Unveiling hidden wavepacket dynamics in time-resolved x-ray scattering data via singular spectrum analysis

Abstract

Time-resolved x-ray liquidography (TRXL) is a powerful technique for directly tracking ultrafast structural dynamics in real space. However, resolving the motion of vibrational wavepackets generated by femtosecond laser pulses remains challenging due to the limited temporal resolution and signal-to-noise ratio (SNR) of experimental data. This study addresses these challenges by introducing singular spectrum analysis (SSA) as an efficient method for extracting oscillatory signals associated with vibrational wavepackets from TRXL data. To evaluate its performance, we conducted a comparative study using simulated TRXL data, demonstrating that SSA outperforms conventional analysis methods such as the Fourier transform of temporal profiles and singular value decomposition, particularly under low SNR conditions. We further applied SSA to experimental TRXL data on the photodissociation of triiodide ( ${\mathrm{I}}_3^{-}$) in methanol, successfully isolating oscillatory signals arising from wavepacket dynamics in ground-state ${\mathrm{I}}_3^{-}$ and excited-state ${\mathrm{I}}_2^{-}$, which had been challenging to resolve in previous TRXL studies. These results establish SSA as a highly effective tool for analyzing ultrafast structural dynamics in time-resolved experiments and open new opportunities for studying wavepacket dynamics in a wide range of photoinduced reactions.

I. INTRODUCTION

Molecular vibrations play a central role in the progress of chemical reactions. In femtosecond time-resolved experiments, ultrashort laser pulses create coherent superpositions of molecular vibrations, known as vibrational wavepackets, and the motion of these wavepackets is often discussed as a key factor in interpreting reaction dynamics. Since tracking the dynamics of wavepackets on potential energy surfaces (PESs) provides crucial insight into reaction pathways, various time-resolved spectroscopic techniques have been actively applied to investigate activated wavepackets in photoinduced reactions.^1–18 These studies focus on detecting temporal oscillations in experimental signals induced by wavepackets and determining the associated vibrational frequencies to identify the activated vibrational modes. More recently, with the advent of x-ray free-electron lasers (XFELs),^19–23 femtosecond time-resolved x-ray liquidography (fs-TRXL), also known as time-resolved x-ray solution scattering, has emerged as a relevant tool for studying wavepacket dynamics in solution-phase reactions.^24–28 In typical TRXL experiments, a laser pulse initiates a photoinduced response, and an x-ray pulse arriving after a controlled time delay measures an x-ray scattering pattern from the transient molecular structure. As a result, by utilizing x-ray scattering signals that are directly sensitive to molecular structure, it becomes possible to simultaneously obtain temporal information (vibrational frequencies) and spatial information (vibrational motions) on coherent molecular vibrations.

Although TRXL has been successfully applied to several molecular systems to analyze wavepacket dynamics, some challenges remain for further applications to a broader range of systems. For example, when the contribution of the oscillatory signal arising from wavepacket motion is relatively minor compared to the overall signal, it becomes challenging to achieve a sufficiently high signal-to-noise ratio (SNR) to resolve these oscillatory features. One straightforward way to capture such small oscillatory signals is by enhancing the experimental temporal resolution, as insufficient resolution reduces the amplitude of oscillation signals due to convolution with the instrument response function (IRF). Improving the IRF can be achieved through technical adjustments, such as shortening the time duration of both pump and probe pulses or decreasing the sample thickness to minimize velocity mismatch. However, these approaches often involve significant experimental challenges. Furthermore, although reducing the sample thickness enhances temporal resolution, it also reduces the intensity of the scattering signal, which in turn requires longer data acquisition times to maintain a sufficient SNR. In this regard, beyond experimental improvements, developing a more effective data analysis method capable of separating oscillatory signals from measured time-resolved data would offer a practical and accessible alternative solution.

In this work, we developed an analytical method to efficiently extract oscillatory signals from time-resolved data using singular spectrum analysis (SSA), a linear algebra-based technique. SSA is a model-free signal decomposition method commonly used in various fields, including climatology, finance, biomedical signal processing, and has also been applied in certain areas of physics to extract meaningful temporal patterns from noisy data.^29–35 SSA decomposes time-resolved data into three components: “trends,” “oscillations,” and “noise.” This method is based on singular value decomposition (SVD),^36,37 a well-established linear algebra technique for analyzing time-resolved experimental data, and both methods share a key feature: they extract major signal components by analyzing the singular values and vectors of the data. One of the main differences is that SSA additionally employs embedding the original data before performing SVD, which enhances the detection of recurrent signals, that is, the oscillation components, in the data. This enables more effective extraction of oscillatory signals in time-resolved data than SVD alone. Furthermore, SSA can extract oscillatory signals whose amplitude or frequency varies over time, making it more suitable than the standard Fourier transform (FT) method.^32,33 For instance, oscillatory signals arising from wavepacket dynamics typically exhibit exponential damping due to vibrational relaxation and time-dependent frequency shifts resulting from the anharmonicity of the PES. Under such conditions, SSA offers a compelling framework for resolving weak, time-varying oscillatory components manifested in time-resolved data.

We applied SSA to simulated TRXL data to benchmark its ability to extract oscillatory components compared to other commonly used analysis methods, including the FT of the time profile of the signal and SVD analysis. Through a comparative study, we demonstrated that SSA extracts oscillatory signals more effectively, particularly under limited SNR conditions. This method was then applied to experimental TRXL data on the photodissociation of the triiodide ( ${\mathrm{I}}_3^{-}$) in methanol. In contrast to other analysis methods, which failed to separate the oscillatory signals in the TRXL data, SSA successfully extracted the oscillation in the signal and identified wavepacket motions in both the ground and excited states. This work demonstrates that SSA can more effectively extract oscillatory signals from TRXL data than commonly used analysis methods. In addition, SSA can be applied to various other time-resolved experimental data, opening the door to studying wavepacket dynamics in many chemical reactions.

II. RESULTS AND DISCUSSION

A. Extraction of oscillatory signals from time-resolved x-ray liquidography data using singular spectrum analysis

In this study, we applied SSA to extract oscillatory signals from TRXL data effectively. SSA is a model-free signal decomposition technique based on SVD. Compared to SVD, SSA enhances the contribution of recurrent signals by transforming the original data into a trajectory matrix through an additional embedding process. This transformed data are then subjected to SVD for component separation, followed by a reconstruction process that converts the decomposed components back into the original data space. The overall process of SSA is schematically illustrated in Fig. 1, and each step is discussed in detail later. As a result, SSA enables the decomposition of a given time-resolved data into three components: “trend,” “oscillation,” and “noise.” The “trend” refers to a slowly varying component without periodicity (non-repeating) over time, while “oscillation” represents a component that exhibits periodic behavior over time. “Noise,” as commonly understood, corresponds to irregular signals characterized by rapid oscillations or a lack of discernible trends. For example, in TRXL data, signals associated with changes in the concentration of chemical species typically follow a non-periodic, gradual change governed by rate laws. Such signals from population kinetics can be decomposed as the “trend” component using SSA. If the data also contain signals originating from molecular vibrations, such as those induced by wavepackets generated in the ground or excited states by femtosecond laser pulses, the signal will show periodic changes corresponding to the activated vibrational modes. In this case, such oscillatory signals are isolated as the “oscillation” component through SSA. Finally, the “noise” component represents random statistical errors inherent in experimental measurements, typically appearing as random fluctuations over time.

FIG. 1.

The four steps of SSA are schematically illustrated. (a) In the first step, “embedding,” a data matrix A is constructed from the difference scattering curves. Then, A is transformed into a trajectory matrix H by sliding a window of length L along the row vectors of A, with each windowed segment forming a column of H. (b) In the second step, “decomposition,” the trajectory matrix H is decomposed by SVD into U, S, and V matrices. (c) In the third step, “grouping,” each SVD component is categorized into one of three groups: trend, oscillation, or noise. The grouping is performed based on the inspection of the singular values and the w-correlation matrix. (d) In the fourth step, “reconstruction,” the SVD components within each group are reconstructed into the original data space by diagonal averaging. See the main text for further details.

By definition, SSA is an analytical method applied to one-dimensional time-resolved data. For example, in a time-resolved transient absorption experiment, SSA can be performed on the temporal evolution of absorbance at a single wavelength. When multiple temporal profiles measured at different wavelengths exhibit common oscillatory features, SSA can be extended to analyze these profiles simultaneously. This approach is referred to as multi-channel SSA (MSSA). MSSA, a multivariate extension of SSA, enables the simultaneous analysis of multiple temporal profiles and has the advantage of more effectively extracting oscillatory components shared across multiple temporal profiles. In typical TRXL experiments, time-resolved difference scattering curves, ΔS(q, t), are obtained as two-dimensional data as a function of both the momentum transfer vector, q = (4π/λ)sin(2θ/2), where λ is the x-ray wavelength and 2θ is the scattering angle, and the time delay between a pump laser pulse and a probe x-ray pulse, t. When oscillatory signals from molecular vibrations are present in TRXL data, they typically appear at multiple q values. Therefore, to effectively extract the oscillatory components embedded in TRXL data, it is advantageous to simultaneously utilize the temporal profiles at various q values by applying a multi-channel SSA approach. Although we employed this multi-channel strategy in this study, for simplicity, we will hereafter refer to this analysis method as SSA.

To describe the results and interpretation of SSA on TRXL data, we generated simulated TRXL data for a model reaction involving the structural change of I₂ in the gas phase (I₂ → I⋯I). In this model, we considered a situation where wavepacket motion occurs in the photoexcited I⋯I species, resulting in the temporal oscillation of the I–I distance of I⋯I, R(I–I). Specifically, the R(I–I) of I⋯I was set to exhibit a damped oscillation around 3.3 Å with an oscillation frequency of 110 cm⁻¹ (corresponding to a period of 300 fs), while the I–I distance of the reactant I₂ was fixed at 2.7 Å. The temporal evolution of R(I–I) used for generating the simulated TRXL data are presented in Fig. 2(a). Theoretical scattering curves for I₂ and I⋯I, denoted as $S_{{\mathrm{I}}_2}(q)$ and $S_{\mathrm{I}\dots \mathrm{I}}(q,\hspace{0.17em}\hspace{0.17em}t)$, respectively, were calculated based on the Debye equation in Eq. (1).

$$S(q)=2F_{\mathrm{I}}^2(q)+2F_{\mathrm{I}}^2(q)\frac{\hspace{0.17em}\text{sin}(qR(\mathrm{I}-\mathrm{I}))}{qR(\mathrm{I}-\mathrm{I})}.$$ (1)

Here, F_I(q) is the atomic form factor of an I atom. The difference scattering curves, qΔS(q, t) [ $=q{S}_{\mathrm{I}\cdots \mathrm{I}}(q,\hspace{0.17em}\hspace{0.17em}t)-q{S}_{{\mathrm{I}}_2}(q)$], were then obtained. Subsequently, the convolution of the signal by the IRF was considered, and random noise was added as follows:

$$\text{IRF}\left(t\right)\otimes q\Delta S(q,\hspace{0.17em}\hspace{0.17em}t)+N(q,\hspace{0.17em}\hspace{0.17em}t).$$ (2)

Here, the IRF(t) was introduced by a Gaussian function with a full width at half maximum (FWHM) of 100 fs, and the noise, N(q, t), was generated as random numbers following a Gaussian distribution, with the standard deviation set arbitrarily. The simulated ΔS(q, t) were generated in the q range from 1.5 to 6.5 Å⁻¹, with time delays ranging from 300 fs to 2 ps. The generated simulated TRXL data, qΔS(q, t), are shown in Fig. 2(b).

FIG. 2.

(a) A model reaction of I₂ in the gas phase (I₂ → I⋯I) considered for generating the simulated TRXL data shown in (b). The time-dependent R(I–I) of I⋯I was modeled using a damped cosine function with an oscillation frequency of 110 cm⁻¹. The model reaction and description of R(I–I) are schematically illustrated at the top. (b) Simulated TRXL data of I₂, qΔS(q, t), used for the SSA analysis. (c) The singular values of the trajectory matrix, H, calculated during the SSA analysis. Based on the magnitudes and features of the singular values, each SVD component was categorized into one of three groups: “trend,” “oscillation,” and “noise.” (d) The w-correlation matrix calculated for the SVD components obtained from the SSA analysis. In (c) and (d), the trend, oscillation, and noise groups are indicated by red, blue, and cyan squares, respectively. (e) and (f) The reconstructed signals of (e) the trend group, qΔS_trend(q, t), and (f) the oscillation group, qΔS_oscillation(q, t), obtained from the SSA analysis. (g) The FT amplitude spectra of qΔS_oscillation(q, t), FT_SSA. The position of 110 cm⁻¹ is indicated by a magenta arrow. (h) Theoretical fits of qΔS_oscillation(q, t) obtained from the structural analysis. The corresponding R(I–I) of I⋯I retrieved from the structural analysis is shown by a red curve in (a).

SSA was applied to the simulated TRXL data of I₂. The process of SSA consists of four main steps, as schematically shown in Fig. 1: (1) embedding, (2) decomposition, (3) grouping, and (4) reconstruction. The processes of SSA applied to TRXL data are discussed in detail below. The procedures of SSA have already been well established, and the detailed procedure is summarized with references to previous works.³³

1. Step 1: Embedding

TRXL data were organized into an M × N matrix, A, whose columns correspond to time-resolved difference scattering curves at time delays of t, qΔS(q, t), with M and N denoting the number of q points and time-delay points, respectively. The simulated data for I₂ were generated at 251 q points (M) and 86 time-delay points (N). The data matrix A can also be expressed using the temporal profile at the mth q point, qΔS(q_m, t), as a row vector as follows:

$$\mathbf{A}(q,\hspace{0.17em}\hspace{0.17em}t)=\left[\begin{array}{c}q\Delta S(q_1,\hspace{0.17em}\hspace{0.17em}t)\\ q\Delta S(q_2,\hspace{0.17em}\hspace{0.17em}t)\\ ⋮\\ q\Delta S(q_M,\hspace{0.17em}\hspace{0.17em}t)\end{array}\right]\in {ℝ}^{M\times N}.$$ (3)

The first step of SSA involves embedding the data matrix A into a high-dimensional vector space, the trajectory matrix H. As illustrated in Fig. 1(a), the row vectors of A are first transformed into an L × K Hankel matrix, H^(m), by sliding a window of length L along the row vectors of A, with each windowed segment forming an H^(m) as follows:

$${\mathbf{H}}^{(m)}=\left[\begin{array}{cccc}q\Delta S(q_m,\hspace{0.17em}\hspace{0.17em}{t}_1)& q\Delta S(q_m,\hspace{0.17em}\hspace{0.17em}{t}_2)& \cdots & q\Delta S(q_m,\hspace{0.17em}\hspace{0.17em}{t}_K)\\ q\Delta S(q_m,\hspace{0.17em}\hspace{0.17em}{t}_2)& q\Delta S(q_m,\hspace{0.17em}\hspace{0.17em}{t}_3)& \cdots & q\Delta S(q_m,\hspace{0.17em}\hspace{0.17em}{t}_{K+1})\\ ⋮& ⋮& \ddots & ⋮\\ q\Delta S(q_m,\hspace{0.17em}\hspace{0.17em}{t}_L)& q\Delta S(q_m,\hspace{0.17em}\hspace{0.17em}{t}_{L+1})& \cdots & q\Delta S(q_m,\hspace{0.17em}\hspace{0.17em}{t}_N)\end{array}\right].$$ (4)

Subsequently, each H^(m) is stacked to construct a trajectory matrix, H, as follows:

$$\mathbf{H}=\left[\begin{array}{cccc}{\mathbf{H}}^{(1)}& {\mathbf{H}}^{(2)}& \cdots & {\mathbf{H}}^{(M)}\end{array}\right]\in {ℝ}^{L\times MK},$$ (5)

where K = N − L + 1, and thus H forms an L × MK matrix.

Here, the window length L is a user-defined parameter that determines the size of the segments of the temporal profile used during the embedding process. If the window length L used for SSA is set too small, oscillatory signals may be over-decomposed into unnecessarily fine structures, while a value of L that is too large may fail to capture the oscillatory signals. Therefore, careful selection of an appropriate L is essential. In general, if the number of time points is sufficiently large, it is recommended to set the window length for SSA to approximately one-half of the total number of time points.³² In this simulation, we used L = 70, as it provided the clearest separation of the oscillatory components. Nevertheless, similar SSA results were obtained over a broad range of L values.

2. Step 2: Decomposition

The next step is to apply SVD to the trajectory matrix H constructed in the embedding process. More details on SVD are described in the supplementary material. Through SVD, H was factorized into three matrices satisfying H = USV^T, as depicted in Fig. 1(b). Here, U and V are matrices whose columns are referred to as the left singular vectors (lSVs) and the right singular vectors (rSVs), respectively, and S is a diagonal matrix whose elements are the singular values, s_i. By applying SVD, H can be expressed as a combination of SVD components, as represented in Fig. 1(b). We note that the term “SVD component” used here refers collectively to the lSV (u_i), rSV (v_i), and corresponding singular value (s_i) of the same index.

3. Step 3: Grouping

Each SVD component of the trajectory matrix H is categorized into one of three groups: trend, oscillation, or noise. This categorization is based on two quantities: the singular values and the w-correlation matrix, as illustrated in Fig. 1(c). Primarily, the singular values are employed to differentiate significant components (trend and oscillation) from noise. The singular values represent the contribution of the corresponding lSV and rSV to the overall data (trajectory matrix H). SVD components categorized as trend or oscillation generally have large singular values, while those grouped as noise have smaller singular values. Although both the trend and oscillation groups consist of SVD components with significant singular values, a key distinction lies in the fact that the oscillation group typically consists of paired components, with each pair representing an oscillatory signal originating from a single oscillation frequency present in the original data (A). As a result, the singular values of SVD components in an oscillation group tend to be similar in magnitude.

SSA was applied to the simulated TRXL data for I₂ shown in Fig. 2(b), and the singular values of the SVD components of the trajectory matrix H are presented in Fig. 2(c). The first singular value has a significant magnitude but does not form a pair with any subsequent singular value, allowing the corresponding SVD component to be tentatively assigned to the trend group. Meanwhile, the second and third singular values have similar magnitudes and are both significant, so the corresponding pair, the second and third SVD components, can be tentatively assigned to the oscillation group. In contrast, the fourth and subsequent singular values are significantly lower than the third, and thus can be classified into the noise group.

As a complementary method for the grouping, a w-correlation matrix can be examined. The w-correlation matrix intuitively visualizes the weighted correlation between SVD components; values close to 1 indicate a strong correlation between two components, while values near 0 indicate a weak correlation. The procedure for calculating the w-correlation matrix is described in the supplementary material. As shown in the w-correlation matrix of the simulated TRXL data for I₂ in Fig. 2(d), the trend group forms a single isolated block. The oscillation group appears as a 2-by-2 block, reflecting the paired nature of the SVD components in the oscillation group. In contrast, the noise components, found among the later SVD components, exhibit numerous irregular mutual correlations, leading to the formation of a large block. Note that the w-correlation matrix indicates whether grouping specific SVD components as the “oscillation” component is reasonable. From the w-correlation matrix of the simulated data for I₂, it can be seen that the correlation between the second and third SVD components is close to 1, indicating that these two SVD components together describe a single oscillatory signal in the data. In conclusion, considering both the singular values and the w-correlation matrix of the simulated data for I₂, we can assign that the first SVD component represents the trend of the data, the second and third components together describe the oscillation, and the remaining SVD components correspond to noise.

4. Step 4: Reconstruction

After grouping the SVD components, each group needs to be reconstructed into the original data space, yielding signals as functions of q and t. In the SSA process, since the decomposition is performed on the trajectory matrix H obtained by the embedding step, this reconstruction involves a dimensional transformation to convert the SVD components of H back into the original data space. As illustrated in Fig. 1(d), this can be achieved by applying diagonal averaging to the matrix $\tilde{\mathbf{H}}$,

$$\tilde{\mathbf{H}}=\sum _{j\hspace{0.17em}\in \hspace{0.17em}I}{s}_j{u}_j{v}_j^{\mathrm{T}},$$ (6)

where j corresponds to the indices of the SVD components belonging to a specific group (I, which is either trend, oscillation, or noise) and s_j, u_j, and v_j denote the jth singular value, lSV, and rSV, respectively. The reconstructed signals of the trend and oscillation groups are termed qΔS_trend(q, t) and qΔS_oscillation(q, t), respectively.

The qΔS_trend(q, t) and qΔS_oscillation(q, t) of the simulated data of I₂ are shown in Figs. 2(e) and 2(f), respectively. In the case of qΔS_trend(q, t), it can be seen that the overall tendency of the data shown in Fig. 2(b) has been extracted. In contrast, an oscillatory feature is observed in qΔS_oscillation(q, t), where the sign of the scattering intensity repeatedly changes over time. To verify whether the oscillatory feature observed in qΔS_oscillation(q, t) is consistent with the oscillation frequency of R(I–I) considered for generating the simulated data of I₂, we performed FT of the temporal profiles at each q point in qΔS_oscillation(q, t) obtained from the SSA, FT_SSA. As shown in Fig. 2(g), distinct peaks appear around 110 cm⁻¹ in the FT_SSA, confirming that the SSA analysis accurately resolved the oscillation frequency introduced in the simulated data. In addition, we performed structural analysis using qΔS_oscillation(q, t) to determine the R(I–I) of I⋯I, as detailed in the supplementary material. The fitted I⋯I structures obtained from the structural analysis successfully reproduce the R(I–I) values that were considered in generating the simulated data, as shown in Fig. 2(a).

B. Comparative evaluation of singular spectrum analysis and other analysis methods for extracting oscillatory signals

In this section, we compared how effectively the oscillatory signals arising from molecular vibrations embedded in TRXL data can be extracted using SSA, relative to other commonly used analysis methods. Specifically, four simulated TRXL datasets with different SNRs were generated based on the same model reaction of I₂ illustrated in Fig. 2(a). As shown in Fig. 3(a), these simulated datasets were prepared with noise levels of 30%, 60%, 90%, and 120%. Here, the noise level refers to the approximate ratio of the standard deviation of Gaussian noise to the amplitude of the oscillatory signal arising from the vibration of I₂. Further details regarding the definition of noise level are provided in the supplementary material. SSA and other analysis methods were then applied to each dataset to evaluate how well the oscillation frequency of 110 cm⁻¹ in R(I–I) could be resolved.

FIG. 3.

(a) Four simulated TRXL datasets, qΔS(q, t), for a model reaction of I₂ (I₂ → I⋯I) with different noise levels (30%, 60%, 90%, and 120%). (b) The FT amplitude spectra of the qΔS(q, t), FT_qΔS. Each spectrum was scaled to better visualize the characteristic features of the FT spectrum. (c) (top) The first right singular vectors (rSVs) of the four datasets, obtained by SVD analyses (black dots). The overall temporal trends of the rSVs were fitted with baseline functions (red curves), and the FT amplitude spectra of the fitting residuals, FT_SVD, are shown (bottom). In (a)–(c), the noise levels of the datasets are indicated at the top right corner of each panel. In the FT spectra shown in (b) and (c), the positions of 110 cm⁻¹, the oscillation frequency of R(I–I) of I⋯I used to generate the simulated data, are indicated by magenta arrows.

Three different analysis methods, including SSA, were applied to each simulated dataset to extract the oscillatory components. The first method involves calculating the q-resolved FT amplitude spectra, FT_qΔS, as shown in Fig. 2(g), by performing FT of the temporal profiles of the simulated data at each q point. This approach is one of the most basic and preliminary ways to identify time-domain oscillatory signals in the various types of time-resolved experimental data. It is analogous to performing FT of signal changes at each probe wavelength in time-resolved spectroscopy. The second method applies SVD to the simulated datasets and examines the temporal behavior of rSVs, which contain information about the time-dependent changes in the data. Further details of the SVD analysis for TRXL data are provided in the supplementary material. SVD has been widely used to analyze TRXL data, particularly to track reaction kinetics. When oscillatory signals are present, they typically appear as oscillatory features in the rSVs.^27,38 We quantitatively inspected the oscillatory features by inspecting FT spectra on the rSVs from SVD analysis, FT_SVD. The final method is to apply SSA, which is the main focus of this study. We obtained qΔS_oscillation(q, t) and examined its oscillatory feature by calculating FT spectra, FT_SSA. In summary, we applied these three analysis methods to the four simulated datasets with different noise levels and compared their efficiencies in extracting the oscillatory signals by checking whether the 110 cm⁻¹ oscillation is resolved in FT_qΔS, FT_SVD, and FT_SSA.

As the first approach, FT_qΔS was calculated for each simulated dataset. Typically, to extract oscillatory components via the FT, it is essential to remove the baseline from the temporal profiles first, for example, by fitting and subtracting an exponential decay. For this purpose, we subtracted the baseline of each temporal profile at every q point and then performed the FT. The resulting FT_qΔS of the four datasets are shown in Fig. 3(b). When the noise level is 30%, peaks at a wavenumber of 110 cm⁻¹ are observed across a wide q range. However, as the noise level increases to 60%, the 110 cm⁻¹ peak gradually becomes weaker in the spectra. In addition, at the noise levels of 90% and 120%, it becomes difficult to identify the corresponding peak.

As the second analysis method, SVD was applied to the TRXL data, and the temporal behavior of the rSV was examined through FT_SVD. The SVD analyses were performed on the four datasets to obtain the first rSVs, and the overall signal trends, which overlapped with the oscillatory component, were removed through baseline fitting. Then, FT was performed on the fitting residuals, and the resulting FT_SVD are represented in Fig. 3(c). It can be seen that using SVD to identify the oscillatory components is more effective than examining the FT_qΔS. Specifically, in the FT_qΔS for the 60% noise level data, it is difficult to determine the peak as shown in Fig. 3(b), but in the FT_SVD, the 110 cm⁻¹ peak is distinguishable. However, starting from the 90% noise level, difficulties arise in identifying the corresponding vibration frequency even in FT_SVD. As the noise level increases, the intensities of artifact peaks other than the 110 cm⁻¹ peak also increase, making it difficult to accurately resolve the 110 cm⁻¹ peak. In fact, at a noise level of 120%, it becomes ambiguous to assign a single vibration frequency.

Finally, the oscillatory signals of the four datasets were extracted through SSA. The second and third SVD components of H were grouped as the “oscillation,” similarly to the results shown in Figs. 2(c) and 2(d). Then, the reconstructed signals of the oscillation group, qΔS_oscillation(q, t), were obtained as shown in Fig. 4(a). The qΔS_oscillation(q, t) of the four datasets exhibit periodic signal changes over time across all the noise levels. For a more quantitative analysis, the FT spectra of qΔS_oscillation(q, t), FT_SSA, are represented in Fig. 4(b). The peaks at 110 cm⁻¹ are consistently observed in the FT_SSA, regardless of the noise levels. As a result, these observations demonstrate that SSA is more effective than other commonly used analysis methods, such as performing FT on temporal profiles or applying SVD, in extracting subtle oscillatory signals embedded in TRXL data.

FIG. 4.

(a) SSA analyses were performed on the four simulated TRXL datasets shown in Fig. 3(a). The reconstructed signals of the “oscillation” groups, qΔS_oscillation(q, t), are shown. (b) The FT amplitude spectra of the qΔS_oscillation(q, t), FT_SSA. The positions of 110 cm⁻¹ are indicated by magenta arrows. In both (a) and (b), the noise levels of the datasets are indicated at the bottom right corner of each panel.

C. Considerations in the interpretation of results of singular spectrum analysis

In TRXL data containing oscillatory signals arising from wavepacket dynamics, both the fundamental frequency of molecular vibrations and their harmonic frequencies (that is, integer multiples of the fundamental frequency) can be observed simultaneously. This phenomenon does not originate from a characteristic or potential artifact of the SSA method, but naturally arises because the temporal behavior of TRXL signals cannot be fully represented by a single periodic function, even when the molecular vibration follows a single vibrational mode. See the supplementary material for more details. However, such harmonic frequencies are often not well resolved in experimental TRXL data due to limited temporal resolution and SNR, as higher-frequency oscillations are more strongly attenuated by the IRF. Nevertheless, if oscillation components at frequencies similar to harmonic frequencies appear alongside the fundamental frequency, careful attention is required when interpreting the SSA results. One possible approach to distinguish them is to reconstruct signals from each oscillation component and perform structural analysis to verify whether the reconstructed signals exhibit characteristic features consistent with theoretically predicted TRXL signals of the molecule under investigation.

Another point to consider is that when SSA is applied to data with high noise levels, the amplitude of the oscillations may become distorted, although the oscillation frequency can still be well determined. As shown in Fig. S1, when significant noise (250% noise level) is added to the simulated dataset, an abnormal phenomenon appears in the amplitude of the reconstructed oscillation signal, where the amplitude first decreases and then increases. This phenomenon is typically observed when analyzing oscillatory signals with damping using SSA, especially at later time delays, where the SNR becomes significantly worse. In such cases, the amplitude-related information, such as the damping time constant, becomes unreliable.

Additionally, while SSA is highly effective at separating oscillatory components, in some unfortunate cases, it may force the extraction of oscillatory components from irregular noise patterns. Typically, such artifact oscillatory components tend to appear as physically implausible high-frequency signals. Therefore, when interpreting the results of SSA, it is essential to consider factors such as the magnitude of the singular values, the w-correlation matrix, and any physical prior knowledge to distinguish between real oscillatory components and artifacts.

Furthermore, careful consideration is needed when applying SSA to early time regions influenced by the IRF. The rapid signal changes near time zero caused by the IRF, as well as those induced by structural changes in the ultrafast time range, can lead to the extraction of an excessive number of trend components. A critical issue is that some of these trend components may contain artifact frequencies that are not associated with actual molecular vibrations. To demonstrate this point, we generated simulated TRXL data by considering the 110 cm⁻¹ stretching vibration of I₂ and assuming an IRF with a FWHM of 100 fs (see Fig. S4). SSA was then applied to two different time ranges: one excluding the IRF-affected region (300 fs to 2 ps), and the other including it (−200 fs to 2 ps). As shown in Fig. S4, SSA applied to the time range from 300 fs to 2 ps successfully extracts the oscillatory signal corresponding to the vibration of I₂. When SSA was applied to the time range including the IRF effect, four additional trend components were extracted, as well as the oscillatory signal, to describe the rapid signal changes caused by the IRF. A major concern in this context is that some of these trend components may contain artifact frequencies unrelated to actual molecular vibrations. For example, the reconstructed signal from one of these trend components shows an oscillatory feature in the time domain, and its Fourier transform reveals two artifact frequencies, irrelevant to the molecular vibration, as shown in Fig. S4(e). We note that such artifact frequencies do not always appear in the trend components; their presence depends on factors such as the noise level and the specific characteristics of the signal. Nonetheless, to effectively utilize the information embedded in the trend components for studying structural dynamics within the IRF-affected time range, further methodological validation through follow-up studies is required.

D. Investigating wavepacket dynamics of triiodide in methanol using singular spectrum analysis on TRXL data

In Sec. II B, we demonstrated that SSA is significantly more effective than commonly used analysis methods. Therefore, we aim to apply this technique to separate the oscillatory signals from actual TRXL experimental data and observe the wavepacket dynamics involved in the photoinduced reaction. As an example, we focus on the photodissociation of the triiodide ( ${\mathrm{I}}_3^{-}$) in methanol. Upon photoexcitation with a 400 nm laser, ${\mathrm{I}}_3^{-}$ is known to decompose into ${\mathrm{I}}_2^{-}$ and an iodine radical (I·)^39–44 within 300 fs, as illustrated in Fig. 5(a). During this process, both the excited-state wavepacket in ${\mathrm{I}}_2^{-}$ and the ground-state wavepacket in ${\mathrm{I}}_3^{-}$ are generated. While the existence and dynamics of the wavepackets in the ground-state ${\mathrm{I}}_3^{-}$ and photoexcited ${\mathrm{I}}_2^{-}$ have already been confirmed through time-resolved spectroscopy,³⁹ the wavepacket dynamics have not been observed in TRXL experimental data,^40–43 due to the limitations in temporal resolution and SNR.

FIG. 5.

(a) Schematic illustration of the ultrafast reaction dynamics of ${\mathrm{I}}_3^{-}$ in methanol. Upon 400 nm laser excitation, ${\mathrm{I}}_2^{-}$ and an iodine radical (I·) are generated, accompanied by the formation of ground-state and excited-state wavepackets in ${\mathrm{I}}_3^{-}$ and ${\mathrm{I}}_2^{-}$, respectively. (b) Experimental difference scattering curves, qΔS_exp(q, t), of ${\mathrm{I}}_3^{-}$. (c) The FT amplitude spectra of qΔS_exp(q, t) of ${\mathrm{I}}_3^{-}$, FT_qΔS. (d) The first rSV of qΔS_exp(q, t) of ${\mathrm{I}}_3^{-}$ (black dots) and its fit by a convolution of the instrument response function (IRF) and an exponential decay function (red curve). (e) The fitting residual corresponding to the fit shown in (d). (f) FT amplitude spectrum of the fitting residual, FT_SVD.

In this regard, by applying the SSA method to the TRXL data of I₃⁻, we aim to extract the oscillatory components resulting from the wavepacket motion of ${\mathrm{I}}_3^{-}$ and ${\mathrm{I}}_2^{-}$ and assign the molecular vibrational modes contributing to these oscillations. The TRXL data of triiodide in methanol, qΔS_exp(q, t), are shown in Fig. 5(b), and the experimental details are provided in the supplementary material.⁴⁵ First, we checked whether the oscillatory signals arising from the wavepacket motions could be captured in FT_qΔS or FT_SVD. As shown in Fig. 5(c), no noticeable peak was observed in the FT_qΔS, indicating that oscillatory signals are not readily observable in the temporal profiles of the raw TRXL data. We also performed SVD on the TRXL data of ${\mathrm{I}}_3^{-}$ and examined the first rSV to check for the presence of oscillatory signals. As shown in Fig. 5(d), oscillations are observed in the first rSV, overlapping with the signal rise caused by the IRF at earlier times (t < 300 fs). However, these oscillations are no longer clearly visible at later times (t > 300 fs). To calculate FT_SVD, the first rSV was fitted with a single exponential decay function, and the residual was obtained by subtracting the fitted curve, as shown in Fig. 5(e). Subsequently, FT was applied to this residual to obtain the FT_SVD. However, the FT_SVD did not reveal any distinct oscillation frequency, as shown in Fig. 5(f). These results indicate that neither the FT_qΔS nor the FT_SVD of the TRXL data of triiodide was able to extract the oscillatory signals associated with the wavepacket dynamics.

Subsequently, we applied SSA to the TRXL data of triiodide, extracting the trend and oscillatory components from the data, as shown in Figs. 6(a) and 6(b), respectively. The SSA was applied to the data after 300 fs, when the dissociation into ${\mathrm{I}}_2^{-}$ and an iodine radical (I·) is completed.^39,40 The reconstructed signal of the trend, qΔS_trend(q, t), shows a gradual variation over time, whereas that for the oscillation, qΔS_oscillation(q, t), exhibits distinct periodic changes, indicating that SSA has successfully decomposed the signal into these two components. The oscillatory feature was quantitatively examined using FT_SSA, the FT spectra of qΔS_oscillation(q, t). As shown in Fig. 6(c), the FT_SSA unambiguously identifies the 110 cm⁻¹ oscillation manifested in the TRXL data of triiodide. These results demonstrate that SSA is particularly effective in uncovering oscillatory signals within the data, which would otherwise be difficult to detect using other analytical methods.

FIG. 6.

(a) and (b) Reconstructed signals of (a) the “trend” group, qΔS_trend(q, t), and (b) the “oscillation” group, qΔS_oscillation(q, t), obtained from the SSA analysis on the experimental qΔS_exp(q, t) of ${\mathrm{I}}_3^{-}$. (c) The FT amplitude spectra of the qΔS_oscillation(q, t), FT_SSA. Peaks around 110 cm⁻¹ are indicated with a magenta arrow. (d)–(f) Theoretical fits of qΔS_oscillation(q, t) obtained from structural analyses considering wavepacket motions in (d) ${\mathrm{I}}_3^{-}$, (e) ${\mathrm{I}}_2^{-}$, and (f) both ${\mathrm{I}}_3^{-}$ and ${\mathrm{I}}_2^{-}$. Among these, only the analysis incorporating both ${\mathrm{I}}_3^{-}$ and ${\mathrm{I}}_2^{-}$, shown in (f), provides satisfactory fits to the experimental qΔS_oscillation(q, t). (g) Time-dependent I–I distances, r₁(t) (cyan), r₂(t) (blue), and r₃(t) (black) of ${\mathrm{I}}_3^{-}$, and r(t) of ${\mathrm{I}}_2^{-}$ (red), determined from the structural analysis. The frequencies of two vibrational modes contributing to qΔS_oscillation(q, t), the symmetric stretching mode of ${\mathrm{I}}_3^{-}$ (110 cm⁻¹) and the stretching mode of ${\mathrm{I}}_2^{-}$ (107 cm⁻¹), are indicated. The definitions of these structural parameters for ${\mathrm{I}}_3^{-}$ and ${\mathrm{I}}_2^{-}$ are illustrated at the top.

To elucidate the origin of the 110 cm⁻¹ oscillation observed in the FT_SSA, structural analyses were performed using qΔS_oscillation(q, t). Since the TRXL signal is sensitive to electronic states involving structural changes, both the wavepackets on the PES of the ground-state ${\mathrm{I}}_3^{-}$ and the excited-state ${\mathrm{I}}_2^{-}$ could serve as the origin of the oscillation frequency. Considering that only one frequency of the 110 cm⁻¹ was observed, we primarily considered the possibility that the vibration of either ${\mathrm{I}}_2^{-}$ or ${\mathrm{I}}_3^{-}$ was detected. Structural analyses were performed considering the symmetric stretching mode of ${\mathrm{I}}_3^{-}$ or the stretching mode of ${\mathrm{I}}_2^{-}$, and the fitting results are shown in Figs. 6(d) and 6(e), respectively. The details of the structural analysis are described in the supplementary material. Compared to the experimental qΔS_oscillation(q, t) in Fig. 6(b), neither of the two results exhibits satisfactory agreement. As a result, it was confirmed that the motion of the wavepacket by the single vibrational mode of either ${\mathrm{I}}_2^{-}$ or ${\mathrm{I}}_3^{-}$ cannot account for the experimental data. We note that the other vibrational modes of ${\mathrm{I}}_3^{-}$, such as the asymmetric stretching and bending modes, exhibited worse agreement with the experimental data compared to the symmetric stretching mode. Then, we performed structural analysis considering both the symmetric stretching mode of ${\mathrm{I}}_3^{-}$ and the stretching mode of ${\mathrm{I}}_2^{-}$. The resulting fitted curves shown in Fig. 6(f) are in good agreement with the experimental data, suggesting that the 110 cm⁻¹ oscillation originates not from a single species, but from both the ground-state ${\mathrm{I}}_3^{-}$ and the excited-state ${\mathrm{I}}_2^{-}$. As a result of the structural analysis, the frequencies of the symmetric stretching mode of ${\mathrm{I}}_3^{-}$ and the stretching mode of ${\mathrm{I}}_2^{-}$ were determined to be 110 cm⁻¹ and 107 cm⁻¹, respectively. Furthermore, the time-dependent evolution of the structural parameters for ${\mathrm{I}}_3^{-}$ and ${\mathrm{I}}_2^{-}$ was obtained, as shown in Fig. 6(g). The frequencies of the two vibrational modes extracted through SSA from the TRXL of triiodide were observed to be similar to those of the symmetric stretching mode of ${\mathrm{I}}_3^{-}$ and the stretching mode of ${\mathrm{I}}_2^{-}$ (111 and 114 cm⁻¹, respectively) observed in femtosecond transient transmission experiments under ethanol solvent conditions.³⁹ In addition, the phases of the two modes are not synchronized, as shown in Fig. 6(g) and Table S2. This observation can be explained by the fact that the initial phase of the ground-state wavepacket, generated via resonant impulsive stimulated Raman scattering, reflects the initial structural displacement in the excited state occurring within the IRF. As a result, the two vibrational modes are not in phase but exhibit a relative phase difference. More detailed results of the structural analyses are summarized in Tables S2–S4.

It is worth noting that while only a single frequency could be resolved when examining the time-domain information through the FT spectra, two closely spaced vibrational frequencies were unambiguously determined by incorporating structural information from the q-dependent scattering signals. These results demonstrate that SSA applied to TRXL data is effective for investigating wavepacket dynamics.

In this study, the symmetric stretching mode of ${\mathrm{I}}_3^{-}$ was observed, whereas the asymmetric stretching and bending modes were not observed. The structural change associated with the bending mode of the ${\mathrm{I}}_3^{-}$ molecule is primarily reflected in the time-dependent variation of r₃ [see Fig. 6(g) for the definition of structural parameters]. However, the variation in r₃(t) is influenced not only by the bending mode but also by the symmetric and asymmetric stretching modes, so that it is difficult to isolate the contribution of a specific vibrational mode. Furthermore, since r₃(t) is inherently longer than r₁(t) and r₂(t), the corresponding scattering signal has a smaller magnitude, leading to a lower SNR. For the asymmetric stretching mode, since r₁(t) and r₂(t) vary in an anti-phase manner, the resulting difference scattering signal is intrinsically weaker than that of the symmetric stretching mode. It is also noteworthy that the wavenumber of the asymmetric stretching mode of ${\mathrm{I}}_3^{-}$ reported in an ethanol solvent environment is 143 cm⁻¹,³⁹ which is higher than that of the symmetric stretching mode. Accordingly, the faster oscillation frequency would have further reduced the detectable signal amplitude due to convolution with the IRF. Consequently, the limited SNR and temporal resolution of the current experimental data likely hindered the precise observation of the bending and asymmetric stretching modes. The relatively small contribution of r₃ to the x-ray scattering signal may also result in greater uncertainty in the determination of θ, the bond angle of ${\mathrm{I}}_3^{-}$, as θ directly determines r₃. For example, a previous TRXL study on ${\mathrm{I}}_3^{-}$ reported a bent equilibrium geometry,⁴⁰ whereas the optimized structure in this study corresponds to a linear geometry. This discrepancy can be attributed to the lower sensitivity of the x-ray scattering signal to variations in θ compared to other structural parameters. Indeed, as shown in Fig. S5, the fit quality to the experimental curves does not vary significantly for θ values between 160° and 180°, which may account for the discrepancy regarding the equilibrium geometry of ${\mathrm{I}}_3^{-}$ between the two studies.

In conclusion, SSA enabled the efficient extraction of oscillatory signals associated with the wavepacket dynamics of ${\mathrm{I}}_3^{-}$, which could not be clearly distinguished using other analytical methods. Even though this study focused on the application of SSA to TRXL data, SSA can be applied to various types of time-resolved experimental data. Therefore, it is expected that the impact of this work will extend beyond reaction dynamics research using TRXL, with significant implications for follow-up studies in various experimental fields. On a final note, technical advancements at XFEL facilities are expected to further enhance the efficiency of extracting oscillatory signals using SSA. For example, increased photon flux, higher repetition rates, or higher-energy x-ray pulses can enable the measurement of TRXL signals over a wider q range, or allow signals to be collected at more time points within the same data acquisition time while maintaining a similar level of SNR. Such experimental improvements would allow SSA to more clearly resolve oscillatory components embedded in experimental data, as demonstrated in Figs. S6 and S7.

III. CONCLUSION

In this study, we propose the application of SSA as an effective strategy to precisely track molecular wavepacket dynamics using TRXL data. Given that TRXL signals are sensitive to multiple molecular species and reaction pathways, with oscillatory components accounting for only a minor fraction of the overall signal, it is often challenging to isolate these components due to limitations in temporal resolution and SNR. We addressed these challenges by introducing an advanced analytical approach for efficiently extracting molecular vibrational signals from TRXL data.

Through a comparative investigation of various analysis methods applied to simulated TRXL data, we demonstrated that SSA is particularly effective in extracting oscillatory components. Specifically, compared to commonly used methods such as FT and SVD, SSA was able to isolate vibrational signals even under low SNR conditions. Subsequently, SSA was applied to the experimental TRXL data for the photoinduced reaction of ${\mathrm{I}}_3^{-}$. As a result, a vibrational component at 110 cm⁻¹, which was difficult to detect using other analysis methods, was successfully extracted. Based on this component, structural analysis was performed, which enabled us to reveal the wavepacket dynamics on the PESs of ground-state ${\mathrm{I}}_3^{-}$ and excited-state ${\mathrm{I}}_2^{-}$, associated with the symmetric stretching mode of ${\mathrm{I}}_3^{-}$ and the stretching mode of ${\mathrm{I}}_2^{-}$. Follow-up studies utilizing TRXL with SSA are expected to provide deeper insights into various photochemical reactions and complex non-equilibrium reaction pathways. Moreover, given the versatility of SSA in analyzing various time-resolved experimental data, it has the potential to serve as a promising analytical tool for advancing research in ultrafast reaction dynamics.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Jaeseok Kim: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (equal); Methodology (lead); Resources (equal); Visualization (equal); Writing – original draft (equal). Hyunwoo Jeong: Data curation (supporting); Formal analysis (supporting). Jae Hyuk Lee: Data curation (supporting); Resources (equal). Rory Ma: Data curation (supporting); Resources (equal). Daewoong Nam: Data curation (supporting); Resources (equal). Minseok Kim: Data curation (supporting); Resources (equal). Dogeun Jang: Data curation (supporting); Resources (equal). Jong Goo Kim: Conceptualization (equal); Funding acquisition (lead); Investigation (equal); Resources (equal); Supervision (lead); Validation (lead); Visualization (equal); Writing – original draft (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.