\left|{\psi }_{i}\left({{\bf{r}}},t\right)\right|}^{2}.$$
(4)

The time-dependent total charge of the molecule is calculated by:

$${q}_{{{\rm{tot}}}}\left(t\right)={N}_{0}-\int n\left({{\bf{r}}},t\right){{\rm{dr}}},$$
(5)

where N0 is the total number of electrons at time zero.

The calculations have been performed in a three-dimensional real-space sphere with a radius of 32 Å and spacing of 0.16 Å. The propagation of Kohn–Sham orbitals in time has been done with a time step of 0.002 fs. To prevent reflections of the electrons from the boundary of the simulation box, the Kohn–Sham orbitals are multiplied with a mask function which is unity in the inner simulation region and is gradually switched to zero at the borders. Since the ionic fragments can be treated as point-charge particles after the dissociation is complete, we have followed the TDDFT calculation at close range by a classical calculation to obtain the asymptotic momentum of all ions. To initiate the classical calculation following the TDDFT simulation, we record the position and velocity of ions at a specific time, 8 fs after the pulse concludes, which corresponds to a total simulation time of 27.25 fs in the TDDFT framework. This chosen timeframe ensures that the ions have sufficiently separated from each other, allowing for distinct orbital representation. This temporal snapshot is selected to capture a moment when the effects on each ion are minimal. Each simulation is computationally demanding and requires 1200 h of processor time, so calculations were carried out on a multiple-core cluster of 60 cores, allowing us to perform each calculation in 20 h.

Results and discussion

Figure 1a shows the laboratory frame component of momentum plotted for oxygen, carbon, and sulfur fragments detected in coincidence, projected on the detector plane, with laser polarization parallel to the detector. The data is arranged so that O2+, C2+, and S2+ lie on the left, in the center, and on the right side, respectively, which starts to indicate the shape of the molecules detected, which are largely orientated along the laser polarization direction. Most of the momentum is contained in the O2+ and S2+ fragments which are at the ends of the molecule and experience repulsion from both of the other two atomic fragments, while the C2+ fragments are distributed in the middle of the plot indicating they have only small momentum, because they originate close to the middle of the molecule and are repelled similarly from both end fragments. The momentum of C2+ is mainly perpendicular to the polarization direction indicating a small bend, because OCS is a linear molecule with geometry close to, but never exactly straight, even in its ground state. This occurs because each of the three vibrational modes (bending symmetric and asymmetric stretching) all have a half quantum of energy in their ground states, known as zero-point energy. The consequence of the vibrational motion is that in any symmetry plane passing through the molecular axis, the molecule must be seen to be vibrating, a phenomenon which can only occur if the vibrational motion is split into two orthogonal modes out of phase by 90°, making the molecule rotate about its axis with a small angle, and giving zero possibility of observing a bend angle of 180°4,62,63.

Fig. 1: Analysis of fragment momentum and kinetic energy release in the concerted break-up of the OCS molecule.
figure 1

a The laboratory frame component of momentum plot for the O2+, C2+, and S2+ fragments detected in coincidence, projected on the detector plane, with laser polarization parallel to the detector, and data arranged so that O2+, C2+, and S2+ lie on the left, in the center, and on the right side, respectively. For the O2+ momentum, we combined all positive and negative components by rotating the positive momentum to align with the negative and correcting for any tilt relative to the polarization angle. b Dalitz plot, which illustrates the concerted break-up and bending of the molecule before dissociation. c The angle distribution between the momenta of O2+ and S2+, as shown in the inset, is taken from the experiment (\({\theta }_{v}\)) (solid line) and compared with the point-charge Coulomb Explosion (CE) model (dashed line). d Distribution of Kinetic Energy Release (KER) for O2+ (dash-dotted line), C2+ (dashed line), and S2+ (dotted line), and total kinetic energy release (TKER) (solid line), comparison with theoretical CE model (dashed vertical line).

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Figure 1b shows a Dalitz plot64 for the (2,2,2) channel of OCS6+. A Dalitz plot is a two-dimensional scatter plot used to study the distribution of energies among three decay products of a mother particle. The plot is highly sensitive to the initial geometry of the molecule and the steps that occur during the fragmentation process. In a Dalitz plot, a central oval signifies concerted processes, with an increase along the y-axis indicating bending, while diagonal lines forming an “X” shape indicate sequential processes6,14,31,65. Intuitively, the fact that no diagonal lines are present and the data forms a single island indicates that only concerted break-up occurs, with both bonds extending together, while the concentration close to −0.3 (C2+ energy close to zero) indicates the predominance of near linear geometries and the limited vertical spread indicates the extent of bending during the ionization process65,66. If significant sequential processes existed along with the concerted process, they could not be used to reconstruct initial geometry without adding significantly to the theoretical treatment. If such processes needed to be distinguished, however, they could be, using the Native Frames67 approach. Therefore, all break-up events observed in this study are attributed to concerted break-up. Nevertheless, we will examine how, even on the timescale of tens of femtoseconds, processes might occur at different times and on different timescales, in order to build up a complete picture of the molecular break-up.

Figure 1c shows the distribution of the angle θV between the momentum vectors of O2+ and S2+ which peaked at 160° and extends from 120° to 180°. The distribution is significantly wider than the Coulomb explosion (CE) simulation which represents instantaneous ionization from the ground state bend distribution, indicating the significance of molecular deformation during the stepwise ionization process4,45,62. Additionally, the simulated distribution is shifted towards larger angles compared to the experimental data. This is also evident in the lower experimental total kinetic energy release (TKER) of Fig. 1d, compared to the Coulomb explosion calculation, assuming instantaneous ionization to OCS6+, and a point-like particle behavior from the most probable equilibrium position of the neutral molecule.

Before we tackle the full distribution of initial molecular geometries, we want to first follow the ionization and dissociation of one model OCS molecule, in order to understand qualitatively and quantitatively, how the Coulomb explosion progresses. For this example, we assume the most probable initial geometry RCO (t = 0) = 1.16 Å, RCS (t = 0) = 1.56 Å, and θOCS (t = 0) = 172°68,69. Both the CO and CS bonds are assumed to be bent equally from the molecular axis in the linear geometry (the gray dotted arrow depicted in the inset of Fig. 2a). The full width at half maximum of the laser pulse is fixed at 7 fs, and the peak intensity and polarization angle of the laser pulse are assumed to be 3.2 PW cm−2 and α = 40° (see Fig. 2a), respectively. Intensity is the only free parameter in the calculation as a range of peak intensities are available within the laser focus, and the intensity is adjusted to maximize the ionization yield to the 6+ charge state in (2,2,2) channel; the intensity value that was found to vary from 2.8 × 1015 W cm−2 to 5 × 1015 W cm−2 as the angle between the laser polarization and the molecular axis varied from 0° to 90°. We have concentrated on the highest charge state available, as it should most closely resemble a purely Coulombic interaction, and allow us to compare our powerful theoretical approach with a simple Coulombic model, and thus reveal how the details of the stepwise ionization processes lead to the current experimental results. The laser polarization angle α is always measured with respect to the gray dotted arrow shown in Fig. 2a.

Fig. 2: Bond length dynamics and charge state of the OCS molecule under laser interaction.
figure 2

a Theoretically calculated inter-nuclear distances as a function of time for the most probable initial geometry (Rco(t = 0) = 1.16 Å, and Rcs(t = 0) = 1.56 Å, θOCS(t = 0) = 172°, and α = 40°). The blue dashed line (red dash-dotted line) represents the evolution of RCO (RCS) in the laser-induced ionization (LII) approach. The green solid line shows the total net charge of the molecule over time. The cyan dotted line (magenta large dotted line) illustrates the evolution of RCO (RCS) in the field-free stepwise ionization (FFSI) approach. The inset shows an OCS molecule with oxygen (red circle), carbon (black circle), and sulfur (gold circle) connected by black lines, along with the bond angle (θOCS) and the E field angle (α) relative to the x-axis (gray dotted arrow). b The variation of the net charge for each atom over time: oxygen (red dashed line with circles), carbon (black dotted line with squares), and sulfur (gold dash-dotted line with triangles). The pink curve represents the driving laser pulse. t1 (t2) denotes the moment when the CO (CS) bond length starts increasing.

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Figure 2a shows the simulated variation of internuclear distances (RCO and RCS) in time over the course of the laser pulse and after, along with the electric field (pink solid line in Fig. 2b) and the total charge (green solid line in Fig. 2a). In addition to the principle “laser-induced ionization” (LII) model in which we incorporate the full effect of the laser field on ion dynamics, we have included a “field-free stepwise ionization” (FFSI) approach, in which we turn off the field between ionization events for comparison. In our LII approach, we combine TDDFT and classical treatment, and the ionization is induced by the laser field. Ionization of the molecule occurs when the laser field reaches its peak intensity during each half cycle. Between these ionization events, the ions are influenced by the laser field, affecting the state of the molecule at the next ionization step. In the FFSI approach, the ground state orbitals of the molecule are propagated in time without applying any external field. At specific times, corresponding to the ionization times recorded from the LII approach as indicated by jumps in the total charge (green solid line in Fig. 2a), the ground states of the molecule are switched to ionized states and then propagated further in time. This procedure is repeated until the molecule reaches the OCS6+ states. At each step, the coordinates and the velocity of the ions are recorded at the end of propagation and used as the initial conditions for the next step. The effect of the laser pulse on the dynamics of the molecule in the Coulomb explosion process becomes visible by comparing LII (blue dashed and red dash-dotted lines) and FFSI (cyan and magenta dotted lines) approaches as shown in Fig. 2a. As can be seen, the bond length variations deviate from that obtained by the LII approach, particularly for RCO, which expands more rapidly in the LII approach, emphasizing the importance of considering the laser field-induced effects on the ion dynamics during dissociative ionization in the tunneling regime. In Supplementary Movies 1–5, animations of the electron and orbital distribution behavior in the field are provided. The time variation of the total net charge of the molecule (green solid curve), displayed in Fig. 2a, shows that the molecule first ionizes to OCS+, and then converts to OCS3+ in a rapid step and finally progressively reaches 6+ in several steps. It is important to note that each step in this curve does not correspond to the removal of a single electron. This is a feature of TDDFT, which treats electrons as wavefunctions; while the electrons are localized, they are not treated as discrete quantities. Bond length variation shows that the stretching of the CO bond starts at t1 = 4.5 fs, earlier than for the CS bond which begins at t2 = 8 fs. From 14 fs on, the CO bond is longer than the CS bond. Figure 2b shows the instantaneous E field along with the charge state of each individual ion as a function of time; the charge oscillates between the oxygen and sulfur ions, out of phase by 180° with the E field, as the overall charge increases due to tunnel ionization. The oscillation of ion charges observed in Fig. 2b is attributed to the oscillation of the laser field. When the laser field reaches its maximum in the positive direction (toward sulfur), the valence electrons shift in the opposite direction (toward oxygen). This shift reduces the electron density around sulfur, increasing its net charge, while the increased electron density around oxygen decreases its net charge. Consequently, the net charge of sulfur follows the direction of the laser field, while the oxygen charge oscillates in the opposite direction. This behavior depends on the orientation of the molecule in the defined coordinate system. If the positions of sulfur and oxygen were exchanged (rotated by 180°), the oscillation of the oxygen charge would follow the laser field, and the sulfur charge would oscillate in the opposite direction. Additionally, the carbon charge always follows the oxygen charge due to its proximity to oxygen. The charge state of the ions approaches 2+ as the system evolves over time and the ions move farther apart. The simulation can follow the electronic and nuclear motion on a sub-cycle level, as the six electrons are removed by the few-cycle laser pulse.

In Fig. 3, the ionization rate for parallel (α = 0°) and perpendicular (α = 90°) configurations and the role of different orbitals of the OCS molecule in the ionization process are investigated. For each R/Req value on the x-axis, a simulation is performed to determine the total ionization; and the ionization probability per optical cycle is calculated by dividing the total ionization by the number of laser cycles. As can be seen from Fig. 3a, the ionization rate when the laser polarization is parallel to the OCS molecular axis (horizontal) is significantly larger than in the perpendicular case, and the ionization probability of the parallel configuration is further enhanced from R = 1.3 Req (RCO = 1.3 RCO−eq, RCS = 1.3 RCS−eq), with Req the equilibrium bond length used in Fig. 2 as the initial bond length at t = 0. Figure 3b, c show energy gaps and transition moments as a function of internuclear distances, respectively. These figures indicate that the primary contributors to the ionization process are the parallel transitions 3σ → 4σ and 1π → 2π. For these transitions, the energy gaps decrease, and the transition moments increase, both of which can lead to an enhancement in the ionization rate. These transitions would manifest as CREI in the event that the molecule expands up to R/Req = 1.6. Both dominant transitions (3σ → 4σ and 1π → 2π) are parallel transitions and result in charge localization, while the other transitions correspond to π-type orbitals that become localized around different atoms as the internuclear distances increase. However, this expansion is restrained due to the short pulse duration. If, hypothetically, the pulse duration was extended to 35 fs as observed for CO232, CREI would emerge as the dominant ionization mechanism. Figure 3d shows the orbitals associated with the most dominant transition (1π to 2π) at R = Req and R = 1.6 Req. As can be seen, by increasing the bond lengths, both π orbitals transition into p orbitals that are primarily concentrated around the oxygen atom. This transformation leads to an enhancement in the dipole transition. The vertical dotted lines indicate the positions of the ions.

Fig. 3: Analysis of ionization dynamics and orbital transitions in OCS3+.
figure 3

a Ionization probability per optical cycle as a function of internuclear distances for OCS3+ and as a function of laser polarization. A six-cycle laser pulse with 800 nm central wavelength and an intensity of 0.4 PW cm−2 has been used. Both RCO and RCS are altered simultaneously. b Energy gaps and c transition moments as a function of internuclear distances. The considered orbitals of OCS3+ are as follows: 3σ, 1π, 2π (HOMO), 3π (LUMO), and 4σ. \({R}_{{eq}}\) is the equilibrium internuclear distance of each bond as used for the initial condition in Fig. 2. d The orbitals corresponding to the most dominant transition (1π to 2π) at R = Req and R = 1.6 Req are shown. Each orbital is normalized to its maximum value, and the vertical dotted lines indicate the positions of the ions from left to right: oxygen, carbon, and sulfur, respectively.

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Given the polarization dependence observed in Fig. 3, we investigate its influence on different trajectories next. Figure 4a, b shows simulated trajectories considering different initial molecular bend angles and orientations with respect to the laser polarization. In the trajectories presented, it is observed that the OC bond begins stretching between t = 4.5–5 fs (as depicted in the inset of Fig. 4a) across all trajectories. On the other hand, the stretching of the CS bond occurs between t = 3.5–8.5 fs, depending on the angle of laser polarization (as shown in the inset of Fig. 4b). Trajectories with initial laser polarization angles close to zero (α = 0°) exhibit a faster and earlier stretching of the CS bond. Trajectories with larger polarization orientations (larger α) exhibit more oscillations of RCS at low t followed by a monotonic increase from around t = 9 fs. The lines of the same color represent identical initial ground state conditions, but differ in laser orientation. The dependence of the trajectories and bending on the initial laser polarization direction relative to the molecular axis are shown explicitly in Supplementary Figs. 1–3. The oscillations are induced in the ionized molecules before their bonds break, and are coherent between the trajectories, because the electric field envelope used in the simulations is the same and therefore CEP (carrier-envelope phase) stable across the simulations. Depending on the initial conditions, each bond breaks at different moments. Bonds that break later experience more oscillations. In most cases, the CO bond breaks earlier, resulting in fewer oscillations for the CO bond compared to the CS bond. Additionally, Fig. 4c plots the temporal evolution of the OCS angles for different initial conditions. Molecules with a larger angle relative to the laser polarization direction at the onset of the interaction undergo more significant bending after the interaction. We employ a “simulated probability density” color bar as a visual indicator to represent both the weighting based on probability densities, where lighter colors indicate lower probabilities and darker colors represent higher probabilities, and color-labeling corresponding to the ground state molecular geometry shown in Fig. 4d. This figure depicts points on the bend distribution of the ground state geometry used as in the starting point of the simulation process, along with their color-coded probability density. To find this distribution we use the approach of Siegmann et al.62, which gives us a result close to previously calculated for OCS4.

Fig. 4: Time evolution of inter-nuclear distances and bond angles for OCS with different initial conditions.
figure 4

Theoretical calculations were performed to determine the inter-nuclear distances over time for a molecule, considering its initial angle ranging from 155° to 180° in 5-degree steps. These calculations were carried out for various polarization orientations, ranging from 0° to 70° in 10-degree steps, which results in groups of lines of the same color, taking into account the probability derived from the simulated probability density of the molecule’s ground state bend distribution, represented in (d), and coded in different colors here according to the color bar. The inter-nuclear distances for both a RCO and b RCS, depicted in Fig. 2a, as well as the c temporal evolution of molecular angles θOCS, were analyzed. Detailed views of the short-term inter-nuclear distance evolution are provided in the insets of (a, b). d Points on the bend distribution of the ground state geometry are used as the starting point of the simulation process, along with their color-coded probability density. The dashed gray line represents the tail of the bending distribution of the ground state below 152.5°, which constitutes less than 0.1% of the possible bend population.

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Finally, to represent the comparison between experiment and theory, we show in Fig. 5a the TKER, and the kinetic energy release (KER) by each fragment, calculated in the simulation, once again with contributions from initial geometries weighted according to ground state probability density. A small bump in the experimental energy distribution above the instantaneous CE level, could be the result of excited states not included in the simulation process, such as through recollision. Also, we can compare the simulated final momenta with those measured experimentally. We do this using a correlated momentum plot called a Newton plot31, which, in this case, gives an intuitive sense of the geometry of the molecule by representing each atomic ion as a distinct island in momentum space. Figure 5b shows the experimental result normalized to the momentum of the O2+ ion, as a reference, its magnitude is one and it has momentum in the y-axis. Any three measured momentum points for O2+, C2+, and S2+ add to zero, so the extent of bending is apparent, with points on the S2+ and C2+ islands furthest from zero correlated with each other and those closest likewise. In a Newton plot, well-defined islands represent concerted processes, and in this case, the common half-ring structures that correspond to sequential processes are not present6,31,65,67,70.

Fig. 5: Fragment Kinetic Energy Release and Newton plot analysis of OCS6+ from experimental results: insights from TDDFT calculations.
figure 5

a The TDDFT calculation of the fragments’ KER and TKER (black solid lines) compared to experimental results (colored solid lines: blue for oxygen, orange for carbon, green for sulfur, and red for TKER). b Normalized Newton plot of OCS6+ for experimental data, relative to the O2+ ion, as a reference. c, d Magnified view of experimental S2+ and C2+ Newton plot islands (with green scale in the background), respectively, compared to TDDFT results (represented by colored data points according to probability density). Insets represent the normalized counts of experimental (EXP) and TDDFT points (SIM) within the mean experimental momentum (MEM) cut of the Newton plot islands.

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In Fig. 5c, d, we overlay these experimental results (green scale) with the results of the selected simulation data points (colored data points) by weighting and color-labeling them according to the probability density of the initial geometry from which they originate, as shown in Fig. 4d. The agreement between the simulation and experiment in Fig. 5c, d is excellent and the experimental results are almost completely obscured by the simulation. To show the high degree of agreement more clearly, we introduce the Mean Experimental Momentum (MEM), derived from the mean momentum of the experimental dataset, binned at intervals of 0.1 (arb. units) along the momentum axis, which is represented by the cyan dotted line. In the insets of Fig. 5c, d, we depict the normalized counts (arb. units) of both experimental and theoretical data points along this MEM cut. All theoretical data regarding polarization orientations (α) relative to the molecular axis were taken into account in these insets. The excellent agreement further emphasizes the similarity between theoretical and experimental results and the rich information provided by the method presented here.

Conclusions

In conclusion, the simulation of multiple ionization of a small polyatomic molecule in an evolving laser field using TDDFT allows us to accurately predict the fragment ion momentum distribution and energy release, from information about the real-space initial geometric distribution of the molecule, effectively allowing us to image its ground state bending distribution. By modeling the effects of the electric field on the break-up process, we have been able to accurately predict the main features of the experimental results with detailed accuracy, while revealing details of the dynamics such as oscillations in the relative bond expansion not possible to infer previously. The approach is not limited to triatomic molecules such as OCS and will be expanded further in the future to investigate larger polyatomic molecules. Because the method is a forward approach, it can be used to compare with experimental results in the tunneling regime, not limited to complete coincidence, in which every break-up fragment is detected, but can be used with incomplete coincidence maps11 or covariance maps12,71. The approach is not limited to imaging a ground state geometry, it could potentially be coupled with an excited state wave function created in a pump-probe experiment, to generate a real-space molecular movie with detailed information and high accuracy, this would be a progression of the method used in acetylene35 where the proton migration was modeled using the full multiple spawning approach72 and compared to experimental populations estimated from Newton plots, the measured Newton plots could instead be directly compared with those generated from the most relevant subset of the simulation results. This along with the development of optimization parameters will allow the approach to be scaled effectively to handle evolving molecular geometries. Although TDDFT calculations are more computationally expensive than other methods; by optimizing simulation parameters and utilizing high-performance computing clusters we can significantly reduce the simulation time. This will allow TDDFT to be applied to larger molecules with lower symmetry, where more possible structures need to be considered, or for a Monte Carlo approach to be used with smaller molecules.