# Signature of Attochemical Quantum Interference upon Ionization and Excitation of an Electronic Wave Packet in Fluorobenzene

Anthony Ferté, Dane Austin, Allan S Johnson, Felicity Mcgrath, João Pedro Malhado, Jon P Marangos, Morgane Vacher

<figure><img src="image_1.png" /></figure>

## To cite this version:

Anthony Ferté, Dane Austin, Allan S Johnson, Felicity Mcgrath, João Pedro Malhado, et al.. Signature of Attochemical Quantum Interference upon Ionization and Excitation of an Electronic Wave Packet in Fluorobenzene. Physical Review Letters, 2024, 133 (20), pp.203201. 10.1103/physrevlett.133.203201. hal-04892444

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Signature of Attochemical Quantum Interference upon Ionization and Excitation of an Electronic Wave Packet in Fluorobenzene

Anthony Ferté,¹ Dane Austin,² Allan S. Johnson,² Felicity McGrath,²

João Pedro Malhado,³ Jon P. Marangos,² and Morgane Vacher¹,※

¹Nantes Université, CNRS, CEISAM, UMR 6230, F-44000 Nantes, France

²Quantum Optics and Laser Science Group, Blackett Laboratory, Imperial College London, London, United Kingdom

³Chemistry Department, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom

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(Received 12 December 2023; revised 5 April 2024; accepted 17 September 2024; published 12 November 2024)

Ultrashort pulses can excite or ionize molecules and populate coherent electronic wave packets, inducing complex dynamics. In this Letter, we simulate the coupled electron-nuclear dynamics upon ionization to different electronic wave packets of (deuterated) benzene and fluoro-benzene molecules, quantum mechanically and in full dimensionality. In fluoro-benzene, the calculations unravel both interstate and intrastate quantum interferences that leave clear signatures of attochemistry and charge-directed dynamics in the shape of the autocorrelation function. The latter are in agreement with experimental high-harmonic spectroscopy measurements of benzenes and fluoro-benzene.

DOI: 10.1103/PhysRevLett.133.203201

With ultrashort light pulses emerging [1,2], the time resolution of measurements is pushed to the intrinsic timescale of electron dynamics in molecules, i.e., the attosecond [3–6]. Exploiting the large pulse bandwidth to coherently excite several electronic states and forming a so-called electronic wave packet [7–9] unveiled the prospect to go beyond the possibilities of standard photochemistry. Coherent superpositions of electronic states can also be produced via strong field ionization [10,11]. In both cases, the interference between the components of an electronic wave packet can in principle be used to steer chemical reactivity. Such attochemical control, also called charge-directed reactivity, was demonstrated theoretically and experimentally in diatomics [12,13]. Extending it to polyatomic molecules is one of the current main prospects of attochemistry [14,15].

Experimentally, generating and observing such dynamics is challenging. Attosecond pump-probe schemes, although becoming possible [16–19], are arduous. As an alternative, high-harmonic generation spectroscopy (HHS) [20] has been successfully applied to probe the subfemtosecond coupled electron-nuclear dynamics on simple systems, retrieving autocorrelation functions [21–28]. Its main disadvantage is the presence of a strong field that may perturb the dynamics of interest, although field-free simulations

succeeded in reproducing experimental observables [22,23]. Recent developments of HHS allowed one to measure the dynamics in polyatomic molecules [29].

Simulating a chemical reaction induced by an electronic wave packet is also challenging [14,30]. Such works were performed with full quantum dynamics methods and focused on small molecules [31,32] or medium-size molecules in reduced dimensionality [33,34]. The models used are mostly suited to describe rigid motions and are thus difficult to apply to reactions such as dissociations or cis-trans isomerizations. Also, reduced dimensionality may not capture all relevant features of the dynamics. Other simulations employed mixed quantum-classical dynamics methods [35–38], not treating electronic coherence—a key property for attochemistry—accurately [30].

In this Letter, the coupled electron-nuclear dynamics upon ionization and excitation of electronic wave packets in benzene derivatives (Fig. 1) is simulated fully quantum mechanically and in full dimensionality. We focus on ionization, which can occur using attosecond pulses

<figure><img src="image_4.png" /><figcaption>FIG. 1. Singly occupied molecular orbitals of the (left) $\Psi_Q$ and (right) $\Psi_A$ cationic states of FBZ.</figcaption></figure>

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

<footnote>*Contact author: morgane.vacher@univ-nantes.fr</footnote>
(mainly produced in the extreme ultraviolet domain) or intense IR fields. Beyond validating the concept of charge-directed reactivity upon population of different superpositions of the two lowest cationic states of fluoro-benzene, the calculations show that several types of quantum interferences leave clear signatures in the autocorrelation function. Importantly, this quantity is accessible experimentally via HHS [29].

Benzene (BZ) and its derivatives are a formidable set of systems to investigate coupled electron-nuclear dynamics induced upon ionization by attosecond or subfemtosecond pulses [37,39–41] before electronic decoherence [42–49]. Indeed, ionization from the $\pi$ system of BZ leads to a conical intersection between the two lowest cationic states. The corresponding two resulting singly occupied orbitals are shown in Fig. 1, and the electronic states are said to be of quinoid ($\Psi_Q$) and antiquinoid ($\Psi_A$) characters, respectively. In fluoro-benzene (FBZ), a slight offset between the neutral structure and the electronic degeneracy point in the cation leads to a finite energy gap at the vertical ionization geometry: $\Delta E_{\text{vert}}^{\text{adia}} \approx 0.27$ eV.

In the present simulations of the postionization dynamics of BZ and FBZ, several initial electronic wave packets were considered: the pure diabatic states $\Psi_Q$ and $\Psi_A$, as well as both equally weighted, real-valued, in-phase, and opposite-phase superpositions, i.e., $(1/\sqrt{2})(\Psi_Q \pm \Psi_A)$. The dynamics was simulated using the single-set formalism of the direct-dynamics variational multiconfigurational Gaussian (DD-vMCG) method, the development of which started two decades ago in the QUANTICS package [50]. In this method, the molecular wave packet is expanded onto a set of variationally coupled time-dependent Gaussian basis functions (GBFs) that evolve quantum mechanically [30,51–53]. It combines the advantages of quantum dynamics accuracy and on-the-fly simulations in full dimensionality. The postionization dynamics of BZ and FBZ molecules were simulated for 10 fs using five and ten GBFs, respectively, to achieve convergence (see Supplemental Material [54]). Electronic structure calculations were performed at the complete active space self-consistent field (CASSCF) (5e, 6o) level via the widely used GAUSSIAN software [55] with an active space including the $\pi/\pi^*$ orbitals and using the 6-31G* basis set. Electronic state diabatization was performed using the regularization method [56,57]. It is noted that the present simulations do not include any external field.

Figure 2 shows the resulting average nuclear motion, i.e., the expectation value of the nuclear position analyzed in the branching space of BZ and FBZ, induced by the different wave packets considered. For both molecules, exciting a pure diabatic state leads to an initial motion oriented solely along the gradient difference vector with opposite directions, while excitation of mixed superpositions $(1/\sqrt{2})(\Psi_Q \pm \Psi_A)$ leads to an initial average motion mostly oriented along the derivative coupling vector, again with

<figure><img src="image_5.png" /><figcaption>FIG. 2. Average quantum nuclear motion over 10 fs analyzed in the branching space, upon ionization of (left) BZ and (right) FBZ molecules and initial population of different electronic wave packets. The large black circle indicates the conical intersection point, while the small black circle marks the vertical ionization geometry (superimposed in BZ). The branching space vectors are represented in the left figure.</figcaption></figure>

opposite directions. These observations are in line with previous simulations [30,35,58,59] validating the control over the molecular motion in the branching space achieved by tuning the initial electronic wave packet composition.

The aims of the present work are (i) to provide an experimental signature of such charge-directed dynamics and (ii) to give physical insights into this phenomenon, using the comparison between BZ and FBZ. More precisely, we explore features of the autocorrelation function,

$$ \mathcal{O}(t) = \langle \Psi^{\text{mol}}(0) | \Psi^{\text{mol}}(t) \rangle, \quad (1) $$

the overlap between the molecular wave packets $\Psi^{\text{mol}}$ at time $t$ and at $t=0$, i.e., the ionization time. Figure 3 reports the autocorrelation functions for both molecules upon excitation to the different electronic wave packets considered. The initial evolution of these autocorrelation functions was fitted using a Gaussian expression $\exp[-(t/\tau)^2/2]$. The decay times $\tau$ are reported in Table I.

In the BZ cation, both adiabatic states are degenerate at the neutral geometry due to symmetry. Any superposition of the two electronic states being a valid eigenstate, BZ is always excited to a pure adiabatic eigenstate. All four initial electronic wave packets yield very similar autocorrelation

<figure><img src="image_6.png" /><figcaption>FIG. 3. Modulus of the autocorrelation functions along the quantum dynamics induced upon ionization of BZ (dashed) and FBZ (solid) and excitation to different electronic wave packets.</figcaption></figure>
<table><caption>TABLE I. Decay times $\tau$ deduced from the fits of the early time evolution of the autocorrelation functions, upon ionization of BZ and FBZ and initial population of different electronic wave packets.</caption><thead><tr><th></th><th>Ψ<sub>Q</sub></th><th>Ψ<sub>A</sub></th><th>(1/√2)(Ψ<sub>Q</sub> + Ψ<sub>A</sub>)</th><th>(1/√2)(Ψ<sub>Q</sub> - Ψ<sub>A</sub>)</th></tr></thead><tbody><tr><td>FBZ</td><td>3.51 fs</td><td>3.56 fs</td><td>3.54 fs</td><td>3.55 fs</td></tr><tr><td>FBZ</td><td>3.05 fs</td><td>3.33 fs</td><td>2.64 fs</td><td>2.64 fs</td></tr></tbody></table>

functions (Fig. 3, dashed curves): It steadily decreases from 1 with a decay time of ≈3.5 fs and reaches a minimum value of about 0.4 around 6 fs. The predicted decay times are significantly longer than the sub-fs value reported in another theoretical work [60]. The absence of significant differences reflects the isotropy of the potential energy surfaces around the symmetry-required $\Psi_Q/\Psi_A$ conical intersection.

In contrast, in FBZ, the two adiabatic states are not degenerate at the vertical ionization geometry, and thus, a nonstationary coherent wave packet can be excited. All electronic wave packets yield autocorrelation functions that drop to significantly lower values (solid curves), almost reaching 0 upon excitation to $\Psi_A$. Except for the two equally weighted superpositions that are indistinguishable due to the symmetry of the derivative coupling, the initially excited electronic wave packets lead each to qualitatively different autocorrelation functions. In particular, a significantly faster decay is obtained in the case of the initially mixed electronic wave packets: 2.6 fs instead of 3.0–3.3 fs.

These predictions are compared with measurements of unaligned BZ and FBZ using the methodology reported previously [29]. In brief, 2D datasets of harmonic spectrum versus laser intensity were recorded. A physically transparent analysis, in the spirit of quantitative rescattering theory [61], was achieved by fitting the yield to the three-factor ansatz: $S(\omega, I_L) = A(\omega)B(I_L)C(\bar{\omega})$, where $\omega$ is the harmonic frequency, and $\bar{\omega}$ is the frequency normalized to the classical cutoff so that the latter occurs at $\bar{\omega} \approx 3.17$. The static factor $A(\omega)$ encodes the photorecombination dipole averaged over participating orbitals and molecular angles, as well as any uncompensated spectral response of the experimental apparatus. The ionization rate factor $B(I_L)$ encodes the angularly and orbital-averaged tunnel ionization rates. The dynamic factor $C(\bar{\omega})$ depends on the normalized frequency which has a one-to-one mapping to the trajectory time, provided short or long trajectories are selected [21,22,62]. In the present measurements, the short trajectories are selected. The phase matching condition in high-harmonic generation strongly favors the electron first returns [23]. This fitting technique implicitly takes into account the different cation potentials in the molecular factors A and B and reduces all dynamic processes, such as the nuclear motion and the electronic dynamics, into a single observable which can be expressed as a function of the time directly $C(t)$. This dynamical factor was shown to be proportional to the nuclear autocorrelation function

<figure><img src="image_7.png" /><figcaption>FIG. 4. Top: dynamical factors C extracted from the HHS of BZ and FBZ. Bottom: ratio of experimental C factors (black points) and of predicted autocorrelation functions upon ionization and excitation to different electronic wave packets (solid colored lines).</figcaption></figure>

(the generalized version of which includes the possibility of multiple cationic states) [22,23,29].

Figure 4 (top) presents the experimental C factors for BZ and FBZ. By extracting C, a proportional measure of the nuclear autocorrelation function is obtained. From these data, a decay constant of $\tau = 4 \pm 1$ fs was previously extracted for BZ [29], in excellent agreement with the present calculation. At later times, C falls more rapidly for FBZ. By determining the ratio of C between two molecules, the ratio of autocorrelation functions is obtained, free from the influence of the various differing static contributions which can otherwise obscure HHS. This is possible thanks to the cancellation of the returning electron flux [63]. Figure 4 (bottom) plots the ratio of the C factors for BZ and FBZ together with the ratio of predicted autocorrelation functions: Good agreement is found for the initial coherent excitation of the two cationic states (see also Supplemental Material [54]). Part of the discrepancy between theory and experiment may be explained by the absence of the field in the simulations.

To gain insights into the charge-directed dynamics phenomenon and in particular understand further the faster decay of the autocorrelation function in the case of mixed electronic wave packets, we analyze in more detail $O^{Q+A}(t)$. The system being in a superposition of the two diabatic states, the autocorrelation function is the coherent sum of two complex terms,

$$O^{Q+A}(t) = \underbrace{\sum_{i,j} C_i^{Q*}(0) C_j^Q(t) \times \langle \mathcal{X}_i(0) | \mathcal{X}_j(t) \rangle}_{\mathcal{O}^{Q+A}(t)} + \underbrace{\sum_{i,j} C_i^{A*}(0) C_j^A(t) \times \langle \mathcal{X}_i(0) | \mathcal{X}_j(t) \rangle}_{\mathcal{O}^{Q+A}(t)}, \quad (2)$$
<figure><img src="image_8.png" /><figcaption>FIG. 5. Modulus of $O^{Q+A}(t)$ the autocorrelation function of FBZ upon initial population of the $(1/\sqrt{2})(\Psi_Q + \Psi_A)$ wave packet (orange) and of its two diabatic components $O_Q^{Q+A}(t)$ (dashed purple) and $O_A^{Q+A}(t)$ (dashed blue). Note that they were doubled here for ease of representation. Incoherent sum $|O_Q^{Q+A}(t)| + |O_A^{Q+A}(t)|$ (black). Modulus of the autocorrelation functions upon initial population of the pure diabatic states $O^Q(t)$ (purple) and $O^A(t)$ (blue).</figcaption></figure>

where $\chi_i$ are the GBFs, and $C_i^{Q/A}$ the expansion coefficients of the molecular wave packet in the diabatic basis “Q/A” and GBF $\chi_i$.

Figure 5 decomposes the full autocorrelation function $O^{Q+A}(t)$ (orange) onto its two diabatic components $O_Q^{Q+A}(t)$ (dashed purple) and $O_A^{Q+A}(t)$ (dashed blue). Also reported is the incoherent sum $|O_Q^{Q+A}(t)| + |O_A^{Q+A}(t)|$ (black). The naive incoherent summation of both diabatic components (black) yields a result very different from the true autocorrelation function (orange): slower decay with no structure. This demonstrates that interferences between the two diabatic components are responsible for the accelerated decay of the autocorrelation function. This effect is named interstate interference hereafter. It is noted that the autocorrelation functions of the diabatic components are very similar to those of the pure diabatic states. In Supplemental Material [54] we use this to model the autocorrelation functions of arbitrary initial wave packets.

For both $O_Q^{Q+A}(t)$ terms to interfere, a phase difference has to accumulate along the dynamics. Importantly, this phase difference is independent of the initial phases of the two diabatic components. It only accounts for the phase difference accumulated during the dynamics which, over a short period of $\tau$ a.u., is approximately $\tau \times \Delta E_{\text{vert}}^{\text{adia}}$. In FBZ, the phase difference initially accumulates at the expected rate given the initial energy gap (see Supplemental Material [54]). Notably, a phase difference of $\pi$ that maximizes interference effects is reached just before 6 fs, which corresponds to the time at which the full autocorrelation function (Fig. 5, orange curve) presents a minimum. In the case of BZ, the two electronic states being degenerate at the vertical ionization geometry, the phase

difference accumulated throughout the dynamics remains extremely small.

After the interstate interference local minimum of $|O^{Q+A}(t)|$ in FBZ, a small increase can be observed. This originates from the antiquinoid diabatic component of the autocorrelation function $O_A^{Q+A}(t)$ (dashed blue) almost reaching zero around the 8.5 fs mark, thus suppressing the interstate interference.

In the following, the important decay of the full autocorrelation function upon excitation to the pure $\Psi_A$ diabatic state $O^A(t)$—a feature also present in $O_A^{Q+A}(t)$—is analyzed in detail. One of the reasons for it is the large diabatic population transfer: final quinoid population $\approx 0.46$. This also explains the direction reversal of the average nuclear motion along the gradient difference vector (Fig. 2, right). However, this alone does not explain the full structure of $O^A(t)$.

Figure 6 reports the autocorrelation function of FBZ upon initial population of the pure $\Psi_A$ diabatic state (light blue) and of its spatial components on either side of the conical intersection seam. More precisely, it is decomposed as

$$O^A(t) = O_+^A(t) + O_-^A(t), \quad (3)$$

where $O_+^A(t)$ and $O_-^A(t)$ are computed by only accounting for GBF whose final position is either on the same side of the conical intersection seam as the vertical ionization geometry, i.e., $E_A^{\text{diab}} - E_Q^{\text{diab}} > 0$ ($O_+^A$, blue) or on the opposite side: $E_A^{\text{diab}} - E_Q^{\text{diab}} < 0$ ($O_-^A$, red) (see Supplemental Material [54]). Around the 8 fs mark, both components have equal modulus but there is a phase difference (black) of about $\pi$ radian: The near cancellation of $O^A(t)$ stems from destructive interference between the space-restricted components $O_+^A(t)$ and $O_-^A(t)$ belonging to the same $\Psi_A$ diabatic state. This effect is named intrastate interference.

<figure><img src="image_9.png" /><figcaption>FIG. 6. Modulus of $O^A(t)$ the autocorrelation function of the FBZ cation upon initial population of the pure $\Psi_A$ diabatic state (light blue) and of its two space-restricted components $O_+^A(t)$ (blue) and $O_-^A(t)$ (red). Phase difference between the two space-restricted components of the autocorrelation function (black).</figcaption></figure>
In summary, we report quantum simulations of the dynamics induced in the BZ and FBZ molecules upon ionization and excitation to different electronic wave packets. In BZ, all considered initial electronic states lead to similar dynamics due to the isotropy of the potential energy surfaces. In stark contrast, the dynamics in FBZ exhibit different types of interference effects. In particular, interstate interference between the two diabatic components leads to an accelerated decay of the autocorrelation function upon excitation of a mixed electronic wave packet. Also, an intrastate interference leads to the almost cancellation of the autocorrelation function upon population of the pure higher lying diabatic state. This is due to the destructive interference between the components of the wave packet that crosses the conical intersection seam during the dynamics and the part that ultimately remains on the same side.

The predicted autocorrelation functions are in very good agreement with experimentally measured dynamics via HHS in both molecules, highlighting the sensitivity of autocorrelation functions to the composition of the initial electronic wave packet. Moreover, in the case of FBZ, this provides a clear signature of not only the ability to influence the induced molecular dynamics through electronic wave packets, but also of the coherent behavior and interference effects that may influence chemical dynamics on ultrafast timescales. The present simulations indicate the coherent excitation of several electronic states in the experiment. However, they also show that the current experimental signal-to-noise ratio does not allow us to draw any decisive conclusions as to the exact composition of the initial wave packet or the ensemble of initial wave packets (see also Supplemental Material [54]). We thus hope that the present study will motivate further experimental investigations of such quantum ultrafast dynamical effects with improved precision, perhaps with aligned sample and controlled polarization to affect the composition of the excited wave packet. Using a longer wavelength would also extend the experimental time window and allow direct observations of the structures in the autocorrelation functions due to interferences.

**Acknowledgments**—We thank the Région des Pays de la Loire who provided postdoctoral funding for A. F. The project is also partly funded by the European Union through ERC Grant No. 101040356 (M. V.). The views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. This work was performed using HPC resources from GENCI-IDRIS (Grant No. 101353) and CCIPL (Le centre de calcul intensif des Pays de la Loire).

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