Diffractive Imaging of Coherent Nuclear Motion in Isolated Molecules

Structural Study of Aqueous Electrolyte Solution by MeV Liquid Electron Scattering

The impact of ions on water has long been a subject of great interest, as it is closely tied to the hydration structure, dynamics, and properties of electrolyte solutions. Over centuries of investigation, the influence of ions on water's structure remains highly debated. Prevailing techniques, such as neutron and X-ray scattering, primarily focus on the microscopic structure of salt solutions at very high concentrations, mostly above 1 mol/L. In this study, we measured the structure of aqueous potassium iodide (KI) and potassium chloride (KCl) solutions using MeV liquid electron scattering (MeV-LES) across a concentration range of 0.10 to 0.75 mol/L. The obtained results provide detailed insights into the variations in ion-oxygen and oxygen-oxygen correlations as a function of concentration. The observed structural differences between KI and KCl solutions are in line with the structure maker/breaker theory, which suggests that iodide ions exert a more pronounced effect than chloride ions on disrupting the water shell. This work demonstrates the potency of MeV-LES for investigating the atomic structure in liquids, augmenting the modern analytical toolbox.

Pushing the limits of ultrafast diffraction: Imaging quantum coherences in isolated molecules

Quantum coherence governs the outcome and efficiency of photochemical reactions and ultrafast molecular dynamics. Recent ultrafast gas-phase X-ray scattering and electron diffraction have enabled the observation of femtosecond nuclear dynamics driven by vibrational coherence. However, probing attosecond electron dynamics and coupled electron-nuclear dynamics remains challenging. This article discusses advances in ultrafast X-ray scattering and electron diffraction, highlighting their potential to resolve attosecond charge migration and vibronic coupling at conical intersections. Novel techniques, such as X-ray scattering with orbital angular momentum beams and combined X-ray and electron diffraction, promise to selectively probe coherence contributions and visualize charge migration in real-space. These emerging methods could further our understanding of coherence effects in chemical reactions.

The photodissociation dynamics and ultrafast electron diffraction image of cyclobutanone from the surface hopping dynamics simulation

The comprehension of nonadiabatic dynamics in polyatomic systems relies heavily on the simultaneous advancements in theoretical and experimental domains. The gas-phase ultrafast electron diffraction (UED) technique has attracted significant attention as a unique tool for monitoring photochemical and photophysical processes at the all-atomic level with high temporal and spatial resolutions. In this work, we simulate the UED spectra of cyclobutanone using the trajectory surface hopping method at the extended multi-state complete active space second order perturbation theory (XMS-CASPT2) level and thereby predict the results of the upcoming UED experiments in the Stanford Linear Accelerator Laboratory. The simulated results demonstrate that a few pathways, including the C2 and C3 dissociation channels, as well as the ring opening channel, play important roles in the nonadiabatic reactions of cyclobutanone. We demonstrate that the simulated UED signal can be directly interpreted in terms of atomic motions, which provides a unique way of monitoring the evolution of the molecular structure in real time. Our work not only provides numerical data that help to determine the accuracy of the well-known surface hopping dynamics at the high XMS-CASPT2 electronic-structure level but also facilitates the understanding of the microscopic mechanisms of the photoinduced reactions in cyclobutanone.

A comparative review of time-resolved x-ray and electron scattering to probe structural dynamics

The structure of molecules, particularly the dynamic changes in structure, plays an essential role in understanding physical and chemical phenomena. Time-resolved (TR) scattering techniques serve as crucial experimental tools for studying structural dynamics, offering direct sensitivity to molecular structures through scattering signals. Over the past decade, the advent of x-ray free-electron lasers (XFELs) and mega-electron-volt ultrafast electron diffraction (MeV-UED) facilities has ushered TR scattering experiments into a new era, garnering significant attention. In this review, we delve into the basic principles of TR scattering experiments, especially focusing on those that employ x-rays and electrons. We highlight the variations in experimental conditions when employing x-rays vs electrons and discuss their complementarity. Additionally, cutting-edge XFELs and MeV-UED facilities for TR x-ray and electron scattering experiments and the experiments performed at those facilities are reviewed. As new facilities are constructed and existing ones undergo upgrades, the landscape for TR x-ray and electron scattering experiments is poised for further expansion. Through this review, we aim to facilitate the effective utilization of these emerging opportunities, assisting researchers in delving deeper into the intricate dynamics of molecular structures.

Electron scattering beyond the independent atom model: Quantum fluctuation of the Coulomb potential

In the past century, electron scattering has mostly served as a powerful tool to measure the microscopic structure of gases, liquids, and solids in either a static or time-resolved manner. One common basis for these works is the independent atom model, which directly relates electron scattering signals to the atomic structure of matter. In this perspective, we explore the information content of electron scattering that goes beyond the independent atom model. We show that the small-angle limit of the electron scattering signal encodes the quantum mechanical fluctuation of the long-range Coulomb potential. This quantum fluctuation, described by the second moment of the dipole operator, is the root cause of the intermolecular van der Waals forces.

Prediction challenge: First principles simulation of the ultrafast electron diffraction spectrum of cyclobutanone

Computer simulation has long been an essential partner of ultrafast experiments, allowing the assignment of microscopic mechanistic detail to low-dimensional spectroscopic data. However, the ability of theory to make a priori predictions of ultrafast experimental results is relatively untested. Herein, as a part of a community challenge, we attempt to predict the signal of an upcoming ultrafast photochemical experiment using state-of-the-art theory in the context of preexisting experimental data. Specifically, we employ ab initio Ehrenfest with collapse to a block mixed quantum-classical simulations to describe the real-time evolution of the electrons and nuclei of cyclobutanone following excitation to the 3s Rydberg state. The gas-phase ultrafast electron diffraction (GUED) signal is simulated for direct comparison to an upcoming experiment at the Stanford Linear Accelerator Laboratory. Following initial ring-opening, dissociation via two distinct channels is observed: the C3 dissociation channel, producing cyclopropane and CO, and the C2 channel, producing CH2CO and C2H4. Direct calculations of the GUED signal indicate how the ring-opened intermediate, the C2 products, and the C3 products can be discriminated in the GUED signal. We also report an a priori analysis of anticipated errors in our predictions: without knowledge of the experimental result, which features of the spectrum do we feel confident we have predicted correctly, and which might we have wrong?

Tracking Molecular Fragmentation in Electron-Ionization Mass Spectrometry with Ultrafast Time Resolution

ConspectusMass spectrometry is a powerful analytical method capable of identifying compounds given a minute amount of material. The fragmentation pattern that results following molecular activation serves as a fingerprint that can be matched to a database compound for identification. Over the past half century, studies have addressed and, in many cases, named the chemical reactions that lead to some of the principal fragment ions. Theories have been developed to predict the observed fragmentation patterns, many of which assume that energy redistributes prior to dissociation. However, the existence of rearrangements and nonergodic processes complicates the prediction of fragmentation patterns and the identification of compounds that have yet to be entered into a curated database. To date, very few studies have addressed the time-dependent nature of the fragmentation of radical cations and, in particular, processes occurring with picosecond or shorter time scales where one expects to find nonergodic reactions.This Account focuses on a novel approach that enables tracking of molecular fragmentation in electron-ionization mass spectrometry with ultrafast time resolution. The two challenges that have prevented the time-resolved studies following electron ionization are the random impact parameter and moment of ionization of each molecule. In addition, medium-sized molecules can produce fragmentation patterns with tens if not hundreds of product ions. Spectroscopically interrogating all of these ions as a function of time is another major challenge. We describe strong field disruptive probing, a method that ionizes molecules on a femtosecond time scale and allows us to track in time the formation of all fragment ions simultaneously.Molecular fragmentation following ionization can occur on a very wide range of time scales. Metastable ions can survive from nanoseconds to microseconds; reactions that depend on vibrational energy redistribution can take picoseconds to nanoseconds; and direct dissociation processes and some rearrangements can take place in femtoseconds to picoseconds. All of these processes depend on the dynamics that occur during attoseconds and femtoseconds following the ionization process. Following a discussion of these time scales, we provide three examples of fragmentations that have been studied with femtosecond time resolution. Each of these examples include unforeseen reaction dynamics that involve a nonergodic process, highlighting the importance of time resolution in mass spectrometry. Finally, we explore future challenges and unresolved questions in mass spectrometry and, more broadly, in the domain of electron-initiated chemical reactions.

Time-resolved x-ray spectroscopy of nucleobases and their thionated analogs

The photoinduced relaxation dynamics of nucleobases and their thionated analogs have been investigated extensively over the past decades motivated by their crucial role in organisms and their application in medical and biochemical research and treatment. Most of these studies focused on the spectroscopy of valence electrons and fragmentation. The advent of ultrashort x-ray laser sources such as free-electron lasers, however, opens new opportunities for studying the ultrafast molecular relaxation dynamics utilizing the site- and element-selectivity of x-rays. In this review, we want to summarize ultrafast experiments on thymine and 2-thiouracil performed at free-electron lasers. We performed time-resolved x-ray absorption spectroscopy at the oxygen K-edge after UV excitation of thymine. In addition, we investigated the excited state dynamics of 2-tUra via x-ray photoelectron spectroscopy at sulfur. For these methods, we show a strong sensitivity to the electronic state or charge distribution, respectively. We also performed time-resolved Auger-Meitner spectroscopy, which shows spectral shifts associated with internuclear distances close to the probed site. We discuss the complementary aspects of time-resolved x-ray spectroscopy techniques compared to optical and UV spectroscopy for the investigation of ultrafast relaxation processes.

Regularized weighted sine least-squares spectral analysis for gas electron diffraction data

Here, we present a new approach for obtaining radial distribution functions (RDF) from the electron diffraction data using a regularized weighted sine least-squares spectral analysis. It allows for explicitly transferring the measured experimental uncertainties in the reduced molecular scattering function to the produced RDF. We provide a numerical demonstration, discuss the uncertainties and correlations in the RDFs, and suggest a regularization parameter choice criterion. The approach is also applicable for other diffraction data, e.g., for x-ray or neutron diffraction of liquid samples.

Ultrafast electron diffraction from transiently aligned asymmetric top molecules: Rotational dynamics and structure retrieval

Ultrafast electron diffraction (UED) from aligned molecules in the gas phase has successfully retrieved structures of both linear and symmetric top molecules. Alignment of asymmetric tops has been recorded with UED but no structural information was retrieved. We present here the extraction of two-dimensional structural information from simple transformations of experimental diffraction patterns of aligned molecules as a proof-of-principle for the recovery of the full structure. We align 4-fluorobenzotrifluoride with a linearly polarized laser and show that we can distinguish between atomic pairs with equal distances that are parallel and perpendicular to the aligned axis. We additionally show with numerical simulations that by cooling the molecules to a rotational temperature of 1 K, more distances and angles can be resolved through direct transformations.

Multichannel photodissociation dynamics in CS2 studied by ultrafast electron diffraction

The structural dynamics of photoexcited gas-phase carbon disulfide (CS₂) molecules are investigated using ultrafast electron diffraction. The dynamics were triggered by excitation of the optically bright ¹B₂(¹Σ_u⁺) state by an ultraviolet femtosecond laser pulse centred at 200 nm. In accordance with previous studies, rapid vibrational motion facilitates a combination of internal conversion and intersystem crossing to lower-lying electronic states. Photodissociation via these electronic manifolds results in the production of CS fragments in the electronic ground state and dissociated singlet and triplet sulphur atoms. The structural dynamics are extracted from the experiment using a trajectory-fitting filtering approach, revealing the main characteristics of the singlet and triplet dissociation pathways. Finally, the effect of the time-resolution on the experimental signal is considered and an outlook to future experiments provided.

Ultrafast Imaging of Molecular Dynamics Using Ultrafast Low-Frequency Lasers, X-ray Free Electron Lasers, and Electron Pulses

The requirement of high space-time resolution and brightness is a great challenge for imaging atomic motion and making molecular movies. Important breakthroughs in ultrabright tabletop laser, X-ray, and electron sources have enabled the direct imaging of evolving molecular structures in chemical processes, and recent experimental advances in preparing ultrafast laser and electron pulses resulted in molecular imaging with femtosecond time resolution. This Perspective presents an overview of the versatile imaging methods of molecular dynamics. High-order harmonic generation imaging and photoelectron diffraction imaging are based on laser-induced ionization and rescattering processes. Coulomb explosion imaging retrieves molecular structural information by detecting the momentum vectors of fragmented ions. Diffraction imaging encodes molecular structural and electronic information in reciprocal space. We also present various applications of these ultrafast imaging methods in resolving laser-induced nuclear and electronic dynamics.

Ultrafast Imaging of Molecules with Electron Diffraction

Photoexcited molecules convert light into chemical and mechanical energy through changes in electronic and nuclear structure that take place on femtosecond timescales. Gas phase ultrafast electron diffraction (GUED) is an ideal tool to probe the nuclear geometry evolution of the molecules and complements spectroscopic methods that are mostly sensitive to the electronic state. GUED is a passive probing tool that does not alter the molecular properties during the probing process and is sensitive to the spatial distribution of charge in the molecule, including both electrons and nuclei. Improvements in temporal resolution have enabled GUED to capture coherent nuclear motions in molecules in the excited and ground electronic states with femtosecond and subangstrom resolution. Here we present the basic theory of GUED and explain what information is encoded in the diffraction signal, review how GUED has been used to observe coherent structural dynamics in recent experiments, and discuss the advantages and limitations of the method. Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Ultrafast Valence-Electron Dynamics in Oxazole Monitored by X-ray Diffraction Following a Stimulated X-ray Raman Excitation

Direct imaging of the ultrafast quantum motion of valence electrons in molecules is essential for understanding many elementary chemical and physical processes. We present a simulation study of valence-electron dynamics of oxazole. A valence-state electronic wavepacket is prepared with an attosecond soft X-ray pulse through a stimulated resonant X-ray Raman process and then probed with time-resolved off-resonant single-molecule X-ray diffraction. We find that the time dependent diffraction signal originates solely from the electronic coherences and can be detected by existing experimental techniques. We thus provide a feasible way of imaging electron dynamics in molecules. Moreover, the created electronic coherences and subsequent electron dynamics can be manipulated by the resonant X-ray Raman excitation tuned to different core-excited states.

Nonadiabatic Dynamics with Coupled Trajectories

In this paper, we discuss coupled-trajectory schemes for molecular-dynamics simulations of excited-state processes. New coupled-trajectory strategies to capture decoherence effects, revival of coherence and nonadiabatic interferences in long-time dynamics are proposed, and compared to independent-trajectory schemes. The working framework is provided by the exact factorization of the electron-nuclear wave function, and it exploits ideas emanating from various surface-hopping schemes. The new coupled-trajectory algorithms are tested on a one-dimensional two-state system using different model parameters which allow one to induce different dynamics. The benchmark is provided by the numerically exact solution of the time-dependent Schrödinger equation.

An adaptive approach to machine learning for compact particle accelerators

Machine learning (ML) tools are able to learn relationships between the inputs and outputs of large complex systems directly from data. However, for time-varying systems, the predictive capabilities of ML tools degrade if the systems are no longer accurately represented by the data with which the ML models were trained. For complex systems, re-training is only possible if the changes are slow relative to the rate at which large numbers of new input-output training data can be non-invasively recorded. In this work, we present an approach to deep learning for time-varying systems that does not require re-training, but uses instead an adaptive feedback in the architecture of deep convolutional neural networks (CNN). The feedback is based only on available system output measurements and is applied in the encoded low-dimensional dense layers of the encoder-decoder CNNs. First, we develop an inverse model of a complex accelerator system to map output beam measurements to input beam distributions, while both the accelerator components and the unknown input beam distribution vary rapidly with time. We then demonstrate our method on experimental measurements of the input and output beam distributions of the HiRES ultra-fast electron diffraction (UED) beam line at Lawrence Berkeley National Laboratory, and showcase its ability for automatic tracking of the time varying photocathode quantum efficiency map. Our method can be successfully used to aid both physics and ML-based surrogate online models to provide non-invasive beam diagnostics.

Quantum state tomography of molecules by ultrafast diffraction

Ultrafast electron diffraction and time-resolved serial crystallography are the basis of the ongoing revolution in capturing at the atomic level of detail the structural dynamics of molecules. However, most experiments capture only the probability density of the nuclear wavepackets to determine the time-dependent molecular structures, while the full quantum state has not been accessed. Here, we introduce a framework for the preparation and ultrafast coherent diffraction from rotational wave packets of molecules, and we establish a new variant of quantum state tomography for ultrafast electron diffraction to characterize the molecular quantum states. The ability to reconstruct the density matrix, which encodes the amplitude and phase of the wavepacket, for molecules of arbitrary degrees of freedom, will enable the reconstruction of a quantum molecular movie from experimental x-ray or electron diffraction data.

Ultrafast diffraction is fundamental in capturing the structural dynamics of molecules. Here, the authors establish a variant of quantum state tomography for arbitrary degrees of freedom to characterize the molecular quantum states, which will enable the reconstruction of a quantum molecular movie from diffraction data.

Filming ultrafast roaming-mediated isomerization of bismuth triiodide in solution

Roaming reaction, defined as a reaction yielding products via reorientational motion in the long-range region (3 - 8 Å) of the potential, is a relatively recently proposed reaction pathway and is now regarded as a universal mechanism that can explain the unimolecular dissociation and isomerization of various molecules. The structural movements of the partially dissociated fragments originating from the frustrated bond fission at the onset of roaming, however, have been explored mostly via theoretical simulations and rarely observed experimentally. Here, we report an investigation of the structural dynamics during a roaming-mediated isomerization reaction of bismuth triiodide (BiI₃) in acetonitrile solution using femtosecond time-resolved x-ray liquidography. Structural analysis of the data visualizes the atomic movements during the roaming-mediated isomerization process including the opening of the Bi-I_b-I_c angle and the closing of I_a-Bi-I_b-I_c dihedral angle, each by ~40°, as well as the shortening of the I_b···I_c distance, following the frustrated bond fission.

Determination of excited state molecular structures from time-resolved gas-phase X-ray scattering

We present a comprehensive investigation of a recently introduced method to determine transient structures of molecules in excited electronic states with sub-ångstrom resolution from time-resolved gas-phase scattering signals. The method, which is examined using time-resolved X-ray scattering data measured on the molecule N-methylmorpholine (NMM) at the Linac Coherent Light Source (LCLS), compares the experimentally measured scattering patterns against the simulated patterns corresponding to a large pool of molecular structures to determine the full set of structural parameters. In addition, we examine the influence of vibrational state distributions and find the effect negligible within the current experimental detection limits, despite that the molecules have a comparatively high internal vibrational energy. The excited state structures determined using three structure pools generated using three different computational methods are in good agreement, demonstrating that the procedure is largely independent of the computational chemistry method employed as long as the pool is sufficiently expansive in the vicinity of the sought structure and dense enough to yield good matches to the experimental patterns.

Noninvasive time-sorting in radio frequency-compressed ultrafast electron diffraction

We demonstrate a noninvasive time-sorting method for ultrafast electron diffraction (UED) experiments with radio frequency (rf)-compressed electron beams. We show that electron beam energy and arrival time at the sample after the rf compression are strongly correlated, such that the arrival time jitter may be corrected through the measurement of the beam energy. The method requires minimal change to the infrastructure of most of the UED machines and is applicable to both keV and MeV UED. In our experiment with ∼3 MeV beam, the timing jitter after the rf compression is corrected with a 35-fs root mean square (rms) accuracy, limited by the 3 × 10 - 4 energy stability. For keV UED with a high energy stability, sub-10 fs accuracy in time-sorting should be readily achievable. This time-sorting technique allows us to retrieve the 2.5 THz oscillation related to coherent A1g phonon in the laser-excited Bismuth film and extends the temporal resolution of UED to a regime far beyond the 100-200 fs rms jitter limitation.

A quantum molecular movie: polyad predissociation dynamics in the VUV excited 3pσ2Σu state of NO2

The optical formation of coherent superposition states, a wavepacket, can allow the study of zeroth-order states, the evolution of which exhibit structural and electronic changes as a function of time: this leads to the notion of a molecular movie. Intramolecular vibrational energy redistribution, due to anharmonic coupling between modes, is the molecular movie considered here. There is no guarantee, however, that the formed superposition will behave semi-classically (e.g. Gaussian wavepacket dynamics) or even as an intuitively useful zeroth-order state. Here we present time-resolved photoelectron spectroscopy (TRPES) studies of an electronically excited triatomic molecule wherein the vibrational dynamics must be treated quantum mechanically and the simple picture of population flow between coupled normal modes fails. Specifically, we report on vibronic wavepacket dynamics in the zeroth-order 3pσ2Σu Rydberg state of NO2. This wavepacket exemplifies two general features of excited state dynamics in polyatomic molecules: anharmonic multimodal vibrational coupling (forming polyads); nonadiabatic coupling between nuclear and electronic coordinates, leading to predissociation. The latter suggests that the polyad vibrational states in the zeroth-order 3p Rydberg manifold are quasi-bound and best understood to be scattering resonances. We observed a rapid dephasing of an initially prepared 'bright' valence state into the relatively long-lived 3p Rydberg state whose multimodal vibrational dynamics and decay we monitor as a function of time. Our quantum simulations, based on an effective spectroscopic Hamiltonian, describe the essential features of the multimodal Fermi resonance-driven vibrational dynamics in the 3p state. We also present evidence of polyad-specificity in the state-dependent predissociation rates, leading to free atomic and molecular fragments. We emphasize that a quantum molecular movie is required to visualize wavepacket dynamics in the 3pσ2Σu Rydberg state of NO2.

Strong-field induced fragmentation and isomerization of toluene probed by ultrafast femtosecond electron diffraction and mass spectrometry

We investigate the fragmentation and isomerization of toluene molecules induced by strong-field ionization with a femtosecond near-infrared laser pulse. Momentum-resolved coincidence time-of-flight ion mass spectrometry is used to determine the relative yield of different ionic products and fragmentation channels as a function of laser intensity. Ultrafast electron diffraction is used to capture the structure of the ions formed on a picosecond time scale by comparing the diffraction signal with theoretical predictions. Through the combination of the two measurements and theory, we are able to determine the main fragmentation channels and to distinguish between ions with identical mass but different structures. In addition, our diffraction measurements show that the independent atom model, which is widely used to analyze electron diffraction patterns, is not a good approximation for diffraction from ions. We show that the diffraction data is in very good agreement with ab initio scattering calculations.

Femtosecond X-ray Liquidography Visualizes Wavepacket Trajectories in Multidimensional Nuclear Coordinates for a Bimolecular Reaction

ConspectusVibrational wavepacket motions on potential energy surfaces are one of the critical factors that determine the reaction dynamics of photoinduced reactions. The motions of vibrational wavepackets are often discussed in the interpretation of observables measured with various time-resolved vibrational or electronic spectroscopies but mostly in terms of the frequencies of wavepacket motions, which are approximated by normal modes, rather than the actual positions of the wavepacket. Although the time-dependent positions (that is, the trajectory) of wavepackets are hypothesized or drawn in imagined or calculated potential energy surfaces, it is not trivial to experimentally determine the trajectory of wavepackets, especially in multidimensional nuclear coordinates for a polyatomic molecule. Recently, we performed a femtosecond X-ray liquidography (solution scattering) experiment on a gold trimer complex (GTC), [Au(CN)₂^-]₃, in water at X-ray free-electron lasers (XFELs) and elucidated the time-dependent positions of vibrational wavepackets from the Franck-Condon region to equilibrium structures on both excited and ground states in the course of the formation of covalent bonds between gold atoms.Bond making is an essential process in chemical reactions, but it is challenging to keep track of detailed atomic movements associated with bond making because of its bimolecular nature that requires slow diffusion of two reaction parties to meet each other. Bond formation in the solution phase has been elusive because the diffusion of the reactants limits the reaction rate of a bimolecular process, making it difficult to initiate and track the bond-making processes with an ultrafast time resolution. In principle, if the bimolecular encounter can be controlled to overcome the limitation caused by diffusion, the bond-making processes can be tracked in a time-resolved manner, providing valuable insight into the bimolecular reaction mechanism. In this regard, GTC offers a good model system for studying the dynamics of bond formation in solution. Au(I) atoms in GTC exhibit a noncovalent aurophilic interaction, making GTC an aggregate complex without any covalent bond. Upon photoexcitation of GTC, an electron is excited from an antibonding orbital to a bonding orbital, leading to the formation of covalent bonds among Au atoms. Since Au atoms in the ground state of GTC are located in close proximity within the same solvent cage, the formation of Au-Au covalent bonds occurs without its reaction rate being limited by diffusion through the solvent.Femtosecond time-resolved X-ray liquidography (fs-TRXL) data revealed that the ground state has an asymmetric bent structure. From the wavepacket trajectory determined in three-dimensional nuclear coordinates (two internuclear distances and one bond angle), we found that two covalent bonds are formed between three Au atoms of GTC asynchronously. Specifically, one covalent bond is formed first for the shorter Au-Au pair (of the asymmetric and bent ground-state structure) in 35 fs, and subsequently, the other covalent bond is formed for the longer Au-Au pair within 360 fs. The resultant trimer complex has a symmetric and linear geometry, implying the occurrence of bent-to-linear transformation concomitant with the formation of two equivalent covalent bonds, and exhibits vibrations that can be unambiguously assigned to specific normal modes based on the wavepacket trajectory, even without the vibrational frequencies provided by quantum calculation.

An assessment of different electronic structure approaches for modeling time-resolved x-ray absorption spectroscopy

We assess the performance of different protocols for simulating excited-state x-ray absorption spectra. We consider three different protocols based on equation-of-motion coupled-cluster singles and doubles, two of them combined with the maximum overlap method. The three protocols differ in the choice of a reference configuration used to compute target states. Maximum-overlap-method time-dependent density functional theory is also considered. The performance of the different approaches is illustrated using uracil, thymine, and acetylacetone as benchmark systems. The results provide guidance for selecting an electronic structure method for modeling time-resolved x-ray absorption spectroscopy.

Mapping the emergence of molecular vibrations mediating bond formation

Fundamental studies of chemical reactions often consider the molecular dynamics along a reaction coordinate using a calculated or suggested potential energy surface^1-5. But fully mapping such dynamics experimentally, by following all nuclear motions in a time-resolved manner-that is, the motions of wavepackets-is challenging and has not yet been realized even for the simple stereotypical bimolecular reaction^6-8: A-B + C → A + B-C. Here we track the trajectories of these vibrational wavepackets during photoinduced bond formation of the gold trimer complex [Au(CN)₂^-]₃ in an aqueous monomer solution, using femtosecond X-ray liquidography^9-12 with X-ray free-electron lasers^13,14. In the complex, which forms when three monomers A, B and C cluster together through non-covalent interactions^15,16, the distance between A and B is shorter than that between B and C. Tracking the wavepacket in three-dimensional nuclear coordinates reveals that within the first 60 femtoseconds after photoexcitation, a covalent bond forms between A and B to give A-B + C. The second covalent bond, between B and C, subsequently forms within 360 femtoseconds to give a linear and covalently bonded trimer complex A-B-C. The trimer exhibits harmonic vibrations that we map and unambiguously assign to specific normal modes using only the experimental data. In principle, more intense X-rays could visualize the motion not only of highly scattering atoms such as gold but also of lighter atoms such as carbon and nitrogen, which will open the door to the direct tracking of the atomic motions involved in many chemical reactions.

Ultrafast x-ray and electron scattering of free molecules: A comparative evaluation

Resolving gas phase molecular motions with simultaneous spatial and temporal resolution is rapidly coming within the reach of x-ray Free Electron Lasers (XFELs) and Mega-electron-Volt (MeV) electron beams. These two methods enable scattering experiments that have yielded fascinating new results, and while both are important methods for determining transient molecular structures in photochemical reactions, it is important to understand their relative merits. In the present study, we evaluate the respective scattering cross sections of the two methods and simulate their ability to determine excited state molecular structures in light of currently existing XFEL and MeV source parameters. Using the example of optically excited N-methyl morpholine and simulating the scattering patterns with shot noise, we find that the currently achievable signals are superior with x-ray scattering for equal samples and on a per-shot basis and that x-ray scattering requires fewer detected signal counts for an equal fidelity structure determination. Importantly, within the independent atom model, excellent structure determinations can be achieved for scattering vectors only to about 5 Å-1, leaving larger scattering vector ranges for investigating vibrational motions and wavepackets. Electron scattering has a comparatively higher sensitivity toward hydrogen atoms, which may point to applications where electron scattering is inherently the preferred choice, provided that excellent signals can be achieved at large scattering angles that are currently difficult to access.

Femtosecond Laser-Assisted Electron Scattering for Ultrafast Dynamics of Atoms and Molecules

The recent progress in experimental studies of laser-assisted electron scattering (LAES) induced by ultrashort intense laser fields is reviewed. After a brief survey of the theoretical backgrounds of the LAES process and earlier LAES experiments started in the 1970s, new concepts of optical gating and optical streaking for the LAES processes, which can be realized by LAES experiments using ultrashort intense laser pulses, are discussed. A new experimental setup designed for measurements of LAES induced by ultrashort intense laser fields is described. The experimental results of the energy spectra, angular distributions, and laser polarization dependence of the LAES signals are presented with the results of the numerical simulations. A light-dressing effect that appeared in the recorded LAES signals is also shown with the results of the numerical calculations. In addition, as applications of the LAES process, laser-assisted electron diffraction and THz-wave-assisted electron diffraction, both of which have been developed for the determination of instantaneous geometrical structure of molecules, are introduced.

Femtosecond gas-phase mega-electron-volt ultrafast electron diffraction

The development of ultrafast gas electron diffraction with nonrelativistic electrons has enabled the determination of molecular structures with atomic spatial resolution. It has, however, been challenging to break the picosecond temporal resolution barrier and achieve the goal that has long been envisioned-making space- and-time resolved molecular movies of chemical reaction in the gas-phase. Recently, an ultrafast electron diffraction (UED) apparatus using mega-electron-volt (MeV) electrons was developed at the SLAC National Accelerator Laboratory for imaging ultrafast structural dynamics of molecules in the gas phase. The SLAC gas-phase MeV UED has achieved 65 fs root mean square temporal resolution, 0.63 Å spatial resolution, and 0.22 Å-1 reciprocal-space resolution. Such high spatial-temporal resolution has enabled the capturing of real-time molecular movies of fundamental photochemical mechanisms, such as chemical bond breaking, ring opening, and a nuclear wave packet crossing a conical intersection. In this paper, the design that enables the high spatial-temporal resolution of the SLAC gas phase MeV UED is presented. The compact design of the differential pump section of the SLAC gas phase MeV UED realized five orders-of-magnitude vacuum isolation between the electron source and gas sample chamber. The spatial resolution, temporal resolution, and long-term stability of the apparatus are systematically characterized.

Scattering off molecules far from equilibrium

Pump-probe gas phase X-ray scattering experiments, enabled by the development of X-ray free electron lasers, have advanced to reveal scattering patterns of molecules far from their equilibrium geometry. While dynamic displacements reflecting the motion of wavepackets can probe deeply into the reaction dynamics, in many systems, the thermal excitation embedded in the molecules upon optical excitation and energy randomization can create systems that encompass structures far from the ground state geometry. For polyatomic molecular systems, large amplitude vibrational motions are associated with anharmonicity and shifts of interatomic distances, making analytical solutions using traditional harmonic approximations inapplicable. More generally, the interatomic distances in a polyatomic molecule are not independent and the traditional equations commonly used to interpret the data may give unphysical results. Here, we introduce a novel method based on molecular dynamic trajectories and illustrate it on two examples of hot, vibrating molecules at thermal equilibrium. When excited at 200 nm, 1,3-cyclohexadiene (CHD) relaxes on a subpicosecond time scale back to the reactant molecule, the dominant pathway, and to various forms of 1,3,5-hexatriene (HT). With internal energies of about 6 eV, the energy thermalizes quickly, leading to structure distributions that deviate significantly from their vibrationless equilibrium. The experimental and theoretical results are in excellent agreement and reveal that a significant contribution to the scattering signal arises from transition state structures near the inversion barrier of CHD. In HT, our analysis clarifies that previous inconsistent structural parameters determined by electron diffraction were artifacts that might have resulted from the use of inapplicable analytical equations.

Molecular movie of ultrafast coherent rotational dynamics of OCS

Recording molecular movies on ultrafast timescales has been a longstanding goal for unravelling detailed information about molecular dynamics. Here we present the direct experimental recording of very-high-resolution and -fidelity molecular movies over more than one-and-a-half periods of the laser-induced rotational dynamics of carbonylsulfide (OCS) molecules. Utilising the combination of single quantum-state selection and an optimised two-pulse sequence to create a tailored rotational wavepacket, an unprecedented degree of field-free alignment, 〈cos2θ2D〉 = 0.96 (〈cos2θ〉 = 0.94) is achieved, exceeding the theoretical limit for single-pulse alignment. The very rich experimentally observed quantum dynamics is fully recovered by the angular probability distribution obtained from solutions of the time-dependent Schrödinger equation with parameters refined against the experiment. The populations and phases of rotational states in the retrieved time-dependent three-dimensional wavepacket rationalises the observed very high degree of alignment.

Ultrafast X-ray scattering reveals vibrational coherence following Rydberg excitation

The coherence and dephasing of vibrational motions of molecules constitute an integral part of chemical dynamics, influence material properties and underpin schemes to control chemical reactions. Considerable progress has been made in understanding vibrational coherence through spectroscopic measurements, but precise, direct measurement of the structure of a vibrating excited-state polyatomic organic molecule has remained unworkable. Here, we measure the time-evolving molecular structure of optically excited N-methylmorpholine through scattering with ultrashort X-ray pulses. The scattering signals are corrected for the differences in electron density in the excited electronic state of the molecule in comparison to the ground state. The experiment maps the evolution of the molecular geometry with femtosecond resolution, showing coherent motion that survives electronic relaxation and seems to persist for longer than previously seen using other methods.

Atomic-resolution imaging of carbonyl sulfide by laser-induced electron diffraction

Measurements on the strong-field ionization of carbonyl sulfide molecules by short, intense, 2 µm wavelength laser pulses are presented from experiments where angle-resolved photoelectron distributions were recorded with a high-energy velocity map imaging spectrometer, designed to reach a maximum kinetic energy of 500 eV. The laser-field-free elastic-scattering cross section of carbonyl sulfide was extracted from the measurements and is found in good agreement with previous experiments, performed using conventional electron diffraction. By comparing our measurements to the results of calculations, based on the quantitative rescattering theory, the bond lengths and molecular geometry were extracted from the experimental differential cross sections to a precision better than ±5 pm and in agreement with the known values.

On the limits of observing motion in time-resolved X-ray scattering

Limits on the ability of time-resolved X-ray scattering (TRXS) to observe harmonic motion of amplitude, A and frequency, ω₀, about an equilibrium position, R₀, are considered. Experimental results from a TRXS experiment at the LINAC Coherent Light Source are compared to classical and quantum theories that demonstrate a fundamental limitation on the ability to observe the amplitude of motion. These comparisons demonstrate dual limits on the spatial resolution through Q_max and the temporal resolution through ω_max for observing the amplitude of motion. In the limit where ω_max ≈  ω₀, the smallest observable amplitude of motion is A = 2 π/ Q_max. In the limit where ω_max≥2 ω₀, A≤2 π/ Q_max is observable provided there are sufficient statistics. This article is part of the theme issue 'Measurement of ultrafast electronic and structural dynamics with X-rays'.

Recent Advances in Ultrafast Structural Techniques

A review that summarizes the most recent technological developments in the field of ultrafast structural dynamics with focus on the use of ultrashort X-ray and electron pulses follows. Atomistic views of chemical processes and phase transformations have long been the exclusive domain of computer simulators. The advent of femtosecond (fs) hard X-ray and fs-electron diffraction techniques made it possible to bring such a level of scrutiny to the experimental area. The following review article provides a summary of the main ultrafast techniques that enabled the generation of atomically resolved movies utilizing ultrashort X-ray and electron pulses. Recent advances are discussed with emphasis on synchrotron-based methods, tabletop fs-X-ray plasma sources, ultrabright fs-electron diffractometers, and timing techniques developed to further improve the temporal resolution and fully exploit the use of intense and ultrashort X-ray free electron laser (XFEL) pulses.

Long-lasting field-free alignment of large molecules inside helium nanodroplets

Molecules with their axes sharply confined in space, available through laser-induced alignment methods, are essential for many current experiments, including ultrafast molecular imaging. For these applications the aligning laser field should ideally be turned-off, to avoid undesired perturbations, and the strong alignment should last long enough that reactions and dynamics can be mapped out. Presently, this is only possible for small, linear molecules and for times less than 1 picosecond. Here, we demonstrate strong, field-free alignment of large molecules inside helium nanodroplets, lasting >10 picoseconds. One-dimensional or three-dimensional alignment is created by a slowly switched-on laser pulse, made field-free through rapid pulse truncation, and retained thanks to the impeding effect of the helium environment on molecular rotation. The opportunities field-free aligned molecules open are illustrated by measuring the alignment-dependent strong-field ionization yield of dibromothiophene oligomers. Our technique will enable molecular frame experiments, including ultrafast excited state dynamics, on a variety of large molecules and complexes.

All-optical field-free three-dimensional orientation of asymmetric-top molecules

Orientation and alignment of molecules by ultrashort laser pulses is crucial for a variety of applications and has long been of interest in physics and chemistry, with the special emphasis on stereodynamics in chemical reactions and molecular orbitals imaging. As compared to the laser-induced molecular alignment, which has been extensively studied and demonstrated, achieving molecular orientation is a much more challenging task, especially in the case of asymmetric-top molecules. Here, we report the experimental demonstration of all-optical field-free three-dimensional orientation of asymmetric-top molecules by means of phase-locked cross-polarized two-color laser pulse. This approach is based on nonlinear optical mixing process caused by the off-diagonal elements of the molecular hyperpolarizability tensor. It is demonstrated on SO2 molecules and is applicable to a variety of complex nonlinear molecules.

Spatially Separating the Conformers of a Dipeptide

Atomic-resolution-imaging approaches for single molecules, such as coherent X-ray diffraction at free-electron lasers, require the delivery of high-density beams of identical molecules. However, even very cold beams of biomolecules typically have multiple conformational states populated. We demonstrate the production of very cold (T_rot ≈2.3 K) molecular beams of intact dipeptide molecules, which were then spatially separated into the individual populated conformational states. This is achieved using the combination of supersonic expansion laser-desorption vaporization with electrostatic deflection in strong inhomogeneous fields. This represents the first demonstration of a conformer-separated and rotationally cold molecular beam of a peptide, which enables the investigation of conformer-specific chemistry using inherently non-conformer-specific techniques. It furthermore represents a milestone toward the direct structural imaging of individual biological molecules with atomic resolution by ultrafast diffractive-imaging methods.

An open-source software ecosystem for the interactive exploration of ultrafast electron scattering data

This paper details a software ecosystem comprising three free and open-source Python packages for processing raw ultrafast electron scattering (UES) data and interactively exploring the processed data. The first package, iris, is graphical user-interface program and library for interactive exploration of UES data. Under the hood, iris makes use of npstreams, an extensions of numpy to streaming array-processing, for high-throughput parallel data reduction. Finally, we present scikit-ued, a library of reusable routines and data structures for analysis of UES data, including specialized image processing algorithms, simulation routines, and crystal structure manipulation operations. In this paper, some of the features or all three packages are highlighted, such as parallel data reduction, image registration, interactive exploration. The packages are fully tested and documented and are released under permissive licenses.

Sources of error in Debye-Waller-effect measurements relevant to studies of photoinduced structural dynamics

We identify and quantify several practical effects likely to be present in both static and ultrafast electron-scattering experiments that may interfere with the Debye-Waller (DW) effect. Using 120-nm thick, small-grained, polycrystalline aluminum foils as a test system, we illustrate the impact of specimen tilting, in-plane translation, and changes in z height on Debye-Scherrer-ring intensities. We find that tilting by less than one degree can result in statistically-significant changes in diffracted-beam intensities for large specimen regions containing > 10⁵ nanocrystalline grains. We demonstrate that, in addition to effective changes in the field of view with tilting, slight texturing of the film can result in deviations from expected DW-effect behavior. Further, we find that in-plane translations of as little as 20 nm also produce statistically-significant intensity changes, while normalization to total image counts eliminates such effects arising from changes in z height. The results indicate that the use of polycrystalline films in ultrafast electron-scattering experiments can greatly reduce the negative impacts of these effects as compared to single-crystal specimens, though it does not entirely eliminate them. Thus, it is important to account for such effects when studying thin-foil specimens having relatively short reciprocal-lattice rods.

Conical-intersection dynamics and ground-state chemistry probed by extreme-ultraviolet time-resolved photoelectron spectroscopy

Time-resolved photoelectron spectroscopy (TRPES) is a useful approach to elucidate the coupled electronic-nuclear quantum dynamics underlying chemical processes, but has remained limited by the use of low photon energies. Here, we demonstrate the general advantages of XUV-TRPES through an application to NO₂, one of the simplest species displaying the complexity of a non-adiabatic photochemical process. The high photon energy enables ionization from the entire geometrical configuration space, giving access to the true dynamics of the system. Specifically, the technique reveals dynamics through a conical intersection, large-amplitude motion and photodissociation in the electronic ground state. XUV-TRPES simultaneously projects the excited-state wave packet onto many final states, offering a multi-dimensional view of the coupled electronic and nuclear dynamics. Our interpretations are supported by ab initio wavepacket calculations on new global potential-energy surfaces. The presented results contribute to establish XUV-TRPES as a powerful technique providing a complete picture of ultrafast chemical dynamics from photoexcitation to the final products.

Imaging CF3I conical intersection and photodissociation dynamics with ultrafast electron diffraction

Conical intersections play a critical role in excited-state dynamics of polyatomic molecules because they govern the reaction pathways of many nonadiabatic processes. However, ultrafast probes have lacked sufficient spatial resolution to image wave-packet trajectories through these intersections directly. Here, we present the simultaneous experimental characterization of one-photon and two-photon excitation channels in isolated CF₃I molecules using ultrafast gas-phase electron diffraction. In the two-photon channel, we have mapped out the real-space trajectories of a coherent nuclear wave packet, which bifurcates onto two potential energy surfaces when passing through a conical intersection. In the one-photon channel, we have resolved excitation of both the umbrella and the breathing vibrational modes in the CF₃ fragment in multiple nuclear dimensions. These findings benchmark and validate ab initio nonadiabatic dynamics calculations.

Electron energy analysis by phase-space shaping with THz field cycles

Time-resolved electron energy analysis and loss spectroscopy can reveal a wealth of information about material properties and dynamical light-matter interactions. Here, we report an all-optical concept for measuring energy spectra of femtosecond electron pulses with sub-eV resolution. Laser-generated terahertz radiation is used to measure arrival time differences within electron pulses with few-femtosecond precision. Controlled dispersion and subsequent compression of the electron pulses provide almost any desired compromise of energy resolution, signal strength, and time resolution. A proof-of-concept experiment on aluminum reveals an energy resolution of <3.5 eV (rms) at 70-keV after a drift distance of only 0.5 m. Simulations of a two-stage scheme reveal that pre-stretched pulses can be used to achieve <10 meV resolution, independent of the source's initial energy spread and limited only by the achievable THz field strength and measuring time.

Monitoring molecular nonadiabatic dynamics with femtosecond X-ray diffraction

Ultrafast time-resolved X-ray scattering, made possible by free-electron laser sources, provides a wealth of information about electronic and nuclear dynamical processes in molecules. The technique provides stroboscopic snapshots of the time-dependent electronic charge density traditionally used in structure determination and reflects the interplay of elastic and inelastic processes, nonadiabatic dynamics, and electronic populations and coherences. The various contributions to ultrafast off-resonant diffraction from populations and coherences of molecules in crystals, in the gas phase, or from single molecules are surveyed for core-resonant and off-resonant diffraction. Single-molecule [Formula: see text] scaling and two-molecule [Formula: see text] scaling contributions, where N is the number of active molecules, are compared. Simulations are presented for the excited-state nonadiabatic dynamics of the electron harpooning at the avoided crossing in NaF. We show how a class of multiple diffraction signals from a single molecule can reveal charge-density fluctuations through multidimensional correlation functions of the charge density.