SH-DARS: DR of SH⁺ via 2Π States

• SH-DARS is the dissociative recombination process of the sulfanylium ion (SH⁺) with electrons, characterized by recombination through excited 2Π states of neutral SH.
• The methodology employs large-scale multireference electronic structure techniques (MRCI, MCSCF, FOCI) and block-diagonalization diabatization combined with MQDT to accurately compute energy landscapes and reaction cross sections.
• The study reveals that excited ionic cores and Rydberg–valence interactions significantly enhance DR rates in cold interstellar environments, aligning theory with storage-ring experimental results.

SH-DARS refers to the dissociative recombination (DR) of the sulfanylium ion (SH$^+$) with electrons, focusing on recombination pathways through the $^2Π$ electronic states of neutral SH. Theoretical investigations, most notably those employing large-scale multireference electronic structure and multichannel quantum defect theory (MQDT), have characterized the energy landscapes, nonadiabatic couplings, and reaction cross sections for this system with quantitative agreement to storage-ring experiment. These studies provide a detailed picture of how SH$^+$ is destroyed in cold interstellar environments, highlighting the central role of excited ionic cores and Rydberg-valence interactions (Kashinski et al., 30 May 2024).

1. Electronic Structure and State Preparation

Multireference configuration interaction (MRCI) and state-averaged multiconfigurational self-consistent field (MCSCF) calculations constitute the foundational electronic-structure step. Five frozen-core orbitals on S and nine active orbitals—encompassing S(3s), the SH bonding/antibonding orbitals, $\pi_x/\pi_y$, and S(3d) functions—define the active space. Highly smooth $R$-dependence was verified through orbital overlap integrals $|⟨\phi_i(R)|\phi_i(R+\Delta R)⟩|\geq0.97$. SOCI (including singles and doubles) with an aug-cc-pVTZ basis (including S, H diffuse functions) provided adiabatic potential energy curves (PECs) for SH$^+$ $^3Σ^-$ and SH $^2Π$, $^2Σ^\pm$, and $^2Δ$. The equilibrium internuclear distance is $R_e=1.360$ Å for the SH$^+$ ion, in agreement with experiment, and the $^2Π \to ^2Σ^+$ excitation of SH is 3.90 eV versus the experimental 3.85 eV (Kashinski et al., 30 May 2024).

For diabatization and the inclusion of many electronic roots, the computational strategy employs full open-shell CI (FOCI) with a sixteen-orbital active space consisting of S(3s,3p,3d), SH, SH*, Rydberg (4s/4p), and H(2s); the resulting FOCI asymptotes obey NIST atomic energy limits to about 0.03 eV for the principal $^2Π$ states.

2. Block-Diagonalization and Diabatic Hamiltonian Construction

To resolve the strongly interacting valence and Rydberg states relevant for DR, a block-diagonalization diabatization is executed. Given $N$ adiabatic configuration state functions (CSFs) $\Phi_j$ and their eigenstates $\Psi_i(R)=\sum_j c_{ij}(R)\Phi_j(R)$, one selects $N_\alpha$ dominant "diabatic" CSFs. The $N_\alpha \times N_\alpha$ matrix $S$ is constructed from their expansion coefficients, and a unitary transformation $T(R) = S^{-1}(R)[S(R)S^T(R)]^{1/2}$ is defined. The diabatic Hamiltonian

$H_d(R) = T^\dagger(R)\,E_{\rm ad}(R)\,T(R)$

(where $E_{\rm ad}$ is the diagonal matrix of adiabatic energies) is then constructed. For SH, the active subspace is a $4\times4$ block comprising two autoionizing $^2Π$ valence curves ($D_1$, $D_2$) and the two lowest Rydberg series ($R_1$ “ground-core”, $R_2$ “core-excited”). The resulting diabatic Hamiltonian has diagonal elements $V_i(R)$ (the diabatic PECs) and off-diagonal elements $V_{ij}(R)$ (valence-Rydberg couplings), fitted with Gaussian peaks and multipole tails to ensure correct asymptotic behavior (Kashinski et al., 30 May 2024).

3. Multichannel Quantum Defect Theory and Cross Section Computation

The collision dynamics employ a stepwise MQDT approach within a fixed total electronic symmetry $\Lambda$ ($^2Π$). Ionization channels are built on SH$^+$ core states ($c_\beta$: ground $^3Σ^-$, excited $^1Δ$), and dissociation channels $d_j$ are defined for SH neutral fragments. The relevant electronic couplings at fixed $R$ are

$$V^{(e)\Lambda}_{d_j,\,c_\beta}(R) = \langle Φ_{d_j}|H_{\rm el}|Φ^{el,\,c_\beta}\rangle,$$

$$V^{(e)\Lambda}_{c_1,\,c_2}(R) = \langle Φ^{el,\,c_1}|H_{\rm el}|Φ^{el,\,c_2}\rangle.$$

These are vibrationally averaged to obtain the short-range couplings. The reaction $K$-matrix is assembled via second-order Lippmann–Schwinger perturbation and diagonalized, yielding eigenphases $\eta_\alpha$ and frame-transformation matrices $C$, $S$. The generalized scattering matrix $X=(C+iS)/(C-iS)$ is formed, and elimination of energetically closed channels gives the physical $S_{oo}$, from which the DR cross section is evaluated:

$$σ_{\rm DR}(E) = \sum_\Lambda \frac{\pi}{4E}\,ρ^Λ\,\sum_j |S_{d_j,\,v_i^+}^Λ(E)|^2,$$

where $ρ^Λ$ is the spin-statistical weight.

4. Thermal Rate Constants and Principal Numerical Results

Thermal rate coefficients $k(T)$ result from Maxwellian averaging:

$$k(T) = \left(\frac{8k_BT}{\pi m_e}\right)^{1/2} \int_0^\infty σ_{\rm DR}(E)\,e^{-E/(k_BT)}\,dE.$$

The cross section and rates emerge as follows:

• Direct channel (ground core + $D_1$): $σ(E)\approx10^{-17}\,\text{cm}^2$ for $E\sim0.1{-}1$ eV.
• Inclusion of core-excited state: raises low-energy $σ$ by $~\times4.5$ and shifts resonances from 0.3 to 5 meV.
• Addition of the second dissociative state $D_2$: for $E>1$ eV, $σ$ increases by another factor of $\sim4$.
• Rydberg-threshold resonances accumulate in the vicinity of vibrational limits.
• Maxwell-averaged $k(T)$ in the full model (including both cores and both dissociative states): | T (K) | $k(T)$ (cm$^3$ s$^{-1}$) | |-------|--------------------------| | 10 | $6\times10^{-8}$ | | 100 | $8\times10^{-8}$ | | 300 | $1.0\times10^{-7}$ | | 1000 | $1.3\times10^{-7}$ | (Direct only: about half these values.)

At $E=0.01$ eV, direct $σ\sim10^{-15}$ cm$^2$, full model $σ\sim5\times10^{-14}$ cm$^2$; at $E=1$ eV, direct $σ\sim10^{-17}$ cm$^2$, full model $\sim5\times10^{-17}$ cm$^2$ (Kashinski et al., 30 May 2024).

5. Comparison with Storage-Ring Experiments

The predicted DR rates were convoluted with the anisotropic electron beam profile of the TSR ion storage ring (parallel energy spread $k_\parallel=25$ μeV, perpendicular $k_\perp=1.65$ meV). The computed rates agreed with experimental measurements to within 20% for collision energies above 10 meV, with broad resonances attributed to core-excited Rydberg capture resolved in both theory and experiment. Below 10 meV, theoretical rates underpredict experiment, likely due to sensitive dependence on potential energy curve crossings, omission of other symmetries ($^4Π$, $^2Σ^+$), and exclusion of rotational couplings. A plausible implication is that further inclusion of these effects would lead to even closer agreement at the lowest energies (Kashinski et al., 30 May 2024).

6. Significance and Astrophysical Context

The destruction rate of SH$^+$ by DR is $k\sim10^{-7}$ cm$^3$s$^{-1}$ in cold plasma, consistent with the effective removal of SH$^+$ in interstellar environments. Theoretical accord with storage-ring measurements underscores the robustness of combining large-scale MRCI/FOCI electronic-structure, block-diagonalization diabatization, and MQDT collision analysis. Excited-core and nonadiabatic (multiple-curve-crossing) effects are necessary for quantitative predictions; omission of such channels leads to substantial underestimation of the cross section and DR rate. This framework is exemplary for related diatomic ions subject to DR in astrophysical and laboratory plasmas.