Photodissociation Processes Involving the SiH⁺ Molecular Ion: New Datasets for Modeling

Abstract

This paper investigates the photodissociation of the ${\mathrm{S}\mathrm{i}\mathrm{H}}^{+}$ molecular ion, a non-symmetric diatomic species composed of silicon and hydrogen. We provide calculated molecular data and characterize electronic states, deriving cross-sections and spectral absorption rate coefficients as functions of temperature (1000–10,000 K) and EUV and UV wavelength. The calculations are performed within a quantum–mechanical framework of bound–free radiative transitions, using ab initio electronic potentials and dipole transition functions as inputs. In addition, we present a straightforward fitting formula that enables practical interpolation of photodissociation cross-sections and spectral rate coefficients, providing a novel closed-form representation of the dataset for modeling purposes. The resulting dataset provides a consistent and accessible reference for advanced photochemical modeling in laboratory plasmas and astrophysical environments.

Dataset: https://doi.org/10.5281/zenodo.15880488.

Dataset License: CC-BY 4.0.

1. Introduction

Given the complexity of electron dynamics in molecules and the sophistication of experimental setups in this field, atomic and molecular (A&M) data and databases play a crucial role in advancing experimental techniques and providing accurate theoretical support [1,2,3,4,5,6]. Additionally, studying the optical properties of various small molecules, along with their corresponding A&M data, remains highly relevant [7,8,9,10,11]. Precision spectroscopy of molecular ions is particularly significant for applications in astrochemistry, quantum-controlled chemical reactions, and fundamental constant measurements [12,13,14,15,16,17,18]. Such high-precision spectroscopy also enables the search for the astrophysical presence of these small molecules [4,19]. Accurate cross-section data and rate coefficients, covering a wide spectral and thermal range, are essential for modeling such systems and for designing reliable diagnostic tools.

Photodissociation is a key process responsible for molecular fragmentation in diffuse interstellar clouds, making accurate rate calculations essential for predicting interstellar molecular abundances [20,21,22]. These processes are critical in modeling chemical interactions across various astrophysical environments, including dense cloud edges near young stars, protoplanetary disk surfaces, stellar envelopes, and galactic-scale molecular clouds (see e.g., [23] and references therein). Photoprocesses involving small molecular ions such as ${\mathrm{S}\mathrm{i}\mathrm{H}}^{+}$ are particularly valuable in this regard, as they serve as sensitive probes of the interplay between radiation and matter in ionized astrophysical media. In particular, the recent multistate nonadiabatic study of ${\mathrm{S}\mathrm{i}\mathrm{H}}^{+}$ photodissociation [24] has provided extensive data involving coupled excited states and demonstrated the occurrence of rovibrationally resolved Feshbach resonances. While such resonance-resolved calculations are highly detailed, a consistent dataset of cross-sections and rate coefficients covering broad temperature and wavelength ranges, in a form readily applicable to plasma-chemistry and radiative-transfer models, is still lacking. The novelty of the present dataset lies in providing smooth, temperature- and wavelength-resolved cross-sections together with compact analytical fits, thereby ensuring direct usability in modeling environments where resonance-resolved data may be impractical. The present contribution thus addresses this gap by offering a practical complement to existing resonance-focused studies.

Silicon-bearing molecules are important trace components in late-type stars and star-forming regions, accounting for nearly 10% of identified molecular species in space [25]. Recent research underscores the significance of the optical properties of small interstellar molecules like $\mathrm{S}\mathrm{i}\mathrm{H}$ and ${\mathrm{S}\mathrm{i}\mathrm{H}}^{+}$, which have been incorporated into key A&M databases [25,26,27,28]. Silicon hydride cations hold astrophysical importance as constituents of interstellar clouds and stellar atmospheres [29]. Protonated silicon ${\mathrm{S}\mathrm{i}\mathrm{H}}^{+}$ is found in astrophysical environments, and ultrafast laser studies could help simulate conditions found in stellar and interstellar media [30]. This also places ${\mathrm{S}\mathrm{i}\mathrm{H}}^{+}$ among the diatomic ions considered for controlled optical manipulation, including laser-cooling candidates in closed-shell ${}^1\mathsf{\Sigma }-{}^1\mathsf{\Sigma }$ systems, as demonstrated in [31]. Recent experiments by Mosnier et al. [32] demonstrated the production of ${\mathrm{S}\mathrm{i}\mathrm{H}}^{+}$ ions via an ECR source with silane, using the photon-ion merged-beam setup at the PLEIADES beamline. Their combined experimental and theoretical results provide crucial data for identifying these ions in astronomical and laboratory plasmas.

Due to the complexity of electron dynamics in molecules, A&M datasets and services are essential for the development of new experimental techniques and reliable theoretical models (see e.g., [32,33,34,35,36]). Our objective is to obtain spectroscopic data on systems involving hydrogen and silicon atoms, ions, molecules, and molecular ions. For instance, ${\mathrm{S}\mathrm{i}\mathrm{H}}^{+}$ molecular ions are of particular scientific interest due to their potential applications in astrophysics and fundamental physics [32,33,34,35,36,37,38,39]. Unlike earlier works that primarily focused on either experimental production or limited state-specific calculations [35,36,37,38,39], the present dataset provides temperature- and wavelength-resolved cross-sections and rate coefficients in a consistent framework. For these systems, we have calculated spectral absorption rate coefficients and average cross-sections across a broad range of temperatures and wavelengths in the extreme ultraviolet (EUV) and ultraviolet (UV). The resulting datasets can be applied to critical areas such as modeling and experiments at the PLEIADES SOLEIL synchrotron [33]. Moreover, the generated data are relevant for diverse applications, including the development of laser-based diagnostics, modeling of laboratory plasmas, and studies of ultrafast light–matter interaction under astrophysically inspired conditions [32,34]. We emphasize that all data supporting the findings of this study, including potential energy curves, corresponding dipole matrix elements, partial and average photodissociation cross-sections for the investigated species, spectral rate coefficients, and fit coefficients for the simplified photodissociation rate formulas, are openly available in the Supplementary Materials on Zenodo under the dataset identifier https://doi.org/10.5281/zenodo.15880488.

A brief theoretical background and the calculated quantities are presented in Section 2. Section 3 describes and discusses the datasets, i.e., results, particularly in relation to laboratory and astrophysical investigations with some representative examples. Finally, Section 4 is devoted to conclusions and future research directions.

2. Theory

This paper investigates the effects of radiative processes on the optical characteristics of weakly ionized laboratory and astrophysical plasmas. The focus is on bound–free photodissociation processes in non-symmetric $\mathrm{S}\mathrm{i}\mathrm{H}+$ molecular ion. Specifically, we consider the processes that can be described as non-symmetric:

$${\epsilon }_{\lambda }+{AB}^{+}\Rightarrow A^{+}+B,$$

(1)

where $B$ denotes a silicon atom ($\mathrm{S}\mathrm{i}$), $A=H$, and ${AB}^{+}$ represents the ${\mathrm{S}\mathrm{i}\mathrm{H}}^{+}$ molecular ion in its ground electronic state. In this section, the nuclear and electronic motions are separated within the Born–Oppenheimer approximation, and photodissociation is treated as an electric-dipole-allowed bound–free transition between adiabatic electronic states. The interaction with the external field is handled in the dipole (length–gauge) approximation, which is appropriate in the EUV/UV spectral ranges considered and for the internuclear separations that dominate the transition probability. Selection rules for electric-dipole transitions in diatomic ($\mathsf{\Delta }\Lambda = 0, \pm 1; \mathsf{\Delta }S = 0$ in the absence of strong spin-orbit mixing; parity change) are thus enforced throughout the formulation.

In this adiabatic framework, the electronic potentials and transition dipole functions define the radial nuclear dynamics. The dominant photodissociation channel in the EUV/UV is captured by an electric-dipole bound–free transition from ${\mathrm{X}}^1{\mathsf{\Sigma }}^{+}$ to the first excited adiabatic state; this yields a self-consistent and tractable description while retaining the physically relevant contributions for modeling purposes. The adopted level of theory enforces the correct selection rules and near-threshold behavior and is compatible with extensions to additional excited states when needed [24,39,40,41,42].

2.1. The Method

Bound–free processes, i.e., the photodissociation of diatomic molecular ions, are of fundamental importance in astrophysics for modeling stellar atmospheres and astrochemical environments [3,4,16,17]. They are also relevant in theoretical investigations and laboratory plasma diagnostics (see e.g., [3,41], and references therein).

We investigate the photodissociation process given by

$${\epsilon }_{\lambda }+{\mathrm{S}\mathrm{i}\mathrm{H}}^{+}\left({\mathrm{X}}^1{\mathsf{\Sigma }}^{+}\right)\Rightarrow \mathrm{S}\mathrm{i}+{\mathrm{H}}^{+}$$

(2)

The rate coefficients and cross-sections are determined within a quantum mechanical framework, where photodissociation is modeled as a radiative transition between the ground state and the first excited adiabatic electronic state of the molecular ion. These transitions occur due to the interaction between the ion-atom system and the external electromagnetic field, treated within the dipole approximation (see e.g., [41]).

A photon of energy ${\epsilon }_{\lambda }=hc/\lambda$promotes a bound rovibrational level $(v,J)$ on ${\mathrm{X}}^1{\mathsf{\Sigma }}^{+}$ into the continuum of the excited adiabatic state. Energy conservation fixes the exit-channel kinetic energy as $E\prime ={\epsilon }_{\lambda }+{\epsilon }_{J,v}$, where $E_{J,v}=E_{dis}+{ϵ}_{J,v}$ with ${ϵ}_{J,v}<0$ the binding energy relative to the dissociation limit $E_{dis}$. Close to threshold the cross-section follows the Wigner law determined by the exit-channel partial wave, whereas at shorter wavelengths it is governed by the Franck–Condon window, i.e., the radial overlap between the bound wavefunction and the continuum scattering state weighted by the transition dipole function $D\left(R\right)$ [3,39,40,41,42]. This provides a transparent physical picture for the wavelength dependence of the bound–free strength.

The molecular input (adiabatic potential energy curves and $D\left(R\right)$) is used to compute bound ${\psi }_{J,v}(R)$ and energy-normalized continuum ${\psi }_{J\pm 1,E\prime }(R)$ nuclear wavefunctions by solving the radial Schrödinger equation with appropriate boundary conditions. Dipole matrix elements $⟨{\psi }_{J,v}(R)|D\left(R\right)|{\psi }_{J\pm 1,E\prime }(R)⟩$ are evaluated on a dense $R\text{-}$grid to ensure convergence with respect to grid spacing and box size; thermal effects are subsequently incorporated by averaging over Boltzmann-populated initial states.

2.2. The Spectral Characteristics

For the species under investigation, the mean thermal photodissociation cross section ${\sigma }^{\left(bf\right)}(\lambda ,T)$ can be represented in the form given by Srećković et al. [4,42]:

$${\sigma }^{\left(bf\right)}\left(\lambda ,T\right)={\displaystyle \frac{\sum _{J,v}\left(2J+1\right)e^{\frac{-E_{J,v}}{kT}}·{\sigma }_{J,v}^{\left(bf\right)}\left(\lambda \right)}{\sum _{J,v}\left(2J+1\right)e^{\frac{-E_{J,v}}{kT}}}}.$$

(3)

Here, ${\sigma }_{J,v}^{\left(bf\right)}\left(\lambda ,T\right)$ represents the population-weighted mean cross section of all populated initial levels under local thermodynamic equilibrium (LTE), ${\sigma }_{J,v}^{\left(bf\right)}\left(\lambda \right)$ is the partial photodissociation cross section from the initial level $(v,J)$, and $E_{J,v}$ is the energy of that level measured from the ground rovibrational state. In this context, $E_{J,v}=E_{dis}+{ϵ}_{J,v}$, with ${ϵ}_{J,v}<0$ the binding energy relative to the dissociation limit $E_{dis}$ (values of ${ϵ}_{J,v}$are determined as described in [42]). Within the electric-dipole approximation, the partial cross sections are given by

$${\sigma }^{\left(bf\right)}\left(\lambda \right)={\displaystyle \frac{8{\pi }^3}{3\lambda }}\left[{\displaystyle \frac{J+1}{2J+1}}{\left|D_{J,v;J+1,E_{imp}^{\prime }}\right|}^2+{\displaystyle \frac{J}{2J+1}}{\left|D_{J,v;J-1,E_{imp}^{\prime }}\right|}^2\right].$$

(4)

The two terms in Equation (4) correspond to the rotational branches with $\mathsf{\Delta }J=+1$($R$ branch) and $\mathsf{\Delta }J=-1$ ($P$ branch). For $\mathsf{\Sigma }\to \mathsf{\Sigma }$ transitions, the associated Hönl–London line-strength factors take their standard forms $S_R=(J+1)/(2J+1)$ and $S_P=J/(2J+1)$; the first and second terms of Equation (4) therefore represent the $R-$ and $P-$ branch contributions, respectively. For other symmetry changes (e.g., $\mathsf{\Sigma }\to \mathsf{\Pi }$), the corresponding Hönl–London factors should be used [24]. The dipole matrix elements are defined as $D_{J,v;J\pm 1,E_{imp}^{\prime }}=⟨in,J,v; R|D_{in,fin}\left(R\right)|fin,J\pm 1,E^{\prime }⟩$, where $D_{in,fin}\left(R\right)=⟨in;R|D\left(R\right)|fin;R⟩$ is the electronic dipole matrix element and $D\left(R\right)$ is the dipole-moment operator of the system. The continuum kinetic energy entering the final state wavefunction is $E^{\prime }={ϵ}_{J,v}+{\epsilon }_{\lambda }$. Physically, ${\left|D_{J,v;J^{\prime },E^{\prime }}\right|}^2$ quantifies the Franck–Condon-like radial overlap modulated by $D\left(R\right)$: the prefactor $1/\lambda$ reflects the photon density of states, while the dependence on $E^{\prime }\left(\lambda \right)$ governs the near-threshold rise (Wigner law) and shorter-wavelength oscillatory modulations of ${\sigma }^{\left(bf\right)}\left(\lambda \right)$ (for details see Ref. [4]).

The photodissociation spectral rate coefficient can be formulated using Equations (3) and (4), as outlined in [42]:

$$K^{\left(bf\right)}\left(\lambda ,T\right)={\sigma }^{\left(bf\right)}\left(\lambda \right)·{\zeta }^{-1}\left(T\right),$$

(5)

where the factor $\zeta$ is given by the expression:

$$\zeta \left(T\right)={\displaystyle \frac{g_1{g}_2}{g_{12}}}{\left({\displaystyle \frac{\mu kT}{2\pi {\hslash }^2}}\right)}^{{\displaystyle \frac{3}{2}}}·{\displaystyle \frac{1}{\sum _{J,v}\left(2J+1\right)e^{\frac{E_{dis}-E_{J,v}}{kT}}}}.$$

(6)

In this context, $g_1$, $g_2$, and $g_{12}$ represent the electronic statistical weights of the species $A$, $B+,$ and $AB+$, respectively. The quantity ${\sigma }^{\left(bf\right)}(\lambda ,T)$ is defined by Equation (3), and $\mu$ denotes the reduced mass of the ion-atom system under consideration.

3. Results and Discussion

3.1. The Obtained Quantities

The results are presented and illustrated in this section through Figure 1, Figure 2 and Figure 3. The dataset covers the UV spectral region and includes a wide temperature range, from 1000 K to 10,000 K. These findings may be valuable for laboratory plasma diagnostics, astrophysical research, and industrial plasma modeling.

The calculated potential energy curves of the molecular ion and the corresponding electronic dipole matrix element, $D_{in,fin}\left(R\right)$, are shown in Figure 1. These quantities are essential inputs for further calculations, such as the partial photodissociation cross-sections described by Equation (4) in Section 2.2. Specifically, we computed the partial cross-sections ${\sigma }_{J,v}^{\left(bf\right)}(\lambda )$ for the rovibrational states of the ${\mathrm{S}\mathrm{i}\mathrm{H}}^{+}$ molecular ion. The adopted potential energy curves and dipole transition functions were obtained from ab initio electronic-structure calculations, and convergence was verified with respect to basis set and grid parameters. Estimated uncertainties of the potential curves and dipole functions, obtained by varying grid and basis parameters, remain within 5–10%, consistent with typical deviations reported in the literature [3,24,29,32,39,40,41,42].

Figure 2 displays t, he computed average thermal photodissociation cross section, ${\sigma }_{{\mathrm{S}\mathrm{i}\mathrm{H}}^{+}}^{\left(bf\right)}(\lambda )$, for the silicon hydride molecular ion across a broad range of temperatures (1000 K ≤ T ≤ 10,000 K) and wavelengths (100 nm ≤ λ ≤ 190 nm). The results show a strong wavelength dependence, and the surface plot reveals a nontrivial interplay between thermal effects and photonic excitation. Distinct maxima appear in the 180–190 nm wavelength range, highlighting the spectral regions of most efficient dissociation.

Furthermore, absorption coefficients as functions of wavelength and temperature are provided. Figure 3 presents the bound–free spectral rate coefficient, $K_{{\mathrm{S}\mathrm{i}\mathrm{H}}^{+}}^{\left(bf\right)}(\lambda ,T)$, illustrating that the rate coefficients exhibit clear temperature- and wavelength-dependent maxima. These results enable direct implementation of spectral absorption rates into radiative-transfer models and can support a wide range of plasma simulations based on specified environmental conditions and composition.

3.2. Analytical Fitting of the Data

To facilitate the convenient and effective use of calculated data in modeling and the interpretation of experimental results in laboratories, we provide a simple and practical fitting formula for photodissociation cross-sections and spectral rate coefficients. This formula is expressed as a second-degree logarithmic polynomial, obtained via the least-squares fitting method:

$$\mathrm{l}\mathrm{o}\mathrm{g}\left(M^{\left(bf\right)}\left(\lambda ;T\right)\right)=\sum _{k=0}^2{\varsigma }_k\left(\lambda \right){\left(\log \left(T\right)\right)}^k.$$

(7)

Here, $M^{\left(bf\right)}\left(\lambda ;T\right)$ denotes either the photodissociation cross section ${\mathsf{\sigma }}^{\left(bf\right)}\left(\lambda ,T\right)$ or the spectral rate coefficient $K^{\left(bf\right)}\left(\lambda ,T\right)$. The fitting expression is valid within the temperature range 1000 K ≤ T ≤ 10,000 K. Although it is possible that the approximation remains reasonable outside this interval, caution is advised when applying it beyond the stated limits.

The main advantage of this representation is its compact analytical form, which allows fast interpolation on modeling grids and easy use in radiative-transfer and plasma-chemistry codes without relying on large tables of data. A second-degree polynomial fit naturally smooths out narrow resonance structures seen in more complex multi-state calculations [24], so it should be regarded as a practical baseline dataset rather than a full description of all fine details. This balance between accuracy and simplicity makes the dataset especially useful: it provides smooth and numerically stable cross-sections and rate coefficients that can be directly applied in large-scale simulations. Compared to the resonance-resolved results of Yang et al. [24], which contain more spectral detail but are harder to implement in modeling codes, our approach gives broad, temperature- and wavelength-resolved data in a simple analytic form.

Table 1. The fitted data, specifically the coefficients ${\varsigma }_k\left(\lambda \right)$ in Equation (7), for the photodissociation cross-section.

$\mathit{\lambda }$	${\mathit{\varsigma }}_0\left(\mathit{\lambda }\right)$	${\mathit{\varsigma }}_1\left(\mathit{\lambda }\right)$	${\mathit{\varsigma }}_2\left(\mathit{\lambda }\right)$
100	−200.194	90.7844	−11.4763
110	−200.498	91.0462	−11.5121
120	−198.356	89.9434	−11.3617
130	−195.859	88.6491	−11.1867
140	−232.501	108.653	−13.8974
150	−233.995	109.533	−14.0158
160	−234.088	109.661	−14.0318
170	−233.496	109.43	−13.998
180	−236.571	111.195	−14.2302
190	−249.407	118.105	−15.1371
195	−257.007	121.971	−15.6338

and

Table 2. Same as in Table 1 but for the spectral rate coefficient for the photodissociation reaction.

$\mathit{\lambda }$	${\mathit{\varsigma }}_0\left(\mathit{\lambda }\right)$	${\mathit{\varsigma }}_1\left(\mathit{\lambda }\right)$	${\mathit{\varsigma }}_2\left(\mathit{\lambda }\right)$
100	−20.6474	−9.03377	0.93579
110	−20.4778	−9.03016	0.934951
120	−20.4994	−8.95449	0.925523
130	−20.4884	−8.89549	0.917051
140	−20.4885	−8.84382	0.911691
150	−20.5493	−8.74374	0.899024
160	−20.6023	−8.63835	0.885957
170	−20.7367	−8.47388	0.866164
180	−21.8542	−7.77415	0.778526
190	−27.3922	−4.83864	0.410436
195	−32.0192	−2.59088	0.133176

list the coefficients ${\varsigma }_k\left(\lambda \right)$ i.e., selected fitting parameters for the photodissociation cross sections and spectral rate coefficients. This smoothed representation is intended as a modelig complement to state-resolved theoretical studies such as Yang et al. [24], where narrow resonance features are resolved but not directly usable in numerical plasma or radiative-transfer codes.

3.3. Astrophysical Importance

We note that many astronomical software tools and numerical codes handle and make use of photodissociation data (e.g., [1,2]). The chemistry of interstellar and circumstellar space depends heavily on UV photons. It has been known that the abundances of atoms and small molecules in diffuse interstellar clouds are regulated by photodissociation and photoionization processes (see e.g., [38,43]). Small molecules found in cometary comae result from the photodissociation of parent compounds, driven by the Sun’s ultraviolet radiation [23,44]. Photodissociation processes play a crucial role in modeling the chemistry of a wide range of astrophysical regions, including dense clouds near young stars, protoplanetary disks, evolved stars’ envelopes, and giant molecular clouds on galactic scales (see e.g., [39,45]). These gas and dust clouds are referred to as photodissociation or photon-dominated areas when photodissociation is the predominant molecular destruction pathway [46].

To assess the reliability of the present data, we performed consistency checks against available theoretical and experimental studies. In particular, the overall magnitude and spectral dependence of our cross-sections agree qualitatively with the uncoupled calculations of Stancil et al. [29], as well as with the merged-beam measurements of Mosnier et al. [32]. The smooth temperature-resolved profiles are complementary to the more detailed nonadiabatic study of Yang et al. [24], which reports state-specific Feshbach resonances. This level of agreement indicates that the present dataset, though simplified, captures the main trends relevant for modeling purposes. Taken together, these consistency checks suggest that the present dataset is reliable within the quoted uncertainty margins, which is sufficient for its intended application in modeling frameworks.

Photodissociation cross-sections and rates obtained through laboratory measurements or theoretical calculations can be incorporated into simulation tools such as CLOUDY or PDR Toolbox to investigate the chemical and thermal structure of interstellar and circumstellar environments (see, e.g., Ferland et al. [2]). Integrating such cross-sections into chemical reaction networks enables more accurate predictions of molecular abundances, spectral features, and the evolution of photon-dominated regions (PDRs) under UV radiation. In this context, the present dataset provides the missing input required for modeling the destruction of $\mathrm{S}\mathrm{i}\mathrm{H}+$, directly affecting silicon-bearing chemistry and radiative signatures in UV-irradiated astrophysical environments.

Radiation Field and Photodissociation Rates

Photodissociation rates, denoted as $k\left(T\right)$, are crucial for modeling the abundance and time-dependent behavior of chemical species in environments such as the interstellar medium, stellar atmospheres, and planetary atmospheres. When a molecule is broken apart by radiation within a wavelength range ${\lambda }_1$ to ${\lambda }_2$ and a spectral flux $F\left(\lambda \right)$, the temperature-dependent photodissociation rate is calculated using:

$$k\left(T\right)=\underset{{\lambda }_1}{\overset{{\lambda }_2}{\int }}F\left(\lambda \right)·\sigma (\lambda ,T)d\lambda .$$

(8)

The results discussed here enable the computation of integrated quantities over wavelength by combining cross sections or rate coefficients with either the stellar or interstellar radiation spectrum.

The particular astrophysical environment determines which flux model is used. For instance, the radiation from a hot star with an effective temperature of 20,000 K can be approximated using a Planck distribution at that temperature.

The interstellar radiation field (ISRF), as fitted by van Dishoeck and Black [47], is described analytically for wavelengths up to 2000 nm by

$$F(\lambda )=3.67·{10}^4·{\lambda }^{0.7}.$$

(9)

Here, $F\left(\lambda \right)$is expressed in units of ${\mathrm{s}}^{-1} {\mathrm{c}\mathrm{m}}^{-2} {\mathrm{n}\mathrm{m}}^{-1}$. We adopt the ISRF as a representative radiation model because it is a standard reference for diffuse interstellar UV fields and provides a convenient basis for comparison with astrochemical simulations. Nevertheless, Equation (8) is general and can equally be applied with other radiation environments, such as stellar blackbody spectra or detailed stellar atmosphere models, by inserting the corresponding flux $F\left(\lambda \right)$. Specifically, in this study, we present temperature-dependent photodissociation rates for ${\mathrm{S}\mathrm{i}\mathrm{H}}^{+}$, as shown in Figure 4. The rate $k\left(T\right)$ is derived using Equation (8), with $F\left(\lambda \right)$ from Equation (9), integrated over the wavelength interval from 100 nm to 200 nm. The resulting $k\left(T\right)$ values are given in units of ${\mathrm{s}}^{-1}$.

3.4. Laboratory Importance

The photodissociation of the ${\mathrm{S}\mathrm{i}\mathrm{H}}^{+}$ hydride ion represents an important process in laboratory plasma physics and molecular spectroscopy. Unlike earlier works that primarily focused on either experimental production or limited state-specific calculations [32,33,34,35,36,37,38,39,40], the present dataset provides temperature- and wavelength-resolved cross-sections and rate coefficients in a consistent framework. Analogies may be drawn with related molecular ions such as $\mathrm{C}\mathrm{H}+$ and $\mathrm{S}\mathrm{i}\mathrm{O}+$, which display similar UV-induced dissociation pathways arising from transitions to repulsive electronic states. In $\mathrm{C}\mathrm{H}+$ this mechanism is well established through high-resolution studies, while in $\mathrm{S}\mathrm{i}\mathrm{O}+$, the heavier silicon atom shifts the absorption features toward longer wavelengths. $\mathrm{S}\mathrm{i}\mathrm{H}+$ occupies an intermediate position between these two systems, both in terms of electronic structure and dissociation dynamics, which places the present results in the broader context of small, ionized hydrides relevant in both astrochemical and laboratory plasma environments. Its investigation contributes to a better understanding of semiconductor processing and ionized plasma environments [32,35,48]. With respect to plasma and semiconductor research, silicon-containing molecules are commonly found in plasma environments, such as those involved in plasma etching and thin-film deposition in semiconductor industries [49]. Gaining insight into the dissociation and ionization dynamics of these systems is essential for optimizing plasma-enhanced chemical vapor deposition processes. In spectroscopic studies, laboratory measurements of photodissociation and photoionization cross-sections provide essential reference data for astrophysical and plasma modeling (see e.g., [32]). Interactions between VUV and EUV and silicon molecular ions are studied to determine bond dissociation energies and possible reaction pathways.

The photodissociation of the $\mathrm{S}\mathrm{i}{\mathrm{H}}^{+}$ hydride ion represents an important process in laboratory plasma physics and molecular spectroscopy. Unlike earlier works that primarily focused on either experimental production or limited state-specific calculations [32,33,34,35,36,37,38,39,40], the present dataset provides temperature- and wavelength-resolved cross-sections and rate coefficients in a consistent framework.

3.5. Further Study

We will develop a VAMDC-compatible data model [3], create a relational database, and write Python 3.13.5/SQL scripts for data conversion. The data will be published on the SerVO website, providing users with a single platform to access atomic and molecular data (see e.g., [11,24,40]). Additionally, we will offer a simple yet accurate fitting formula for the photodissociation cross-sections and spectral rate coefficients, represented as a second-degree polynomial fit to the numerical results. This will facilitate practical use of the data and will be integrated into the website.

3.6. Development

To further advance this research, we plan to extend our approach to larger and more complex molecules. Addressing these systems will require improvements in the methods and calculations related to molecular properties. This will enhance the interpretation of experimental results in high-energy plasma environments [18,33] and provide a broader dataset for astrochemical modeling [50,51].

4. Summary

This work investigates the photodissociation of ${\mathrm{S}\mathrm{i}\mathrm{H}}^{+}$ molecular ion, focusing on the average cross-sections and spectral rate coefficients of protonated silicon over a temperature range of 1000–10,000 K and across the EUV and UV wavelength spectrum. All data from this study, including potential energy curves, corresponding dipole matrix elements, partial and average photodissociation cross-sections for the investigated species, spectral rate coefficients, and fit coefficients for the simplified photodissociation rate formulas, are openly available in the Supplementary Materials on Zenodo under the dataset identifier https://doi.org/10.5281/zenodo.15880488. The data has implications for both laboratory and astrophysical applications, including spectroscopy, synchrotron studies, and modeling of weakly ionized regions in stellar and interstellar environments. The ${\mathrm{S}\mathrm{i}\mathrm{H}}^{+}$ molecular ion plays a significant role in interstellar chemistry, with attosecond dynamics offering insights into space modeling. Its behavior under the attosecond pulses in laser-driven plasmas and in confined molecular systems provides insights into interstellar dust chemistry. In this study, we calculate photodissociation rates using the ISRF as a representative astrophysical example. The present dataset is restricted to a single dominant adiabatic channel and does not explicitly include nonadiabatic couplings, but this simplification allows us to provide smooth cross-sections and rate coefficients with compact analytic fits that can be directly implemented in modeling. As the next step, we aim to investigate absorption processes for additional species in diverse environments. Moreover, incorporating this data into an A&M database would be highly beneficial. We plan to integrate these datasets into a searchable A&M data repository-VAMDC (http://vamdc.eu (accessed on 5 November 2025)) by expanding one of its nodes, hosted by the SerVO (http://servo.aob.rs (accessed on 5 November 2025)). In future extensions, the approach should be broadened to higher-lying electronic states and coupling effects, which would further increase its predictive power and comparability with advanced theoretical and experimental studies.