# Imaging conical intersection dynamics during azobenzene photoisomerization by ultrafast X-ray diffraction

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Contributed by Shaul Mukamel, November 23, 2020 (sent for review October 23, 2020; reviewed by Thomas Elsaesser and R. J. Dwayne Miller)

## Significance

Vibronic coherences are unique features that emerge in the decisive moments of photochemistry and photophysics. Here we demonstrate how X-ray diffraction can access fundamental information about these coherences. This is enabled by recent developments at free-electron lasers, working toward temporal resolution and single-molecule accessibility of scattering-based measurements. Time-resolved diffraction patterns of the isomerization of azobenzene are simulated. The process is monitored in time via the momentum-space signal, with a special focus on the mixed elastic/inelastic scattering from coherences. We give practical ideas on how this relatively weak contribution to the signal can potentially be retrieved. A direct connection to the real-space movie of the conical intersection dynamics is made, enabling the observation of quantum coherences that direct chemical processes.

## Abstract

X-ray diffraction is routinely used for structure determination of stationary molecular samples. Modern X-ray photon sources, e.g., from free-electron lasers, enable us to add temporal resolution to these scattering events, thereby providing a movie of atomic motions. We simulate and decipher the various contributions to the X-ray diffraction pattern for the femtosecond isomerization of azobenzene, a textbook photochemical process. A wealth of information is encoded besides real-time monitoring of the molecular charge density for the *cis* to *trans* isomerization. In particular, vibronic coherences emerge at the conical intersection, contributing to the total diffraction signal by mixed elastic and inelastic photon scattering. They cause distinct phase modulations in momentum space, which directly reflect the real-space phase modulation of the electronic transition density during the nonadiabatic passage. To overcome the masking by the intense elastic scattering contributions from the electronic populations in the total diffraction signal, we discuss how this information can be retrieved, e.g., by employing very hard X-rays to record large scattering momentum transfers.

Diffraction signals reveal the momentum transfer experienced by an incident photon through interference with one or more scatterers. Traditionally, X-ray diffraction is used to determine the structure of crystalline matter. A momentum-space image, provided by the scattered photons, is recorded, allowing for the reconstruction of the real-space crystal structure. Diffraction patterns are dominated by the Bragg peaks, arising from the interaction of multiple scatterers in a long-range crystalline order. The relevant material quantity is the electronic charge density, from which the X-ray photons elastically scatter.

The advent of free-electron X-ray light sources (1, 2) has added two game-changing ingredients to diffraction experiments. The first is a peak brilliancy that can exceed third-generation synchrotron sources by nine orders of magnitude (3). The high number of photons in the beam allows for a significant reduction of the sample size, while still achieving the necessary ratio of scattered photons per object to record diffraction patterns. Structure determination of nanocrystals (4, 5), aligned gas-phase molecular samples (6), and macromolecular structures (7, 8) were reported. Attempts toward imaging single molecules at free-electron lasers were made (9), although some challenges remain (8). The second ingredient is temporal resolution, enabled by subfemtosecond pulse durations (1 fs =

Diffraction from matter in nonstationary states, such as molecules undergoing a chemical transformation, involves physical processes that go beyond elastic scattering from instantaneous snapshots of the electronic ground state density. This is especially true for excited molecules, being in a time-evolving superposition of many-body states. Inelastic scattering from different electronic states (25⇓–27), as well as electronic and vibrational coherences (27⇓⇓–30), contributes to the signal.

Here we study theoretically the diffractive imaging of vibronic coherences that emerge at conical intersections (CoIns). These are regions of degeneracy in electronic potential energy surfaces (PESs), where the movement of electrons and nuclei become strongly coupled, enabling ultrafast radiationless relaxation channels back to the ground state (31, 32). Processes like the primary event of vision (33) or the outcome of photochemical reactions (23), among many others, are determined by the occurrence and properties of CoIns. A textbook example of CoIn dynamics is the optical switching between *cis* and *trans* azobenzene (Fig. 1*A*): azobenzene can be switched selectively on a femtosecond timescale between both isomers (34⇓–36) with high quantum yield, it exhibits interesting photophysics like a violation of Kasha’s rule (37, 38), and it has found broad application as the photoactive unit in switching the activity of pharmaceutical compounds (39) or neurons (optogenetics) (40). There is a large body of experimental and theoretical literature on azobenzene, and the previous references are by no means exhaustive. For a deeper overview of scientific progress about azobenzene photophysics, we refer the reader to ref. 38 and section 8 of its supplement.

Three-vibrational-mode PESs for the photoisomerization of azobenzene were reported in ref. 41. The nuclear space consists of the carbon–nitrogen–nitrogen–carbon (CNNC) torsion angle and the two CNN bending angles between the azo unit and the benzene rings. CNNC torsion is the reactive coordinate that connects the *cis* and *trans* minima at 0° and 180°, respectively, while the two CNN angles are responsible for the symmetry breaking that leads to the minimum energy CoIn seam (41). We perform nuclear wave packet simulations of the

## Dissecting the Diffraction Signal into Pathways

We follow the derivation of the single-molecule time-resolved X-ray diffraction (TXRD) in refs. 27 and 42, using electronic r and nuclear R degrees of freedom. The single-molecule diffraction signal allows us to circumvent the domination of sharp Bragg peaks and retrieve the continuous patterns from individual molecules. An incident photon is scattered from the molecular electron density **1** is an exact representation which does not invoke the adiabatic approximation and can be similarly formulated in, e.g., a diabatic basis.

The one-molecule TXRD signal of a sample with N noninteracting molecules reads (27)**3** represent elastic scattering from the electronic ground and excited state, respectively, and use the state densities **3**, spanning lines five and six, represents scattering from vibronic coherences. It contains the nuclear wave packet of the two different states in

There are a few important points to note. First, as can be seen from Eq. **3**, scattering does not simply occur from charge densities but from expectation values over products of charge density operators

The second point to note is that Eq. **3** is exact and contains all contributions to the signal. This requires us to explicitly calculate **3**, which is our main focus. These have never been singled out from diffraction patterns, which allow for a direct observation of quantum coherences that determine the fundamental mechanisms of chemical reactions. For this purpose, we follow the complete formulation in Eq. **3** and compute *Materials and Methods* and Eq. **9**).

The third point connects to the structure factor that is responsible for the emergence of Bragg peaks. Diffraction patterns are usually recorded on an organized molecular assembly, i.e., a crystal. In these cases, the two scattering events in the diffraction signal (Eq. **3**) do not occur from the electron density of a single molecule but from two different molecules α and β. Multiplication of two **3** and as investigated here, is conceptually different.

## Azobenzene Quantum Dynamics

Our effective Hamiltonian for azobenzene *trans* geometry is located at 180° of torsion and the *cis* minimum close to 0°. The other two degrees of freedom are the CNN bending angles between the azo unit and the two respective carbon rings, where both the *cis* and *trans* minima are located between 115° and 120°. The two electronic states are the ground state S_{0} and the _{1} state, involving electronic excitation of a nitrogen lone-pair electron. The two potential minima at the *cis* (CNNC = 0°) and *trans* (CNNC = ±180°) geometry in S_{0} are separated by a 1.5-eV barrier. Barrierless isomerization is possible in S_{1} with a CoIn located between the minima, where the electronic states become very close in energy with a significant nonadiabatic coupling. For the

We perform exact nuclear wave packet dynamics on these adiabatic PESs by solving the time-dependent Schrödinger equation (*Materials and Methods*). Assuming impulsive excitation, the wave packet starts in S_{1} at the *cis* geometry (Fig. 2*A*, *Left*) and is initially localized in both nuclear degrees of freedom. Within the first few femtoseconds, it evolves to higher CNN angles and starts to spread along the CNNC torsion (Fig. 2*A*, *Middle*). Around 90 fs, it reaches the CoIn region and starts to relax to S_{0}. A vibronic coherence emerges that is initially localized at the CoIn, visible through the wave packet overlap in the two states (Fig. 2*A*, *Middle*). The coherence magnitude, along with the population dynamics, is drawn in Fig. 2*B*. After evolving away from the CoIn, the wave packet and thus the vibronic coherence are no longer local in the CNN angle but exhibit a spread there as well. This delocalized structure is visible in Fig. 2*A*, *Right*, showing both the local coherence before and around the CoIn and the spreading in CNN thereafter. Around 170 fs, the first significant parts of the wave packet have reached the *trans* geometry. After reaching the product minimum, the wave packet is absorbed at CNNC = ±180°, and the photochemical reaction is completed. As can be seen in Fig. 2*B*, the product yield represented by the red line emerges slightly before 200 fs and accumulates thereafter. Some parts of the wave packet exit the nuclear grid at CNN ≥ 180°, a process that is not further captured by our Hamiltonian and thus acts as a loss channel (blue line in Fig. 2*B*). The important part of the CoIn dynamics happens before 300 fs, with the subsequently highly delocalized wave packet slowly completing the relaxation before 1 ps. Absorbing the nuclear wave packet at CNNC = ±180° is an artificial measure that prevents it from leaving the product minimum again. In reality, most of the nuclear wave packet will be trapped there by, e.g., vibrational relaxation to other modes not contained in our Hamiltonian. Wave packet recurrence to the CoIn region might still happen, a process that will influence the observed kinetics in Fig. 2*B* and especially the coherence magnitude

To image this process by TXRD, we computed the state and transition electron densities across our two-dimensional nuclear space. Examples are depicted in Fig. 1*B* at a geometry that is in the vicinity of the CoIn (CNNC = 92°, CNN = 140°). The state densities **3** consist of products of

## Diffraction Pattern Snapshots during Isomerization

In the following, we assume diffraction from an oriented molecule. This can be achieved (6), although not to a perfect degree, but is experimentally more demanding than a randomly oriented sample. This means that in any case, the diffraction patterns presented in the following are ideal cases that in reality will be not as clear. In ref. 51, we have compared oriented versus nonoriented molecules in the simulation of a spectroscopic signal that targets vibronic coherences. The signal was visible in the isotropic case as well since one of the spatial directions was dominating the others. Diffractive imaging of coherences, as presented here, may exhibit a similar behavior. For a randomly oriented molecular sample, rotational averaging of the presented data must be performed, where we expect the diffraction patterns to look not as clean but still exhibiting the key features. We examine different q-space directions separately to distinguish between their contributions. The molecular axes are defined in Fig. 1*A*, with the x axis perpendicular to the molecular plane of the *trans* geometry, the y axis going through both carbon rings and the

The complete three-dimensional diffraction pattern is presented in Fig. 3, dissected into the different contributions of Eq. **3** and labeled i through v accordingly. An exemplary snapshot at 170 fs is chosen because all contributions are nonvanishing there, while a complete movie is provided as electronic supplement (Movie S2). The two most interesting observations are the relative strengths of the individual terms and the distinct image of the coherence contribution. The dominant term is elastic scattering from the excited state ii since *B*). Ground state elastic scattering (term i) is weaker, although well visible. The inelastic scattering components (terms iii and iv) in Fig. 3 are around three orders of magnitude weaker than their elastic counterparts. This can be attributed to their scattering from the product of two transition density operators, each being around 100 times smaller than the state densities (1 electron versus 96 electrons). The coherence term v in Fig. 3, i.e., scattering from the vibronic coherence, is 2,000 times weaker than the total signal and equally strong as the inelastic scattering terms. In contrast to all other terms, the coherence scattering exhibits negative (blue) as well as positive (red) contributions. It also exhibits the most pronounced dynamic behavior (compare Movie S2) and will be discussed in the following.

To get a better understanding of the diffraction patterns, two-dimensional projections obtained by integration over *A* for the pure *cis* and *trans* geometries, without considering the nuclear wave packet. In all three planes, *cis* and *trans* exhibit distinctly different diffraction patterns. Snapshots at 0, 130, and 170 fs during the wave packet dynamics, and as calculated from Eq. **3**, are depicted in Fig. 4*B*. At 0 fs, the pattern resembles the pure *cis* one because the wave packet starts here. This changes once major parts of the wave packet have moved away from the *cis* region toward the CoIn (130 fs). At 170 fs, where significant parts of the wave packet have reached the *trans* product minimum, the diffraction pattern exhibits the same features as the one for the pure *trans* geometry. In the

The evolution of the coherence term v in Eq. **3** in the two-dimensional projections is depicted in Fig. 5. It vanishes before 80 fs because the wave packet is purely in S_{1}, and there is no vibronic coherence yet. The latter emerges around 90 fs, when the CoIn is reached. The signal is strongest in the

## Time-Resolved Diffraction Traces

In the following, we present time-resolved traces of the diffraction pattern along **2**). The fast oscillations in the coherence contribution are captured by the 10-fs probe as well.

All contributions to the TXRD signal are presented in Fig. 6. The prominent features along *cis* region and through the CoIn, the pattern changes. The 1.2 _{0}. The main features are located at 0, 0.8, and 5.5 _{1} pattern after 120 fs. This is not surprising, as *B*), and the nuclear wave packet explores similar regions in R. The inelastic scattering contributions iii and iv in Fig. 6 are much weaker. Their strongest feature shows up at 4

Our nuclear wave packet simulations assume an initial optical excitation of 100% into S_{1}. Experimentally, this can require very high pulse intensities, where multiphoton absorption channels not only contribute but dominate. In reality, it is rather a small fraction that will be excited to S_{1} to stay in the linear regime (18, 19). This means that unlike our simulations, the ground state diffraction contribution is not zero before 100 fs but more likely the dominating one. This does not affect the qualitative features in the excited state diffraction pattern, and especially in the coherence contribution that we focus on, but can add a large experimental background that needs to be addressed.

## Isolating the Coherence Term

The coherence term, which emerges as a feature unique to the CoIn, is shown in Fig. 6. Again, it exhibits positive (red)/negative (blue) temporal oscillations at each *F* with a logarithmic color scale. At *A*), and thus having a wider spread in q, which is a feature that can be generally expected in many molecules. Going to these momentum transfer amplitudes requires pulse energies between 30 and 40 keV. Currently, up to 25 keV are possible using superconducting accelerators at the European X-ray Free Electron Laser in Hamburg and the Stanford Linear Coherent Light Source (1).

Another tool that may be used to retrieve the coherence term is frequency-resolved detection. By frequency-dispersing the scattered photons, inelastic (Stokes and Anti-Stokes) contributions can be measured separately at different frequencies. In the single-scatterer case described by Eq. **2**, the coherence term is a mixed elastic/inelastic process. This means that such separation is not straightforward, since one of the two photon amplitudes is scattered elastically and the other one inelastically. They thus arrive at the detector with different frequencies, and the coherence term can only be recorded by detecting both photons with a broadband detector. Separation of the coherence term could be achieved by measuring the total signal with a broadband detector and subsequently subtracting the purely elastic and purely inelastic events measured in a separate experiment with a frequency-resolved detection. Other possibilities entail stochastic covariance-based measurements (52, 53) or using entangled photons (54).

Frequency-resolved measurement to single out the coherence term can be more easily performed in the two-molecule diffraction signal (27, 42). This dominates when long-range molecular order is present in the sample and is not applicable in our example since the Hamiltonian for azobenzene isomerization was constructed for the gas phase, with no environmental influence. Nevertheless, the molecular response in the TXRD signal for the two-scatterer case reads (27, 42)**7** can be obtained by solving the phase problem. The only inelastic contribution in Eq. **7** is the coherence term in line four, and in this case it does not include elastic contributions. It can more easily be separated than in the one-scatterer case, although care has to be taken with the influence of the structure factor and the physics of CoIns with intermolecular interactions.

In ref. 51, we have introduced the Wigner time/frequency spectrogram of a spectroscopic signal that reveals detailed information about the coherence pathway on the molecular PES and the energetic distribution of the participating vibronic states. The diffraction image of the vibronic coherence in Fig. 6 (term v) similarly exhibits temporal oscillations with a certain frequency. In this case, it is not the energy splitting between the adiabatic states that is encoded but the phase evolution of the electronic transition density, from which the scattered photons are recorded. The Wigner spectrogram is given by

## Real-Space Reconstruction

The ultimate goal of ultrafast photochemistry is to record movies of CoIn dynamics in real space. As mentioned above, the inverse Fourier transform of the diffraction image does not yield the real-space image. It yields the Patterson function (43), which shows correlations between atomic positions or interatomic distances rather than the full real-space molecular geometry. Additionally, for a correct inversion of the diffraction pattern, the phases of the photons that are lost upon measurement need to be recovered (47). Nevertheless, we can perform a naive inverse Fourier transform of the coherence term v and compare it to the real-space image that we have access to through simulations (although not experimentally observable and not given by the Patterson function). This is shown in Fig. 8, with the coherence term further noted as *A*. Fig. 8*B* displays the naive inverse Fourier transform along *C*, we compare this to the real-space picture of the transition density, which we have calculated by *A* and *B*. This is also visible in the complete three-dimensional snapshots of *D*–*F*. Looking along y, the phase between positive (orange) and negative (teal) density changes from 155 to 195 fs and again to 220 fs. This matches the observations at these times in Fig. 8 *A*–*C*. Thus, by imaging the diffraction pattern of the vibronic coherence, fundamental information about the CoIn itself can be retrieved.

## Summary

We have simulated the TXRD for the

Within 200 fs, the time-resolved diffraction signal changes from exhibiting pure *cis* isomer features to closely resembling the *trans* isomer. This clearly monitors the photochemical reaction process. The signal is dominated by elastic diffraction from the electronic S_{1} and S_{0} states. Mixed elastic/inelastic scattering from the vibronic coherence is significantly weaker, but its ratio can be amplified by two orders of magnitude when looking at high values of scattering momentum transfer. This requires very hard X-ray pulses above 20 keV, motivating further development at the large free-electron laser facilities. The high energy and short wavelengths needed to observe this contribution are currently readily available with electron diffraction (15, 23, 24). Scattering of electrons exhibits similar terms to the X-ray case and contains additional information due to scattering from nuclei. Electron diffraction analogs of the present study are being developed.

The momentum-space coherence contribution to the diffraction signal exhibits distinct temporal oscillations. These closely resemble the real-space evolution and phase modulation of the transition charge density during CoIn relaxation. Although this cannot be directly retrieved from the diffraction pattern, the fundamental physical information is encoded in this coherence term. Recording this feature will constitute a large step toward imaging movies of CoIn dynamics. This image goes beyond the accessibility of vibrational coherences by optical methods and will help in deciphering the basic mechanisms that direct photochemical reactions.

## Materials and Methods

### Ab Initio Quantum Chemistry.

The PES for azobenzene photoisomerization was reported in ref. 41 and mapped in a space of three nuclear degrees of freedom: the CNNC torsion (*cis* and *trans* minimum at 0° and ±180°, respectively) and two CNN bending angles between the _{0} and the first excited state S_{1} involving an

Nonadiabatic coupling matrix elements, accounting for the CoIn dynamics, were computed on the same level of theory, by displacing the molecular structure along the two internal coordinates at each grid point. They are nonvanishing in the region between CNNC = 80° to 110° and CNN = 133° and 147°. Instead of a single point, an extended CoIn seam is present with slight asymmetry in the CNNC torsion, and that is reached by symmetry breaking of the CNN angles.

### Electronic Densities.

Following the electronic structure calculations, the state and transition densities *A*).

### Wave Packet Simulations.

Exact quantum dynamical simulations in the reduced-dimensional space of two reactive coordinates were performed by solving the time-dependent Schrödinger equation_{0} after impulsive excitation to S_{1} with a time step of 0.05 fs. The TXRD signal in Eqs. **2**–**3** was evaluated with a 1-fs time step. The kinetic energy operator **10** is set up according to the G-Matrix formalism (58) in _{1} state, periodic boundary conditions are employed along the CNNC torsion. A Butterworth filter (60) was used to absorb the nuclear wave packet at the borders in the CNN direction to prevent artifacts. In S_{0} the filter was also employed at 0° and 360° of torsion, to absorb the parts of the wave packet that have reached the product minimum.

## Data Availability.

All study data are included in the article and *SI Appendix*.

## Acknowledgments

We gratefully acknowledge support from the Chemical Sciences, Geosciences, and Bio-Sciences Division, Office of Basic Energy Sciences, Office of Science, US Department of Energy, through award DE-SC0019484. D.K. gratefully acknowledges support from the Alexander von Humboldt foundation through the Feodor Lynen program. We thank Markus Kowalewski for providing his QDng quantum dynamics code.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: smukamel{at}uci.edu.

Author contributions: D.K. and S.M. designed research; D.K., F.A., F.S., B.G., and A.N. set up the ab initio Hamiltonian; D.K. and J.R.R. performed research; D.K., F.A., J.R.R., F.S., B.G., A.N., M.G., and S.M. analyzed data; and D.K., J.R.R., and S.M. wrote the paper.

Reviewers: T.E., Max-Born-Institute for Nonlinear Optics and Short Pulse Spectroscopy; and R.J.D.M., University of Toronto.

The authors declare no competing interest.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2022037118/-/DCSupplemental.

Published under the PNAS license.

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