# Monitoring molecular nonadiabatic dynamics with femtosecond X-ray diffraction

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Contributed by Shaul Mukamel, May 10, 2018 (sent for review April 5, 2018; reviewed by Majed Chergui and Thomas Elsaesser)

## Significance

X-ray crystallography has long been used to determine the structure of crystals and molecular samples. More recent advancements in light sources and computational methods made it possible to routinely determine the structure of large proteins. The introduction of X-ray free-electron lasers opens up the possibility to track the dynamics of molecular structures on a femtosecond time scale and to create molecular movies of chemical reactions. The theory of time-independent diffraction is well known. However, time-resolved diffraction techniques pose not only new challenges to experiments but also to their interpretation. In this work, we present a unified theoretical framework that will aid experimental interpretations as well as predictions of types of X-ray diffraction experiments.

## Abstract

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

The term diffraction denotes the interference of waves elastically scattered from different positions in space (1). Since the phase difference between waves originating from different spatial locations encodes the sample geometry, the diffraction of waves can be used to infer the spatial pattern of the arrangement of scatterers. This technique has long been used with off-resonant X-rays to reveal the atomic structure of crystalline solids, where the long-range order amplifies the diffraction signal for certain values of the momentum transfer scattering vector q, known as the Bragg peaks. The location pattern of the Bragg peaks then reveals the long-range crystal structure, while their intensity pattern reflects the unit-cell structure through the classical diffraction signal

Coherent X-ray light sources capable of producing bright, ultrafast pulses have been developed [e.g., the Stanford Linear Coherent Light Source produces pulses with

In this work, we provide a unified quantum electrodynamical (QED) description of time-resolved diffraction signals from gas-phase samples (or single molecules) and from systems that have longer-range structural order such as crystals and liquids. We show that the two types of signals are dominated by different terms and thus have a fundamentally different character. It is tempting to describe the time-resolved signals by simply replacing **4** and refs. 30 and 31). Moreover, while picosecond diffraction (32, 33) is well established and can be interpreted by using kinetic models for the evolving charge density, femtosecond diffraction with FEL sources involves electronic coherences that must be treated with care. Although we will speak throughout of X-ray diffraction, all results are equally applicable to the diffraction of femtosecond electron pulses. This is an emerging technology that can also probe the electronic charge density of material samples (27, 34, 35). We further discuss X-ray scattering resonant with core atomic transitions, which reveals correlations of core and valence electrons, and comment on multidimensional diffraction involving photon coincidence detection where higher-order intensity correlation functions of light are detected (13, 36).

## X-Ray Scattering and the Electronic Charge Density

Infrared or visible light spectroscopies may be adequately described by invoking the dipole approximation in which the field-matter interaction energy is given by the dot product of the external field and a material quantity, the transition dipole. This is the first in a series of higher-order contributions to the field-matter coupling known as the multipolar expansion (37). Retaining only the lowest (dipolar) term is well-justified as long as the radiation field amplitudes do not vary appreciably over the relevant material length scales. This condition may not hold in the hard X-ray regime, and a more general treatment is required. Rather than patching up the dipolar approximation with higher-order multipoles, it is simpler to recast the problem in the framework of the minimal-coupling Hamiltonian wherein the exact coupling of matter to the radiation field is obtained by the substitution *SI Appendix*, Eqs. **S13** and **S14**). Scattering occurs when a vacuum mode of the electromagnetic field is populated due to the matter interaction with the incoming light field. Calculating the total number of photons produced in a given signal mode *SI Appendix*) gives*SI Appendix*, Eq. **S23**). The expectation value

Hereafter, we focus on the off-resonant regime, where the X-ray photon is tuned away from core transitions while the extension to resonant scattering will be presented in *Summary and Future Outlook*. We thus substitute the definition of **2** and only retain the **2**, this can be formalized by taking the Wigner spectrogram of the X-ray A-field to be broad and flat). Writing Eq. **3** in terms of the electric field **3**) in the power of the α and *SI Appendix*, Eq. **S33** in *SI Appendix*). We henceforth omit this prefactor for brevity.

Although formally exact when expressed in terms of the total sample electron-density operator **3** then generate a double sum over molecules α and β, which can be separated into one-molecule (**5**) matches the intuitive form of the classical time-resolved diffraction signal discussed in the introduction, while the single-molecule contribution (Eq. **4**) does not.

In crystals, *SI Appendix*), can be used to interpolate between ordered and disordered samples.

In the case of liquids, *SI Appendix*, Debye–Waller Factor) or by taking the continuum limit of Eq. **6** and integrating over all space, assuming a homogenous distribution of molecules to obtain a delta function

When the terms in the structure factor **5**). In contrast, the signal between the Bragg peaks or from a sample lacking long-range order, such as a gas, is dominated by the single-molecule signal (Eq. **4**) since **4** (28, 31, 39).

The time dependence of the charge-density operators in Eqs. **4** and **5** can be simplified by expansion in system eigenstates. Such expansions are given in *SI Appendix*, but these full electronic+vibrational eigenstates are too expensive to calculate for any but the simplest systems. In the following section, we will instead expand the time-dependent wavefunction in adiabatic electronic eigenstates and keep the nuclear configuration in a real-space wave packet representation (rather than using vibronic eigenstates).

## Time-Resolved Diffraction Movies of Nonadiabatic Dynamics

Conical intersections (CoIns) can be found in nearly every polyatomic molecule and dominate the outcome of many photochemical reactions (40). CoIns provide fast, sub-100-fs nonradiative decay channels that are defined by a strong coupling between nuclear and electronic degrees of freedom. Their direct spectroscopic detection has not yet been demonstrated experimentally. However, we argue that the strong mixing of the nuclear and electronic degrees of freedom creates an electronic coherence that generates clear spectroscopic signatures (41).

In the following, we will investigate the effect of electronic coherences on the diffraction pattern in ordered as well as unordered samples. In ultrafast, time-resolved optical pump/X-ray probe diffraction experiments, the system is pumped into an excited state, and the subsequent coupled electronic and nuclear dynamics is probed after a variable time-delay T. This is depicted diagrammatically in Figs. 1 and 2 where a preparation process by an arbitrary pulse sequence is represented by the shaded box (an example of such a preparation is shown in *SI Appendix*, Fig. S1), the following free-propagation period is represented by a checkered box, and arrows represent the interactions with the X-ray probe. The indices i, j, or k refer to the general case representing an arbitrary number of states, and R refers to and arbitrary number nuclear degrees of freedom. The time-dependent wave function is then expanded as

Electronic operators, such as the charge density, generally depend on the nuclear configuration too, so that

### Time-Dependent Diffraction from Ordered vs. Unordered or Single-Molecule Samples.

For a sample possessing long-range order, so that the structure factor is nonvanishing, the signal is dominated by the two-molecule scattering Eq. **5**, which we now recast as**7** gives the time-resolved scattering signal**11**. Terms *i* and *j* in the amplitude are, when squared, simply the elastic ground- and excited-state scattering, respectively. Their coefficients are

Terms *i* and *j* of Eq. **11** also generate cross-terms when the amplitude is squared. These come as

Finally, terms *k* and *l* in Eq. **11** arise from the combination of inelastic scattering and electronic coherences. Depending on the dynamics, the electronic coherences may rapidly decay, rendering this third term negligible so that the scattering is given only by the ground- and excited-state diffraction and their heterodyne interference. Moreover, the coherence

We note that, even though our discussion has focused on electronic coherences, the same formalism applies to vibrational coherences. Indeed, vibrational wave packets are unavoidably created in the ground state via Raman processes during the pumping (illustrated in *SI Appendix*, Fig. S1).

In the absence of long-range intermolecular order, the vanishing structure factor **7****13** corresponds to a particular diagram in Fig. 2. Terms *a* and *b* represent the elastic and inelastic scattering contributions (Fig. 2 *A* and *B*) from the ground state, while *c* and *d* are the equivalent terms for the excited state (Fig. 2 *C* and *D*). Terms *e*, *f*, *g*, and *h* represent mixed elastic–inelastic processes, which scatter off electronic coherences. Each of these terms originates from two diagrams, which are complex conjugates, and are grouped by the final state (Fig. 2 *E*–*H*).

X-ray diffraction is ordinarily taken to be purely elastic, and the possibility of the inelastic and mixed terms in Eq. **13** is rarely considered (28, 30, 43, 45). In most experimental circumstances, the majority of the molecular charge can be definitively assigned to particular atoms. Only few electrons participate in chemical bonds. This inspires the commonly used independent atom approximation for the molecular charge density**14** yields**13**, this expression neglects the electronic coherences, since

As can be seen from Eqs. **12** and **10**, the *i*) their scaling behavior with respect to the particle number and (*ii*) the connection of the diffraction pattern with charge-density matrix elements. (*i*) The *ii*) The

## Monitoring the Nonadiabatic Avoided-Crossing Dynamics in NaF

We illustrate the various contributions to the diffraction signal for sodium fluoride. This molecule possesses a similar electronic structure to NaI, which was studied in the first femtochemistry experiments (50): an avoided crossing between the ionic and covalent state (potential energy curves can be found in *SI Appendix*). Although well known for facilitating population transport between adiabatic electronic states, the passage of nuclear wave packets through the region where electronic states are degenerate or near-degenerate also generates electronic coherences. The resulting coherent oscillations can be monitored with, e.g., photoelectron or Raman signals and reveal the time-evolving electronic energy gap as well as information on the differential topology of the electronic surfaces via the decoherence time (51, 52). Here, we explore the consequences of these dynamics for ultrafast time-resolved X-ray diffraction in gas-phase NaF (44). Iodine is a strong X-ray scatterer, and its large nuclear charge leads to a charge-density distribution heavily dominated by its core electrons. While this is still the case for molecular form factors of lighter element compounds, they have a relatively more prominent contribution of valence electrons compared with the core electrons, which is why we chose NaF.

In the following, we present simulations and analysis of the time-dependent diffraction patterns for the gas phase (*SI Appendix*).

The signal is finally obtained by evaluating Eq. **12** and inserting the time-dependent wavefunctions and density operators (*SI Appendix*, Fig. S4). The effective electronic coherence is obtained from the combined electronic-nuclear wavefunction as the overlap of the nuclear wave packets, which represents decoherence and the electronic density matrix elements

The wave packet dynamics in the excited-state potential (*A*. The nuclear wave packet passes through the avoided crossing between 200 and 240 fs and reaches its outer turning point at ∼500 fs.

Fig. 3*B* shows the time-dependent excited-state population alongside the magnitude of the electronic coherence. When the wave packet passes through the avoided crossing for the first time,

While we have used a fully quantum description of nuclei and electrons to calculate the molecular dynamics, this approach might not be feasible for polyatomic molecules with a large number of vibrational degrees of freedom. Semiclassical simulations methods like ab initio multiple spawning (53) or surface hopping (54, 55) can be used instead. However, special care has to be taken to correctly include electronic coherences, which are neglected in the surface-hopping protocol.

### The Time-Resolved Single-Particle Diffraction Signal.

The **12**) shown in Fig. 4 is dominated by the oscillating ground-state wave packet that was created by the 10-fs UV pump pulse (*SI Appendix*, Fig. S1*A*).

Fig. 5 shows **13**. The elastic ground-state to ground-state contribution in Fig. 5*A* is in close resemblance to the total *C* (*A*) makes it clear that the features of the nuclear wave-packet motion can be retrieved approximately from the elastic excited-state contribution (a detailed interpretation is given in *SI Appendix*). Fig. 5 *E*–*H* depicts the contribution of the electronic coherences (corresponding to diagrams in Fig. 2 *E*–*H*). At ≈225 fs and at ≈800 fs, as the wave packet enters the avoided crossing region, a coherence is created (Fig. 3*B*): The pattern at 8 Å is at the position of the avoided crossing. The contribution at 800 fs is caused by the returning spatially elongated wave packet passing the avoided crossing on its way back to the Franck–Condon region. The recurrence event itself in the Franck–Condon is not well resolved due to the rather large energy gap (≈4 eV) and is averaged out by the 2.5-fs probe pulse. The coherence contribution is ≈ 3 orders of magnitude weaker than the excited-state density (Fig. 4*C*). The contribution stemming solely from the transition densities (*b* and *d*) is *B* and *D* and *SI Appendix*, Fig. S6). It carries no information about the electronic coherence but is dominated by the shape and magnitude of the transition density *SI Appendix*, Figs. S3 and S6).

The actinic pump pulse creates a wave motion not only in the excited-state potential but also in the ground-state potential due to a Raman process (*SI Appendix*, Fig. S1). A small fraction of the population is moved from the ground to the excited state, creating a “particle” excited state wave packet and a “hole” in the ground state. The respective signals in reciprocal momentum space and in real space given in Fig. 5*A* clearly show that the pump-pulse creates an oscillating wave packet in the ^{1}X state.

In conclusion, the simulated gas-phase diffraction signal of NaF undergoing nonadiabatic avoided crossing dynamics in a nonstationary state is dominated by ground- and excited-state wave packet motions and shows some weak signatures of the electronic coherence created at the avoided crossing. The interatomic distance can be extracted directly from the diffraction signal, and, as such, the shape of the nuclear wave packet can be qualitatively retrieved without further phase reconstruction. For diatomic molecules, this allows us to create a molecular movie. The coherence contributions do not merely indicate that a coherence has been created but also give a hint of where it has been created. They are significantly weaker than elastic scattering processes and appear as a rapid oscillation on top of the diffraction pattern.

The actinic pump pulse also creates a nonstationary nuclear wave packet in the electronic ground state. This must be taken into account in the interpretation of diffraction patterns since the Raman and the excited-state signals are of comparable magnitudes (*SI Appendix*, Fig. S1). Separating the ground- and excited-state contributions has been a long-standing open challenge in nonlinear spectroscopy (56).

Virtually all photophysical and photochemical processes in polyatomic molecules with two or more vibrational coordinates take place via CoIns (57). Observing them in diffraction experiments is an interesting open question. Once the molecule reaches a CoIn, a short-lived electronic coherence is created which can in principle be spectroscopically detected (51, 58) by soft X-rays. One example of a photochemical prototype reaction, which is mediated by a CoIn and has been studied by X-ray diffraction, is the ring-opening reaction in cyclohexadiene (59, 60). Potential signatures in time-resolved X-ray diffraction signals might also be useful to measure the Berry phase (61), which has so far eluded detection in molecules.

### The Time-Resolved Two-Particle Diffraction Signal.

We now turn to diffraction signals generated from samples possessing intermolecular order such as crystals. We will display the two-molecule contribution without the structure factor **11**. The

Fig. 6 shows the diffraction pattern excluding the Bragg peaks as a function of the probe delay according to Eq. **18**. The diffraction pattern is dominated by the electronic ground state and the oscillating wave packet created by the Raman interaction with the pump pulse (for details, see *SI Appendix*, Fig. S1).

Fig. 7 depicts the time evolution of the *A*.

Note that the diffraction intensity not only depends on the population of the electronic state but also on the width of the nuclear wave packet. Since the spread of the wave packet increases with time, its intensity in the diffraction pattern decreases significantly.

Fig. 8 depicts the evolution of the *K* and *L*), including the electronic coherence and transition charge density, contributes to the signal. At 200 fs when the molecule crosses the intersection for the first time, the diffraction pattern contains a clear signature of the crossing. At ∼800 fs, the exited-state wave packet returns and passes the avoided crossing for a second time. The wave packet is now more spread out, resulting in a lower intensity (note that the pattern ∼800 fs is multiplied by a factor 10 to be visible). Between 1.0 and 1.1 ps, the wave packet returns to the Franck–Condon point. A strong coherence is created by the remaining wave packet in the electronic ground state and the returning nuclear wave packet in the excited state.

The inverse Fourier transform of Fig. 8*A* is shown in Fig. 8*B*. It gives a clear indication of the interatomic separations at which the electronic coherences are created: The features at ≈ 8 Å can be attributed to passing through the avoided crossing. The feature at ≈2 Å corresponds to the aforementioned revival event where the remaining ground-state wave packet overlaps with the returning wave packet in the excited state.

### The S 1 vs. S 2 Signals.

The single- and two-molecule contributions have generally different features. While only the two-molecule signal carries intermolecular information via the structure factor

Finally, we note that, since

## Summary and Future Outlook

In this work, we have presented a unified account of time-resolved X-ray scattering starting with the minimal coupling field-matter interaction Hamiltonian. We focused our attention on pulses off-resonant from the core transitions, where the radiative coupling is proportional to the electronic charge density. In ordered samples, the scattering amplitudes of the various molecules add coherently (Eq. **5**), yielding the Bragg peak pattern of the sample. Disorder causes a diffuse scattering due to fluctuations from the average structure that exists between the Bragg peaks as well, but that is also due to the coherent addition of scattering amplitudes from different molecules (Eq. **5**). These two-molecule terms offer heterodyne detection in time-resolved diffraction of excited-state charge densities. In contrast, single-molecule scattering, which is all that remains in a totally unordered sample such as a gas, contains no heterodyne interference between molecular ground- and excited-state electronic charge densities in contradiction to claims made in ref. 29.

In stationary X-ray diffraction experiments, the phase of the ground-state charge density in q-space is often determined by oversampling (2⇓–4). Once complete ground-state information is known, the excited-state amplitude and phase can be obtained in a heterodyne detection in samples with long-range order (13), while oversampling can again be used in gas-phase samples after subtracting off the ground-state charge density.

The present first-principles QED treatment further reveals the role of inelastic and mixed elastic–inelastic terms that are connected to the presence of electronic coherences and carry information on the spatial distribution of the electrons involved in valence molecular excitations. This includes ordinary diagonal as well as off-diagonal transition charge densities. Although weaker than elastic scattering, these terms may be observable with sufficiently bright pulses of duration comparable to electronic time scales.

We have demonstrated the various off-resonant scattering terms with the example of NaF, which possesses an avoided crossing between ionic

For simplicity, we have so far assumed that the X-ray pulse is tuned off-resonant from core transitions. The diffraction signal then solely monitors the ground state and valence excitations. Resonant pulses will probe core excitations as well, and such signals are dominated by the **2** and retaining only terms proportional to j, we obtain upon expanding to lowest order in an external probing field *SI Appendix*, we give expressions for resonant scattering from a nonstationary state prepared via a four-wave-mixing process as an example of a resonant scattering technique.

We further note that we have also not explicitly specified the molecular preparation process (gray area in Fig. 2 diagrams) and kept it general. A large variety of schemes for preparing the molecule in an excited-state population or coherence are possible and have been analyzed, including nonperturbative propagation in the presence of a strong field (44, 51) as well as direct dipole coupling and resonant (64, 65) or off-resonant Raman processes (68). These latter signals involve effective polarizabilities expressed with the current densities given by:

In another exciting future development, it should be possible to carry out multiple diffraction measurements on a single molecule and detect the various diffracted photons in coincidence. Several photon-scattering events can occur and be detected, thus leading to multiple photon-counting signals (39, 69, 70) that are sensitive to multipoint correlation functions of the charge-density operator that probe spontaneous charge-density fluctuations. For example, for a two-photon counting measurement, these signals are given by ref. 68

## Acknowledgments

This work was supported by Chemical Sciences, Geosciences, and Biosciences Division, Office of Basic Energy Sciences, Office of Science, US Department of Energy (DOE) Award DE-FG02-04ER15571; and National Science Foundation Grant CHE-1663822. M.K. was supported by the Alexander-von-Humboldt Foundation through the Feodor-Lynen program. K.B. was supported by the above-mentioned DOE grant.

## Footnotes

↵

^{1}K.B. and M.K. contributed equally to this work.↵

^{2}Present address: Department of Chemistry, University of California, Berkeley, CA 94720.↵

^{3}Present address: Department of Physics, Stockholm University, AlbaNova University Center, 10691 Stockholm, Sweden.- ↵
^{4}To whom correspondence should be addressed. Email: smukamel{at}uci.edu.

This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2015.

Author contributions: K.B., M.K., and J.R.R. designed research; K.B., M.K., J.R.R., and S.M. analyzed data; and K.B., M.K., J.R.R., and S.M. wrote the paper.

Reviewers: M.C., Ecole Polytechnique Fédérale de Lausanne; and T.E., Max Born Institut für Nichtlineare Optik und Kurzzeitspektroskopie.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1805335115/-/DCSupplemental.

Published under the PNAS license.

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