Ultrafast structural dynamics of carbon–carbon single-bond rotation in transient radical species at non-equilibrium

Abstract

Bond rotation is an important phenomenon governing the fate of reactions. In particular, heterogeneously substituted ethane derivatives provide distinct structural conformations around the bond, empowering them as ideal systems for studying the rotation along carbon-containing single bonds. However, structural dynamics of ultrafast single-bond rotation, especially along C–C• bonds, have remained elusive as tracking the detailed changes in structural parameters during the rotational isomerization is challenging with conventional spectroscopic tools. Here, we employ femtosecond time-resolved X-ray liquidography to visualize the rotational isomerization between anti and gauche conformers of tetrafluoroiodoethyl radical (C₂F₄I•) and 1,2-tetrafluorodiiodoethane (C₂F₄I₂), simultaneously. The TRXL data captures perturbations in conformer ratios and structures of each reacting species, revealing that the rotational isomerization of C₂F₄I• and C₂F₄I₂ follows anti-to-gauche and gauche-to-anti paths with time constants of 1.2 ps and 26 ps, respectively. These findings also align with the computational predictions. This work offers an atomic-level insight into the kinetics and structural dynamics of single-bond rotation.

Introduction

The internal rotation of molecules is foundational in chemistry, governing the reactivity, stereochemistry, and fate of reactions¹. In particular, the rotation of single bonds containing carbon atoms plays an indispensable role in synthetic and biological chemistry. The rotational degree of freedom around carbon–carbon (C–C) single bonds provides flexibility to the three-dimensional orientation of atoms in a molecule, giving rise to the concept of conformational isomerism. The conformational isomerism determines the stereochemistry of the products in some key reactions^2,3 and affects the three-dimensional structure of proteins by altering interactions between residues^4,5,6,7. Therefore, understanding the chemical origins, structural dynamics, and potentials of manipulation for the internal single-bond rotation processes is vital for synthesizing complex molecules with desired functionalities, improving the adaptability of proteins, and facilitating the binding of substrates^{8,9,10,11,12,13}.

The conformer behaviors involved in single-bond rotation are decisively determined by dynamic equilibrium, where the forward and reverse transitions have equal rates. The dynamics of equilibration occurring between multiple conformers are settled by electronic, vibrational, and rotational eigenstates, as well as the conformational landscape of a molecule. The simplest molecule featuring rotational isomerism along C–C bond is ethane, where the rotation occurs between two conformers: staggered and eclipsed^1,14,15. Accordingly, ethane and its derivatives have gathered spotlights as models for studying single-bond rotation in optical spectroscopy^{16,17,18,19,20} and computational chemistry^{1,14,15,21,22,23}. However, non-substituted ethane has drawbacks as a candidate system to resolve the structural aspects of single-bond rotation. Notable experimental techniques directly sensitive to molecular structures, such as scattering or diffraction, are not very sensitive to extremely light atoms like hydrogens. Hence, the lack of structural sensitivity for the six hydrogen atoms around the C–C bond makes it challenging to retrieve the structural snapshots of bond rotation in ethane (see the sensitivity plot in Supplementary Fig. 2²⁴).

On the other hand, the two staggered conformers in heterogeneously substituted ethane derivatives (anti and gauche) render alternative avenues to study the C–C single-bond rotation, as they possess distinctive substituents which serve as fingerprints for monitoring single-bond rotation from structural perspectives. Early theoretical studies have shown that the rotational isomerization of n-butane, one of the simplest ethane derivatives, occurs with time constants ranging from 1 ps to 100 ps at room temperature under varying chemical environments^22,23. A two-dimensional infrared vibrational echo spectroscopy study on another ethane derivative (1-fluoro-2-isocyanato-ethane) revealed a time constant of 43 ps for the C–C bond rotation¹⁶. Subsequently, time-resolved infrared and nuclear magnetic resonance spectroscopy studies reported time constants ranging from 6.3 ps (along C–C bond)¹⁷, 47 ps (along C–C• bond)¹⁸, and 3 ps to 27 ps (along C–N bond)¹⁹ under various steric environments around the rotating bond. Nevertheless, time-resolved optical spectroscopy and nuclear magnetic resonance commonly highlight either ultrafast temporal resolution or atomic-level structural sensitivity, but hardly feature both simultaneously^25,26. Meanwhile, an accurate retrieval of structural dynamics requires a simultaneous interpretation of both realms. Accordingly, the ultrafast evolution of atomic-level molecular structures during single-bond rotation remains largely unexplored in solutions.

Time-resolved X-ray liquidography (TRXL), or time-resolved X-ray solution scattering (TRXSS), provides direct and detailed information on structural dynamics^27,28,29,30, complementing the insights obtained from optical spectroscopy. To fully leverage this structural sensitivity, we chose a derivative with three hydrogen atoms at each carbon site of ethane replaced with one iodine and two fluorine atoms. As iodine and fluorine atoms have larger scattering form factors, the structural dynamics of single-bond rotation can be directly visualized by tracing how these substituted atoms move. The photodynamics of this molecule, 1,2-tetrafluorodiiodoethane (C₂F₄I₂), start with the dissociation of one carbon–halogen bond, thereby breaking the symmetry between the two carbon atoms (one carbon becomes CF₂I and the other becomes CF₂•) and making the rotational isomerization along C–C• bond in C₂F₄I• easily resolvable.

Each of C₂F₄I₂ and C₂F₄I• has two stable conformers, anti and gauche, whose population ratio can change after excitation. In fact, a study using density functional theory (DFT) calculations suggested that anti and gauche conformers of C₂F₄I₂ exhibit different absorption cross-sections. For example, in one computational study^31,32, the relative anti-to-gauche ratio of the excitation probability to the electronic states accessible by 267 nm photon (¹B_u in anti, ¹B(1) in gauche) ranges from 80:20 to 89:11, indicating that the excitation of anti-C₂F₄I₂ dominates over gauche-C₂F₄I₂. These excitation ratios from C₂F₄I₂ to C₂F₄I• are determined by the electronic properties of each C₂F₄I₂ conformer. Meanwhile, the dynamic equilibrium ratios of C₂F₄I₂ and C₂F₄I• are determined by the thermodynamic properties, implying that the excitation ratio needs not to be the same as the equilibrium ratio. These considerations suggest the possibility to trigger and capture the ultrafast single-bond rotation along C–C and C–C• bonds by observing the re-equilibration of photoinduced perturbations in anti-to-gauche ratios of C₂F₄I₂ and C₂F₄I• (Fig. 1). Although the photodissociation of C₂F₄I₂ has been extensively studied in gas^33,34,35 and solution phases^36,37, tracking the time-dependent anti-to-gauche ratios during the rotational dynamics has not been fully achieved.

Fig. 1: Photoinduced structural dynamics of 1,2-diiodotetrafluoroethane (C₂F₄I₂).

a Two-step photoinduced dissociation pathway of C₂F₄I₂. It is known that C₂F₄I₂ absorbs an ultraviolet photon at 267 nm to undergo the two-step I• dissociation to form the radical intermediate (C₂F₄I•) and the final product (C₂F₄). In cyclohexane, the secondary dissociation from C₂F₄I• to C₂F₄ is known to occur with a time constant of 292 ps, but the photodynamics before 100 ps remains unknown (emphasized with red bar). b The unknown ultrafast structural dynamics of C₂F₄I₂, depicting possible species involved in the reaction. Before photoexcitation, the C₂F₄I₂ molecules present as a mixture of anti (blue) and gauche (green) conformers. Photoinduced primary iodine dissociation depletes each conformer by different portions (purple-bordered box) depending on absorption cross-section and quantum yields. This also leaves the remaining conformational mixture C₂F₄I₂ whose anti-to-gauche ratio is perturbed from the dynamic equilibrium (magenta-bordered box). c, d The rotational isomerization of (c) C₂F₄I₂ and (d) C₂F₄I•. These perturbed anti-to-gauche ratios recover their dynamic equilibrium through the rotational isomerization of (c) C₂F₄I₂ and (d) C₂F₄I•, while preserving the overall concentration depicted by the area of solid-bordered boxes in (b).

In this work, we conduct a femtosecond TRXL (fs-TRXL) experiment to investigate the structural dynamics of single-bond rotation occurring within 100 ps after photoexcitation. The experimental setup for fs-TRXL is illustrated in Supplementary Fig. 1. Since the fraction of molecules participating in the rotational isomerization is far smaller than those undergoing the dissociation path, we utilize ultrabright X-ray pulses from an X-ray free-electron laser facility to capture subtle transient signals hidden beneath the dissociation dynamics. The high sensitivity of TRXL on heavy atom positions (see Supplementary Fig. 2 for the sensitivity plot²⁴) enables the accurate retrieval of structural motions during rotational isomerization along single bonds (C–C and C–C•) with sub-Å precisions. Moreover, the femtosecond resolution of fs-TRXL enables the capture of sub-ps kinetics, which is particularly relevant to the rotation around C–C• bonds. By analyzing the fs-TRXL data in both reciprocal and real space, we successfully reveal the time-dependent structural dynamics, setting the stage for a detailed exploration of the single-bond rotation.

Results and discussion

Femtosecond TRXL of C₂F₄I₂

To investigate the photoinduced structural dynamics of rotation and dissociation along carbon-containing single bonds, we measured time-resolved two-dimensional X-ray scattering images of the C₂F₄I₂ solution. These scattering images were decomposed into one-dimensional isotropic and anisotropic curves by utilizing the anisotropic signal decomposition method (see section 1-2 of Supplementary Information (SI) for details)³⁸. The one-dimensional isotropic X-ray scattering curves (S₀) collected after the photoexcitation were subtracted by those measured without the laser to obtain difference isotropic X-ray scattering curves. The difference isotropic X-ray scattering curves originate from the atomic pairs (1) within the solute, (2) between the solute and solvent (also known as the cage term) and (3) between the solvent molecules. Here, only the first two contributions (solute-only and cage) encrypt the structural dynamics of the solute and are termed “solute-related”. We quantitatively eliminated the contributions of (3), ΔS₀^heat(q, t), which originate from solvent heating (thus labeled as the solvent term), from the difference isotropic X-ray scattering curves through the projection to extract the perpendicular component (PEPC) method³⁹, leaving only the solute-related data (qΔS₀(q, t), see Fig. 2a). The time-dependent changes of qΔS₀(q, t) are indicative for the solute-related structural dynamics. We constructed a comprehensive structural dynamic model, whose details are discussed throughout the following paragraphs, to disclose the origins of these features. The simulated TRXL curves (qΔS₀′(q, t), see Fig. 2b) coincide well with the experimental curves.

Fig. 2: Kinetics extracted from femtosecond time-resolved X-ray liquidography (fs-TRXL) data.

a The isotropic TRXL data (qΔS₀) of C₂F₄I₂ in cyclohexane. b The simulated TRXL data, qΔS₀′. Isotropic TRXL data can be attributed to the molecular structural dynamics, such as primary and secondary dissociation of I• and rotational isomerization. c The five significant right singular vectors (RSVs, circles with vertical error bars) from the singular value decomposition (SVD) analysis of the experimental data. The error bars were estimated using the method described in Section 2-3 of the SI, where the LSVs in Supplementary Fig. 5a were used to obtain the time-dependent coefficients (S×RSV) as parameterized variables with standard errors. We conducted a global fit analysis with the sum of four exponentials, each with time constants of 130 fs, 1.2 ps, 26 ps (yellow shades), and 292 ps (grey shades to imply that the last dynamics happens slowly after the latest time delay), convoluted with a Gaussian IRF function with a width of 172 fs.

Rotational isomerization along C–C and C–C• bonds

To extract the kinetic information of the solute-related dynamics, we applied singular value decomposition (SVD)⁴⁰ on the TRXL curves. SVD decomposes the data into a product of left singular vectors (LSV(q)), singular values (S), and right singular vectors (RSV(t)). Here, LSV(q) contains the representative signals in q-space that encode the structural information of each chemical species, RSV(t) describes their temporal evolution, and S quantifies the relative contribution of each rank to the data (see Methods and section 1-6 of SI for details). From SVD (in Supplementary Fig. 5), we identified a total of five major components participating in the photodynamics of C₂F₄I₂. To extract the solute-related kinetics, the corresponding five RSVs were globally fitted with a sum of four exponential functions, characterized by the time constants of 130 ± 50 fs, 1.2 ± 0.4 ps, 26 ± 2.4 ps, and 292 ± 136 ps, convoluted with an instrument response function (IRF) with a full-width at half maximum (FWHM) of 172 ± 47 fs representing the temporal resolution of the experiment (Fig. 2c).

To assess the dynamic origins of these time constants, we analyzed the decay-associated difference scattering curves (DADS(q)), akin to a famous concept of decay-associated spectra in time-resolved spectroscopy. Here, each DADS except for the last one (DADS_k(q)) stands for the difference between scattering curves from the solution before and after the dynamics, S_before – S_after, occurring with the time constant of τ_k, while the last DADS accounts for the difference between initial and final states, S_final – S_initial (see section 3-2 of SI). The boldface indicates the matrix form; for instance, the k^th column vector of DADS(q) is DADS_k(q). We constructed the matrix of normalized decays (C_DADS′(t)) and extracted DADS(q) such that the matrix product DADS(q) × C_DADS′(t)^T, which equals the simulated curves (ΔS₀′(q, t)), best agrees with ΔS₀(q, t). We extracted DADSs for a total of four time constants (130 fs, 1.2 ps, 26 ps, and 292 ps).

To attribute the dynamic origins to each DADS_k(q), we simulated difference isotropic scattering curves for various candidate structural transitions (Supplementary Fig. 6) that could take place upon the photoexcitation of C₂F₄I₂. For example, DADS₂(q) was compared with the simulated difference X-ray scattering curves corresponding to (i) the rotational isomerization of C₂F₄I₂, (ii) the rotational isomerization of C₂F₄I•, (iii) the dissociation of two iodine atoms from C₂F₄I₂ to form C₂F₄ + 2I•, and (iv) the formation of a hypothetical isomer (I₂…C₂F₄, suggested in one previous study⁴¹) from either C₂F₄I₂, C₂F₄I• + I•, or C₂F₄ + 2I•. We identified the best candidates, which have the smallest reduced chi-square (χ_ν²) values, to describe the four DADSs (Fig. 3a). χ_ν² is widely utilized as a statistical criterion to evaluate the fitness of models^26,27,42, whose details can be found in section 3-1 of SI. The comparison of each DADS with all possible pathways is demonstrated in Supplementary Figs. 8–10. Based on the statistical considerations, we identified the following candidates that best describe the DADSs: (1) the primary photodissociation of an anti and gauche mixture of C₂F₄I₂ to a perturbed mixture of C₂F₄I• (for DADS₁(q) with 130 fs), (2) the anti-to-gauche rotational isomerization of C₂F₄I• (for DADS₂(q) with 1.2 ps), (3) the gauche-to-anti rotational isomerization of C₂F₄I₂ (for DADS₃(q) with 26 ps), and (4) the secondary dissociation of C₂F₄I• to C₂F₄ and I• (for DADS₄(q) with 292 ps). These “correct” pathways are summarized in Fig. 3a. We confirmed the statistical significance of each dynamic pathway by analyzing the changes in discrepancies between experimental and simulated TRXL curves when each pathway was excluded the model (Supplementary Fig. 11).

Fig. 3: Ultrafast structural dynamics of C–C and C–C• bond rotation.

a The reaction processes occurring in the reacting solution of C₂F₄I₂ within 100 ps after photoexcitation, corresponding to the time constants of 130 fs, 1.2 ps, 26 ps, and 292 ps. b The experimental decay-associated difference scattering curves (qDADS(q), black lines) corresponding to the four time constants, superimposed with respective simulated DADSs (qDADS′(q), red lines). We modeled various difference scattering curves corresponding to the candidate transition processes that can occur upon photoexcitation of C₂F₄I₂ (see Supplementary Figs. 8–10 for details) and identified that the four time constants are best attributable to the structural transitions shown in (a). The experimental error and fit residuals are indicated with vertical bars and blue lines, respectively. c The r-space representation of the experimental (black) and simulated (red) DADSs. In the 2^nd DADS, interatomic distances in the anti-C₂F₄I• (light blue lines, r_FI^a = 3.5 Å and 3.8 Å) and gauche-C₂F₄I• (light green lines, r_FI^g = 3.3 Å and 4.3 Å) vanish and emerge, respectively, implying the anti-to-gauche isomerization of the radical. Likewise, in the 3^rd DADS, interatomic distances in the anti-C₂F₄I₂ (blue lines, r_II^A = 5.0 Å, r_FI^A = 2.9 Å and 3.2 Å) and gauche-C₂F₄I₂ (green lines, r_II^G = 4.0 Å, r_FI^G = 2.8 Å and 3.4 Å) emerge and vanish, respectively, supporting the gauche-to-anti isomerization of C₂F₄I₂. We note that only the contributions of solute-related dynamics to the scattering signal are shown in (b) and (c). d Structural parameters of anti-C₂F₄I₂, gauche-C₂F₄I₂, anti-C₂F₄I•, and gauche-C₂F₄I• used for the iterative optimization. The simulated curves in (b) and (c) were generated using these optimized structural parameters. e The time-dependent concentration profiles with respect to the four components: anti-C₂F₄I• (red), gauche-C₂F₄I• (blue), the gauche-to-anti rotation of C₂F₄I₂ (purple), and C₂F₄ (gray). The error bars were estimated using the method described in section 2-3 of the SI, where the SADSs in Supplementary Fig. 12b were used to obtain the concentration profiles as parameterized variables with standard errors. We note that the third component accounts for the time-dependent concentration change of both anti-C₂F₄I₂ and gauche-C₂F₄I₂ simultaneously, as detailed in the methods section. In (e), the results of linear combination fit (LCF) and the least-χ² optimization based on the kinetic modeling are shown with bars and solid lines, respectively.

The integrated analysis of DADS_k(q)s with the previously assigned time constants shed light on the comprehensive structural dynamics. Firstly, upon photoexcitation, the primary I• dissociation occurs in both C₂F₄I₂ conformers with the apparent time constant of 130 fs (DADS₁(q)). We use the term “apparent” to emphasize that the ultrafast I• dissociation from C₂F₄I₂ in fact occurs through a complicated mechanism rather than a mere exponential kinetics. This is already hinted in the discrepancy between experimental and simulated RSVs (especially in the second RSV, see section 1-5 and Supplementary Fig. 5 of SI). We described this ultrafast temporal range by considering coherent and nonadiabatic real-time atomic motions detailed in section 5 and Supplementary Fig. 14 of SI. Nevertheless, DADS₁(q) can approximate the changes in scattering curve associated with the transition from the two conformers of C₂F₄I₂ to the transient mixture of C₂F₄I₂ and C₂F₄I• at which the coherent and nonadiabatic atomic motions—or the transition occurring with the time constant of 130 fs in the exponential kinetics picture—complete. In other words, while we need an in-depth analysis (in section 5 of SI) to describe how the intricate sub-ps dynamics occurs, the changes in the scattering curve before and after the ultrafast dissociation are incorporated in DADS₁(q). Once the dynamics related to DADS₁ completes, the amount of dissociated anti and gauche conformers are initially transferred intact to the anti and gauche C₂F₄I• mixture. However, the anti-to-gauche ratios of both remaining C₂F₄I₂ and the newly-formed C₂F₄I• immediately after the primary photodissociation differ from those in dynamic equilibrium. Thus, the anti and gauche C₂F₄I• approach their dynamic equilibrium via anti-to-gauche isomerization along C–C• bond with the time constant of 1.2 ps (DADS₂(q)). The non-dissociated C₂F₄I₂ mixture also returns to the dynamic equilibrium via gauche-to-anti isomerization along C–C bond with the time constant of 26 ps (DADS₃(q)). Finally, secondary dissociation occurs in 30% of the equilibrated C₂F₄I• mixture with a time constant of 292 ps, resulting in a 70:30 mixture of C₂F₄I• and C₂F₄ (DADS₄(q)), consistent with the previous ps-TRXL study³⁷.

In general, the observed rotational isomerization time constants of 1.2 ps and 26 ps are comparable to those reported in several previous studies, which range from a few picoseconds to tens of picoseconds. However, we note that the time constant of 1.2 ps for the C–C• bond is drastically faster than the time constant of 26 ps for the C–C bond. This discrepancy may arise from different distributions of vibrational and rotational states contributing distinctively to the pre-exponential factor, and the energetically hot nature of C₂F₄I• (see section 7 of SI for details). To theoretically validate the observed kinetics, we applied Rice-Ramsperger-Kassel-Marcus (RRKM) theory^43,44,45 to estimate the time constants for the observed rotational motions. To do this, we first calculated the shape of the potential energy surface (PES) with respect to the dihedral angle (φ_dihed) for C₂F₄I₂ and C₂F₄I• and identified two transition states (T1 and T2) along φ_dihed (Fig. 4a and Supplementary Fig. 16). Here, it becomes evident that the rotational isomerization of both C₂F₄I₂ and C₂F₄I• proceeds through the T2 state (located between anti and gauche). To quantitatively assess the time constants, we computed the molecular geometry, free energies of ground and excited states, vibrational frequencies, and rotational moments of inertia for the three conformers (anti, gauche, and T2) of C₂F₄I₂ and C₂F₄I•. Incorporating these data into RRKM theory provided the time constants of 20.2 ps for the gauche-to-T2-to-anti rotation of C₂F₄I₂ and 1.15 ps for the anti-to-T2-to-gauche rotation of C₂F₄I•. These estimated values show close accordance with the experimental time constants of 26 ps and 1.2 ps (see Table 1).

Fig. 4: Photodynamics of C₂F₄I₂ as revealed by time-resolved X-ray liquidography.

a A schematic diagram of primary iodine dissociation and rotational isomerization on the potential energy surface (PES) with respect to r_CI and φ_dihed. UV photon irradiation majorly excites anti-C₂F₄I₂ molecules to the Franck–Condon region (red arrow). Thee excited molecules (denoted with an asterisk) rapidly dissociate into anti-C₂F₄I• and I• (blue arrow). Along the PES of C₂F₄I• (magenta coordinate), an excess amount of anti-C₂F₄I• molecules undergo anti-to-gauche rotational isomerization (magenta arrow), overcoming the energy barrier of 3.4 kcal/mol with a time constant of 1.2 ps. Meanwhile, along the PES of C₂F₄I₂ (violet coordinate), a depleted portion of anti-C₂F₄I₂ molecules re-equilibrate by gauche-to-anti rotational isomerization of C₂F₄I₂ (violet arrow), surpassing the energy barrier of 4.0 kcal/mol with a time constant of 26 ps. We note that, at the initial step, only the anti-to-anti excitation is depicted in the diagram for simplicity. b Time-dependent concentrations of anti-C₂F₄I₂, gauche-C₂F₄I₂, anti-C₂F₄I•, and gauche-C₂F₄I•. Upon photodissociation, 5% of anti (blue) and 2% of gauche (green) C₂F₄I₂ convert to their respective conformers of C₂F₄I•, perturbing the anti-to-gauche ratios of both C₂F₄I₂ (to 45.0 mM:12.4 mM = 78:22) and C₂F₄I• (to 2.3 mM:0.2 mM = 91:9). Subsequently, 0.5 mM of anti-C₂F₄I• converts to gauche-C₂F₄I• with a time constant of 1.2 ps (upper-right panel), converging to a dynamic equilibrium anti-to-gauche ratio of C₂F₄I• (to 1.8 mM:0.7 mM = 71:29). Likewise, 0.3 mM of gauche-C₂F₄I₂ isomerizes to anti-C₂F₄I• with a time constant of 26 ps (lower-right panel), recovering a dynamic equilibrium (to 45.3 mM:12.1 mM = 79:21). Finally, 30% of the equilibrated C₂F₄I• undergoes secondary dissociation into the product (C₂F₄) with a time constant of 292 ps, while the remaining C₂F₄I• persist up to 100 ps (as can be seen in Fig. 3a–c).

Table 1 Experimental and theoretical kinetic constants assigned to structural dynamics

Structural dynamics of rotational isomerization

To fully leverage the high sensitivity of TRXL on molecular structures, based on the sensitivity plot²⁴ (see Supplementary Fig. 2), we iteratively optimized four selected structural parameters—the C–I distance (r_CI), the CCI angle (θ_CCI), the planar angle between C–I axis and FCF plane (θ_pl), and φ_dihed—describing the molecular geometry of anti-C₂F₄I₂, gauche-C₂F₄I₂, anti-C₂F₄I• and gauche-C₂F₄I•. Moreover, we introduced four concentration-related parameters to account for population dynamics of excitation and equilibration. Using these structural and concentration-related parameters, the simulated DADSs (DADS′(q)) were computed and superimposed with DADS(q) (Fig. 3b). This rigorous analysis allows retrieving structural (Table 2) and concentration-related (Table 3) parameters for each reacting species. The signs of the first three DADSs were reversed so that the negative and positive signs match with the “depletion” and “formation” of the species. The methodological details are described in the “Structure refinement” section of Methods and section 3 of SI.

Table 2 Optimized structural parameters of the anti and gauche conformers of the two reacting species, C₂F₄I₂ and C₂F₄I•

Table 3 Concentration-related parameters of the photodissociation and rotational isomerization dynamics

TRXL is renowned for its ability to visualize structural changes in real space with sub-Å precision. In real (r) space, the formation and depletion of interatomic pair distances can be more directly visualized by positive and negative signs of the oscillating r-space intensities. To do this, both DADS′(q) and DADS(q) in reciprocal space were Fourier sine transformed to the real space (DADS′(r) and DADS(r), see Fig. 3c). In DADS₂(r), the anti-to-gauche rotation of C₂F₄I• is depicted by the depletions of interatomic distances in anti-C₂F₄I• (r_FI^a = 3.54 Å and 3.76 Å) and the formations in gauche-C₂F₄I• (r_FI^g = 3.28 Å and 4.25 Å). Likewise, in DADS₃(r), the gauche-to-anti rotation of C₂F₄I₂ is expressed by the depletions of interatomic distances in gauche-C₂F₄I₂ (r_II^G = 4.00 Å, r_FI^G = 2.76 Å and 3.43 Å) and the formations in anti-C₂F₄I₂ (r_II^A = 4.99 Å, r_FI^A = 2.87 Å and 3.20 Å). DADS₁(r) and DADS₄(r) show a combination of formation and depletion, as both conformers of C₂F₄I₂ and C₂F₄I• contribute to the primary and secondary iodine dissociation dynamics.

Time-dependent population dynamics revealed by TRXL

To retrieve the time-dependent concentrations of each component, we conducted the linear combination fit (LCF) analysis. From the combined analysis of SVD and DADS, we found five major species (anti-C₂F₄I•, gauche-C₂F₄I•, anti-C₂F₄I₂, gauche-C₂F₄I₂, and C₂F₄) contributing to ΔS₀(q, t). The LCF results (in Fig. 3e) highlight both gauche-to-anti rotation of C₂F₄I₂ and anti-to-gauche rotation of C₂F₄I• that take place with the time constant of 26 ps and 1.2 ps, respectively. These concentration profiles align well with the kinetics identified from the DADS analysis, validating our model and optimized structures.

The optimized concentration-related parameters (Table 3) and the LCF results (Fig. 3e) combined with the theoretical considerations provide a comprehensive picture of single-bond rotation dynamics. Initially, in the 60 mM C₂F₄I₂ solution, both conformers comprise the dynamic equilibrium with the anti-to-gauche ratio of 79( ± 4.8):21. The 267 nm photon activates the antibonding orbitals along the C–I bond in both conformers, exciting the molecule to the Franck-Condon state. Within ultrafast time, the intersystem crossing occurs to the dissociative states (see Supplementary Fig. 15), where one C–I bond ultimately dissociates with the apparent time constant of 130 fs. The total amount of excited molecules is 6.8 mM, where 4.3 mM relaxes back to the ground state and 2.5 mM follows the dissociative path (see section 6-4 and Supplementary Fig. 17 of SI). When the C–I bond completely dissociates, 5% of anti-C₂F₄I₂ and 2% of gauche-C₂F₄I₂ convert to their corresponding C₂F₄I• conformers. At this point, the C₂F₄I• mixture contains 2.3 mM of anti-C₂F₄I• and 0.2 mM of gauche-C₂F₄I•, with an anti-to-gauche ratio of 91( ± 11):9. Soon, the C₂F₄I• mixture reaches dynamic equilibrium with a time constant of 1.2 ps, altering the radical composition to 1.8 mM of anti-C₂F₄I• and 0.7 mM of gauche-C₂F₄I•, leading to an equilibrium anti-to-gauche ratio of 71( ± 8.9):29. Meanwhile, the unbalanced depletion of C₂F₄I₂ leaves 45.0 mM of anti-C₂F₄I₂ and 12.4 mM of gauche-C₂F₄I₂, modifying the anti-to-gauche ratio of C₂F₄I₂ to 78( ± 3.6):22. This perturbed mixture returns to dynamic equilibrium with a time constant of 26 ps, as captured by the conversion of 0.3 mM of gauche-C₂F₄I₂ to anti-C₂F₄I₂. The relatively larger gauche fraction of C₂F₄I• compared to C₂F₄I₂ at dynamic equilibrium implies that the free energy difference between the two conformers is smaller in C₂F₄I• than in C₂F₄I₂ (see Supplementary Fig. 16). Finally, 30% of the equilibrated C₂F₄I• undergoes secondary dissociation to form C₂F₄ with a time constant of 292 ps, consistent with the previous ps-TRXL results. A comprehensive overview of population dynamics is presented in Fig. 4.

Implications of reconstructing rotational isomerization from structural perspectives

In this work, we designed an experiment using an ultrashort laser pulse to transiently generate chemical environments far from equilibrium. We applied fs-TRXL on an illustrative model system of C₂F₄I₂ to elucidate the C–C and C–C• single-bond rotation dynamics coupled with rotational isomerization between multiple conformers. Using a UV pump laser pulse, the non-equilibrium states of C₂F₄I₂ and C₂F₄I• were prepared to allow several unique observations on their journey back to dynamic equilibria. For example, the rotational isomerization among the non-photoreacted molecules could be captured in this work by perturbing the mixture of C₂F₄I₂, in sharp contrast to the typical pump-probe experiments where only the dynamics starting from the photoexcited molecules could be investigated. Moreover, the non-equilibrium mixture of C₂F₄I• features the single-bond rotation along the C–C• bond with the exceptionally faster time constant (1.2 ps) than typical values in single-bond rotation kinetics. Our analysis involved a systematic extraction of major components from the TRXL data, followed by a numerical comparison with various simulated models to account for the structural origins of the dissociation and rotational dynamics. We identified how the population and structure of the conformers, as well as their relative fractions governing each reaction step, change over time. These experimental findings are also well supported by quantum calculations performed at various levels. As a final remark, we note that traditional equilibrium assumptions often do not hold for chemical and biological processes occurring far from equilibrium. Nevertheless, direct observations of such phenomena, particularly concerning rotational coherences, remain in their infancy. In this regard, the insights from this study will pave a way for a deeper comprehension of ultrafast dynamics related to internal bond rotation, such as the coherent motions associated with re-equilibration from states far from dynamic equilibrium.

Methods

Time-resolved X-ray liquidography at PAL-XFEL

We prepared a 60 mM solution of 1,2-tetrafluorodiiodoethane (C₂F₄I₂; 97%, SynQuest) in cyclohexane (C₆H₁₂; Aldrich, 99.9%), and used it for a femtosecond time-resolved X-ray liquidography (fs-TRXL) measurement. We also conducted a separate fs-TRXL measurement with a 6.4 mM solution of 4-bromo-4′-(N,N-diethylamino)-azobenzene (HANCHEM, 99.9%), also referred to as Azo-Br, dissolved in cyclohexane (C₆H₁₂; Aldrich, 99.9%) to elucidate the effect of the response of bulk solvent (see section 2-3 of SI). The fs-TRXL experiments were carried out at the XSS-FXL beamline of PAL-XFEL in Korea.

In the experiment, the C₂F₄I₂ solution was excited with a femtosecond laser pulse at 267 nm to initiate the photodynamics. After a selected time delay, the X-ray scattering images, which encrypt the structural dynamics of the sample, were generated by the femtosecond X-ray pulses from the XFEL. The 267 nm optical laser pulses, generated as the third-harmonics of a Ti:Sapphire laser, were focused to a 132 × 132 μm² spot size (full-width-at-half-maximum, FWHM) with a fluence of 2.20 mJ/mm² at the sample position. The temporal width of the optical laser pulse was 130 fs (FWHM). The X-ray pulses had a narrow bandwidth (ΔE/E ≃ 0.15%) centered at 12.7 keV and were delivered at a repetition rate of 30 Hz. These X-ray pulses were focused to a spot size of 23 × 19 μm² (FWHM) at the sample position. The temporal width of the X-ray pulse was less than 50 fs (FWHM). The optical pump and X-ray probe were spatiotemporally overlapped at the stable position of the liquid jet, with a crossing angle of 10˚. The temporal widths of optical laser and X-ray pulses, the velocity mismatch caused by the width of the capillary jet, and the timing jitter altogether decide the temporal resolution of the experiment, expressed by instrument response function (IRF). We determined the IRF value of 172 ± 47 fs from the data analysis.

To ensure a continuous supply of fresh molecules at the interacting point, the sample solution was circulated through a 100-µm-thick capillary nozzle at a linear velocity faster than 3 m/s. Also, the sample solution was refreshed every two to three hours to maintain the integrity of the sample and to exclude the effects of irreversible products and radiation damages during the experiment. The X-ray scattering images were collected on the Rayonix MX225-HS area detector located 39 mm downstream of the circulating jet position. We accurately calibrated the X-ray beam center position and sample-to-detector distance by measuring the X-ray diffraction pattern of the reference sample, lanthanum hexabromide (LaB₆), and comparing this pattern with the known diffraction peak positions.

The TRXL images were collected at various time delays ranging from −0.3 ps to 3 ps with a linear step of 50 fs, plus an additional 31 log-scaled points per decade up to 100 ps, resulting in a total of 98 time points. To maintain the temporal stability between the laser and X-ray pulses, we employed a timing feedback tool in the PAL-XFEL. To improve the signal-to-noise ratio of the data, a number of images ( ~ 8500 images) were repeatedly collected and averaged at each time delay. A typical TRXL experimental setup is depicted in Supplementary Fig. 1. Further details on the TRXL experiment can be found in previous publications^{26,27,28,29,30,37,46}.

Reducing the data to ΔS₀(q, t)

The real space vectors, referenced at the X-ray beam center on the detector plane, were converted into reciprocal space vectors, whose magnitudes are defined by q = (4π/λ)sin(θ), where λ and 2θ denote the X-ray wavelength and scattering angle, respectively. We performed a series of corrections on the measured scattering images to address four crucial factors: (1) the solid angle of each detector pixel, (2) the polarization direction of the X-rays, (3) the alignment of the detector plane, and (4) attenuation arising from the phosphor screen of the detector. We used pyFAI⁴⁷, a well-known python library for X-ray scattering and diffraction image processing, to perform these corrections. These corrected images were converted into the static 1D isotropic scattering curves (S₀) by removing the anisotropic component (S₂). Afterwards, we normalized the static S₀ curves by dividing them with the sum of scattering intensities within the q range of 3–6.5 Å⁻¹. These normalized S₀ curves were further scaled to properly represent the absolute X-ray scattering signal from one solute molecule. The S₀ curves taken without the laser (S₀(q, off)) were subtracted from those taken at a time delay t (S₀(q, t)) to generate the difference X-ray scattering data (ΔS₀^raw(q, t)) encrypting the structural dynamics information. Details of the initial data processing scheme are provided in section 1 of SI.

Determining the number of intermediates and kinetic constants

We decomposed ΔS₀^raw(q, t) into the solute-related components, ΔS₀(q, t), and the heat-related components, ΔS₀^heat(q, t), by implementing the projection to extract the perpendicular component (PEPC) method. The solvent heating and artifact signals are represented with H(q), a matrix obtained from the TRXL data of Azo-Br. The retrieval of H(q) and the assessment of the temperature rise of the solution are detailed in section 1-3 of SI. ΔS₀^raw(q, t) and ΔS₀^heat(q, t) are demonstrated in Supplementary Figs. 3–4.

To determine the number of intermediates and kinetic constants, we employed singular value decomposition (SVD), an algebraic technique that decomposes the data into a product of left singular vectors (LSV(q), time-independent components), right singular vectors (RSV(t), temporal evolutions) singular values (S, relative contribution of each rank). The SVD analysis of ΔS₀(q, t) revealed five components contributing to ΔS₀(q, t) (Supplementary Fig. 5). These five RSVs were globally fitted by a sum of multiexponential functions convoluted by a Gaussian-shaped IRF with w = 172 ± 47 fs. Satisfactory fit results were obtained using four Gaussian-convoluted exponential functions, each characterized by the shared time constants of 130 ± 50 fs, 1.2 ± 0.39 ps, 26 ± 2.4 ps, and 292 ps. The discrepancy between experimental and fitted curves in the sub-ps time domain of the second RSV can be improved by replacing 130 ± 50 fs with two exponential constants, 105 ± 13 fs and 114 ± 13 fs. Nevertheless, we used four exponentials for the kinetic analysis, as rationalized in the main text (see section 1-5 of SI for details).

Analysis of decay-associated difference scattering curves (DADSs)

To analyze the detailed structural origins of the extracted time constants, we retrieved the experimental DADSs (denoted as DADS(q)) from ΔS₀(q, t) using the following equation.

$$\Delta {{{{\rm{S}}}}}_{0}\left(q,t\right)={{{\bf{DADS}}}}\left(q\right){{{\times }}}{{{{{\bf{C}}}}}_{{{{\bf{DADS}}}}}^{\prime}\left(t\right)}^{{{{\rm{T}}}}}={\sum }_{{{{\rm{k}}}}=1}^{4}{{{{\rm{DADS}}}}}_{{{{\rm{k}}}}}\left(q\right)\times {{{{\rm{C}}}}}_{{{{{\rm{DADS}}}}}_{{{{\rm{k}}}}}}^{{\prime} }{\left(t\right)}^{{{{\rm{T}}}}}$$

(1)

Here, C_DADSk′(t) is a hypothetical kinetic profile corresponding to the process associated with DADS_k(q). The superscript T indicates the transpose of a matrix. Using Eq. 1, we retrieved DADS(q) that minimizes the χ²-distance between ΔS₀(q, t) and DADS(q) × C_DADS(t)^T. The details are provided in section 3 of SI. We extracted and analyzed DADS_k(q)s for a total of four time constants (130 fs, 1.2 ps, 26 ps, and 292 ps). C_DADSk′(t) arises from zero to unity within IRF and decaying again to zero with a time constant of τ_k. It is expressed by the convolution of an exponential decay with a normalized Gaussian function centered at zero, ${{{\mathscr{N}}}}$(w, 0; t), as in the following equations.

$${{{{\rm{C}}}}}_{{{{{\rm{DADS}}}}}_{{{{\rm{k}}}}}}^{{\prime} }\left(t\right)={{{\mathscr{N}}}}\left(w,0{;t}\right)\otimes \left(\exp \left(-\frac{t}{{\tau }_{{{{\rm{k}}}}}}\right)\right)$$

(2)

$${{{\mathscr{N}}}}\left(w,0{;t}\right)=\frac{1}{\sqrt{2\pi }w}\exp \left(-\frac{{t}^{2}}{2{w}^{2}}\right)$$

(3)

Here, ⊗ indicates convolution, and an IRF of w = 172 fs was used to construct ${{{\mathscr{N}}}}$(w, 0; t).

After retrieving DADS(q), we simulated decay-associated difference scattering curves, DADS′(q), corresponding to each potential candidate pathway in Supplementary Fig. 6. The resulting set of candidate DADS′(q) were compared with each DADS_k(q). By identifying the pathway—a pair of chemical species for initial and final states—whose difference scattering curve best matches with each DADS_k(q), we can attribute the dynamic pathway to each kinetic constant (τ_k). We also conducted the structure refinement when constructing each DADS_k′(q) as detailed in the next section. We examined the candidate pathways in Supplementary Fig. 6 for DADS₁(q) (Supplementary Fig. 8), DADS₂(q) (Supplementary Fig. 9), and DADS₃(q) (Supplementary Fig. 10), and attributed them to the primary I• dissociation, the anti-to-gauche rotational isomerization of C₂F₄I•, and the gauche-to-anti rotational isomerization of C₂F₄I₂, respectively. For DADS₄, we simulated the scattering curve using the known species and concentrations at 100 ps, and subtracted the scattering curve of the photodissociated mixture of C₂F₄I₂ (Fig. 3). Details of the DADS analysis is provided in section 3 of SI.

Structure refinement

While analyzing the DADS to identify the structural transitions corresponding to each DADS, we simultaneously determined the molecular geometry of the species involved in the reaction through structure refinement. We started with the molecular geometries of the species optimized by DFT calculations and iteratively varied their structural parameters so that the simulated difference scattering curves, DADS′(q), generated from the refined structures best matched the experimental counterpart, DADS(q). The parameters, r_CI, θ_CCI, θ_pl, and φ_dihed, defined in Fig. 3d (with more details provided in Supplementary Fig. 7) were varied near the DFT-optimized values (Table 2). In C₂F₄I₂, the two carbon atoms share the four structural parameters to represent symmetry. On the other hand, in C₂F₄I•, the structural parameters at the –CF₂I site were taken to be equivalent to those for C₂F₄I₂ of the respective conformer (denoted with the superscripts “A” or “G”), whereas those at the –CF₂• site were defined with new parameters (denoted with the superscripts “a” or “g”).

We utilized the reduced chi-square for the k^th DADS, denoted as χ²_νk, as a statistical criterion to decide the best-matching candidate of DADS_k′(q) with DADS_k(q) for k = 1–3 (see section 3-1 of SI for details of the statistical criteria). DADS′(q) were calculated using well-established formulations from scattering theory, such as the Debye equation (see section 2-1 of SI for details). The simulated DADS′(q) were converted to the r-space, DADS′(r), through the Fourier sine transform (in eqs. S7–S8). We adjusted the amplitude of DADS′(q) to be on the exact scale by multiplying the scattering signal from one solute molecule by the relative molar quantity corresponding to each transition. As the volume of the solution at the interaction point remains identical for every species, we simply used concentrations (c_DADSk) to express relative molar quantities. The scale factors were determined from the kinetic model. Here, the first two equations were used to construct the scale factors, and the remaining four were introduced to build an expression for each of the four DADS_k′(q)s.

$${{{{\rm{c}}}}}_{{{{\rm{DADS}}}}1}={{{{\rm{c}}}}}_{{{{\rm{DADS}}}}2}={{{{\rm{c}}}}}_{{{{\rm{DADS}}}}4}={{{{\rm{c}}}}}_{0}{\chi }_{{{{\rm{exc}}}}}$$

(4)

$${{{{\rm{c}}}}}_{{{{\rm{DADS}}}}3}={{{{\rm{c}}}}}_{0}\left(1-{\chi }_{{{{\rm{exc}}}}}\right)$$

(5)

$${{{{\rm{DADS}}}}}_{1}^{{\prime} }\left(q\right)={{{{\rm{c}}}}}_{{{{\rm{DADS}}}}1}\cdot \left\{\begin{array}{c}\left({f}_{{{{\rm{exc}}}}}\cdot {{{{\rm{S}}}}}_{{{{\rm{A}}}}}\left(q\right)+\left(1-{f}_{{{{\rm{exc}}}}}\right)\cdot {{{{\rm{S}}}}}_{{{{\rm{G}}}}}\left(q\right)\right)\,\hfill\\ -\left({f}_{{{{\rm{exc}}}}}\cdot \left({{{{\rm{S}}}}}_{{{{\rm{a}}}}}\left(q\right)+{{{{\rm{S}}}}}_{{{{\rm{I}}}}}\left(q\right)\right)+\left(1-{f}_{{{{\rm{exc}}}}}\right)\cdot \left({{{{\rm{S}}}}}_{{{{\rm{g}}}}}\left(q\right)+{{{{\rm{S}}}}}_{{{{\rm{I}}}}}\left(q\right)\right)\right)\end{array}\right\}$$

(6)

$${{{{\rm{DADS}}}}}_{2}^{{\prime} }\left(q\right)={{{{\rm{c}}}}}_{{{{\rm{DADS}}}}2}\cdot \left\{\left({f}_{{{{\rm{Rad}}}}}-{f}_{{{{\rm{exc}}}}}\right)\cdot \left({{{{\rm{S}}}}}_{{{{\rm{a}}}}}\left(q\right)-{{{{\rm{S}}}}}_{{{{\rm{g}}}}}\left(q\right)\right)\right\}$$

(7)

$${{{{\rm{DADS}}}}}_{3}^{{\prime} }\left(q\right)={{{{\rm{c}}}}}_{{{{\rm{DADS}}}}3}\cdot \left\{\left({f}_{{{{\rm{rem}}}}}-{f}_{{{{\rm{GS}}}}}\right)\cdot \left({{{{\rm{S}}}}}_{{{{\rm{G}}}}}\left(q\right)-{{{{\rm{S}}}}}_{{{{\rm{A}}}}}\left(q\right)\right)\right\}$$

(8)

$${{{{\rm{DADS}}}}}_{4}^{{\prime} }\left(q\right)={{{{\rm{c}}}}}_{{{{\rm{DADS}}}}4}\cdot \left\{\begin{array}{c}0.7\cdot \left({f}_{{{{\rm{Rad}}}}}\cdot \left({{{{\rm{S}}}}}_{{{{\rm{a}}}}}\left(q\right)+{{{{\rm{S}}}}}_{{{{\rm{I}}}}}\left(q\right)\right)+\left(1-{f}_{{{{\rm{Rad}}}}}\right)\cdot \left({{{{\rm{S}}}}}_{{{{\rm{g}}}}}\left(q\right)+{{{{\rm{S}}}}}_{{{{\rm{I}}}}\cdot }\left(q\right)\right)\right)\\+0.3\cdot \left({{{{\rm{S}}}}}_{{{{\rm{P}}}}}\left(q\right)+2{{{{\rm{S}}}}}_{{{{\rm{I}}}}}\left(q\right)\right)-\left({f}_{{{{\rm{exc}}}}}\cdot {{{{\rm{S}}}}}_{{{{\rm{A}}}}}\left(q\right)+\left(1-{f}_{{{{\rm{exc}}}}}\right)\cdot {{{{\rm{S}}}}}_{{{{\rm{G}}}}}\left(q\right)\right)\end{array}\right\}$$

(9)

Here, f_GS is the population fraction of anti-C₂F₄I₂ in the total amount of C₂F₄I₂ at equilibrium, f_Rad is the population fraction of anti-C₂F₄I• in the total amount of C₂F₄I• at equilibrium, χ_a is the population fraction of anti-C₂F₄I₂ initially dissociated to anti-C₂F₄I•, χ_g is the population fraction of gauche-C₂F₄I₂ initially dissociated to gauche-C₂F₄I•, χ_exc is the total dissociation ratio from C₂F₄I₂ to C₂F₄I•, f_exc is the fraction of anti-C₂F₄I₂ in the dissociating population of C₂F₄I₂, and f_rem is the fraction of anti-C₂F₄I₂ in the remaining population of C₂F₄I₂ after dissociation. c₀ indicates the initial C₂F₄I₂ concentration ( = 60 mM). S_subscript(q) represents the simulated X-ray scattering curve from the species denoted with the subscript, where the subscripts (A, G, a, g, P, and I•) stand for the abbreviation of anti-C₂F₄I₂, gauche-C₂F₄I₂, anti-C₂F₄I•, gauche-C₂F₄I•, C₂F₄, and I•, respectively. Among the concentration-related parameters, the first four (f_GS, f_Rad, χ_a, and χ_g) were iteratively optimized, while the later three (χ_exc, f_exc, and f_rem) were calculated from them using the equations in section 3-3 of SI. The method to construct simulated scattering curves from the molecular structure is described in section 2-1 of SI. The theory for the DADS analysis is summarized in section 3 of SI.

Analysis using linear combination fit (LCF)

Using the optimal model determined from the analysis of the DADSs, we constructed the simulated species-associated difference scattering curves (SADS′(q)). To obtain SADS′(q), we took the following five species into consideration: (1) anti-C₂F₄I•, (2) gauche-C₂F₄I•, (3) anti-C₂F₄I₂, (4) gauche-C₂F₄I₂, and (5) C₂F₄. Then, SADS′_k(q)s were calculated by subtracting the scattering curve for the initially reacting mixture of C₂F₄I₂ from that corresponding to the k^th species.

$${{{{\rm{SADS}}}}}_{{{{\rm{k}}}}}^{{\prime} }\left(q\right)={{{{\rm{S}}}}}_{{{{\rm{k}}}}}\left(q\right)-\left({f}_{{{{\rm{exc}}}}}{{{{\rm{S}}}}}_{{{{\rm{A}}}}}\left(q\right)+\left(1-{f}_{{{{\rm{exc}}}}}\right){{{{\rm{S}}}}}_{{{{\rm{G}}}}}\left(q\right)\right)$$

(10)

Here, f_exc is the fraction of anti-C₂F₄I₂ in the dissociating population of C₂F₄I₂. Also, S(q) indicates the simulated X-ray scattering curve from the species denoted with the subscript, where the subscripts A and G mean anti-C₂F₄I₂ and gauche-C₂F₄I₂, respectively. Initially, the ground state ensemble consists of a mixture of anti and gauche C₂F₄I₂. Upon photoexcitation, the portion of C₂F₄I₂ with an anti-to-gauche ratio of f_exc:(1−f_exc) are depleted from the original equilibrium composition and start to follow the photoreaction pathway, as described in the subtracted term of Eq. 10. Here, it is noteworthy that the SADSs for (3) and (4) have identical shapes but opposite signs, as both components are mathematically represented by the difference between anti and gauche C₂F₄I₂. Therefore, we evaluated the scattering contributions from both anti-C₂F₄I₂ and gauche-C₂F₄I₂ collectively and merged the two components into one (explained as the “third” species, k = 3; therefore, the SADS for C₂F₄ becomes the “fourth” species, k = 4).

$$\begin{array}{c}{{{{{\rm{SADS}}}}}^{{\prime} }}_{{{{\rm{A}}}}}\left(q\right)\times {{{{{\rm{C}}}}}_{{{{\rm{SADS}}}},{{{\rm{A}}}}}\left(t\right)}^{{{{\rm{T}}}}}+{{{{{\rm{SADS}}}}}^{{\prime} }}_{{{{\rm{G}}}}}\left(q\right)\times {{{{{\rm{C}}}}}_{{{{\rm{SADS}}}},{{{\rm{G}}}}}\left(t\right)}^{{{{\rm{T}}}}}\\=\left({{{{\rm{S}}}}}_{{{{\rm{A}}}}}\left(q\right)-\left({f}_{{{{\rm{exc}}}}}{{{{\rm{S}}}}}_{{{{\rm{A}}}}}\left(q\right)+\left(1-{f}_{{{{\rm{exc}}}}}\right){{{{\rm{S}}}}}_{{{{\rm{G}}}}}\left(q\right)\right)\right)\times {{{{{\rm{C}}}}}_{{{{\rm{A}}}}}\left(t\right)}^{{{{\rm{T}}}}}\\+\left({{{{\rm{S}}}}}_{{{{\rm{G}}}}}\left(q\right)-\left({f}_{{{{\rm{exc}}}}}{{{{\rm{S}}}}}_{{{{\rm{A}}}}}\left(q\right)+\left(1-{f}_{{{{\rm{exc}}}}}\right){{{{\rm{S}}}}}_{{{{\rm{G}}}}}\left(q\right)\right)\right)\times {{{{{\rm{C}}}}}_{{{{\rm{G}}}}}\left(t\right)}^{{{{\rm{T}}}}}\\=\left({{{{\rm{S}}}}}_{{{{\rm{A}}}}}\left(q\right)-{{{{\rm{S}}}}}_{{{{\rm{G}}}}}\left(q\right)\right)\times {\left({{{{\rm{C}}}}}_{{{{\rm{A}}}}}\left(t\right)-{f}_{{{{\rm{exc}}}}}\left({{{{\rm{C}}}}}_{{{{\rm{A}}}}}\left(t\right)+{{{{\rm{C}}}}}_{{{{\rm{G}}}}}\left(t\right)\right)\right)}^{{{{\rm{T}}}}}\\={{{{{\rm{SADS}}}}}^{{\prime} }}_{3}\left(q\right)\times {{{{{\rm{C}}}}}_{{{{\rm{SADS}}}}3}\left(t\right)}^{{{{\rm{T}}}}}\end{array}$$

(11)

Here, Eq. 11 has a similar form as Eq. 10 since it is expressed as a multiplication of the time-dependent concentration part and the time-independent q-space part.

Subsequently, we retrieved the LCF concentration matrix, C_SADS(t) (see Fig. 3e), that minimizes the reduced χ² value between the solute-related experimental data (ΔS₀(q, t)) and the simulated data (SADS′(q) × C_SADS(t)^T). Details of the LCF analysis are assessed in section 4 of SI. Since the amplitude of the residual signal is negligible (see Supplementary Fig. 13), we concluded that the LCF model adequately explains the TRXL data and effectively summarizes the structural dynamics of C₂F₄I₂. We also provide an alternative analytical approach based on SADSs in Supplementary Fig. 12, whose details are described in section 3-3 of SI.

Ab-initio and density functional theory (DFT) calculations

Molecular geometries of all reacting species, including anti-C₂F₄I₂, gauche-C₂F₄I₂, anti-C₂F₄I•, gauche-C₂F₄I•, C₂F₄, and C₂F₄…I₂ (species “X”) were optimized using DFT. The ωB97X functional⁴⁸, well-known for its ability to optimize geometry in line with experimental observations^{26,31,32,37,46}, was utilized. These DFT-optimized structures served as the initial points for the structure refinement (see Supplementary Data 1).

To explore the nature of excited states of C₂F₄I₂, potential energy curves (PECs) with respect to the C–I distance were calculated for anti and gauche C₂F₄I₂ in extended multi-state complete active space second-order perturbation (XMS-CASPT2) theory⁴⁹, from the SA(9 singlets, 11 triplets)-CAS(12e, 10o)SCF wavefunctions. Diabatic PECs were obtained, with the semi-diabatization process from the calculated adiabatic PECs. For obtaining accurate energy barriers during the C–C bond rotation, Gibbs free energies were calculated for the two transition states, namely T1 and T2, for the four species involved in rotational isomerization (anti-C₂F₄I₂, gauche-C₂F₄I₂, anti-C₂F₄I•, and gauche-C₂F₄I•). Furthermore, PECs with respect to the dihedral angles (ICCI for C₂F₄I₂ and ICCF for C₂F₄I•) were calculated with three different levels of theory (CCSD(T), XMS-CASPT2, and DFT (ωB97X functional)) for the four species. The results were demonstrated in Table 2. We note that (7e, 5o) was employed for the active space of C₂F₄I•, analogous to the active space of C₂F₄I₂.

To calculate vertical excitation energies and PECs of C₂F₄I₂ with respect to the C–I distance, ANO-RCC-VDZP all-electron basis set was employed. We utilized the LANL2DZ basis set for CCSD(T) and XMS-CASPT2 calculations, and the def2-TZVPP basis set for DFT calculations. Openmolcas 8.6⁵⁰ was used for the multireference calculations, ORCA 5.0.4⁵¹ was used for the CCSD(T) calculations, and Gaussian 16⁵² was used for DFT calculations.

Molecular dynamics simulation

Molecular dynamics (MD) simulations were conducted via the MOLDY⁵³ program to model the interaction between each chemical species involved in the reaction and surrounding solvent molecules. Here, we employed a virtual cubic box composed of one solute molecule surrounded by 512 rigid solvent molecules. The internal structures of each molecule (anti-C₂F₄I₂, gauche-C₂F₄I₂, anti-C₂F₄I•, gauche-C₂F₄I•, and I•) were fixed to the optimized ones from the DFT method, as detailed in the “Density functional theory calculation” section. The partial charge distributions of each molecule were determined by natural population analysis (NPA) using the optimized structures. In the simulation, the molecular dynamics were governed by intermolecular interactions modeled with Coulombic forces and Lennard-Jones potentials. All simulations took place at an ambient temperature of 300 K and a solvent density of 0.779 g/cm³. We initiated system equilibration via coupling to a Nose-Hoover thermostat over a 20 ps duration. The simulations were conducted within the NVT ensemble, utilizing a time step of 100 as, and trajectories were traced for a duration of 1 ns. During the simulation, molecular snapshots were saved at intervals of 100 ps. From these molecular snapshots, we first extracted the pair distribution functions (PDFs) and calculated the theoretical X-ray scattering curves for the cage term, S_cage. This process is described in section 2-1 of SI.

Simulating the rotational isomerization dynamics

We used the Rice-Ramsperger-Kassel-Marcus (RRKM) theory^43,44,45 to calculate the theoretical kinetic constants for the rotational isomerization of C₂F₄I₂ and C₂F₄I•. We assumed that the rotational isomerization dynamics of both C₂F₄I₂ and C₂F₄I• takes a path trespassing either of the two transition states, T1 and T2. We utilized the optimized molecular structures, vibrational frequencies, electronic energy levels, and free energies for anti, gauche, T1, and T2 states of C₂F₄I₂ and C₂F₄I•. The dynamic equilibrium of the forward (anti-to-T2-to-gauche) and the reverse (gauche-to-T2-to-anti) pathways leads to identical difference scattering signals with opposite signs, quenching each other in the kinetic picture. Therefore, only the dynamics arising from the re-equilibration of the perturbed state to the dynamic equilibrium are observed in our experiment. The potential energy surface with respect to the dihedral angle implies that the rotational isomerization takes place through the T2 transition state. In this context, only the theoretical kinetic constants for the assigned rotational dynamics (gauche-to-anti for C₂F₄I₂, anti-to-gauche for C₂F₄I•) passing through the corresponding T2 states were compared with experimentally extracted values in Table 3. All computations based on the RRKM theory were conducted using the ChemRate software^54,55,56.

In the RRKM calculation, the rate constant is expressed by the following equation.

$$k={a}_{1}\times \exp \left(-\frac{{a}_{2}}{{RT}}\right)$$

(12)

Here, the gas constant R is 8.314 J·mol⁻¹·K⁻¹, the temperature T was assumed to be 300 K, and a₁ and a₂ were computed through the RRKM theory. From the computation results, we found that a₁ = 9.48×10⁹ and a₂ = −4117.71 in the gauche-to-T2-to-anti pathway of C₂F₄I₂, and a₁ = 4.56×10¹¹ and a₂ = −1613.45 in the anti-to-T2-to-gauche pathway of C₂F₄I•. At 300 K, these values lead to the time constants of 20.2 ps for the C₂F₄I₂ rotation and 1.15 ps for the C₂F₄I• rotation, as discussed in the main text.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.