Theoretical study of laser-cooled SH^– anion

The potential energy curves, dipole moments, and transition dipole moments for the <inline-formula><tex-math id="M13">\begin{document}${{\rm{X}}^1}{\Sigma ^ + }$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M13.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M14">\begin{document}${{\rm{a}}^3}\Pi $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M14.png"/></alternatives></inline-formula>, and <inline-formula><tex-math id="M15">\begin{document}${{\rm{A}}^1}\Pi $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M15.png"/></alternatives></inline-formula> electronic state of sulfur hydride anion (SH^–) are calculated by using the multi-reference configuration interaction method plus Davidson corrections (MRCI+<i>Q</i>) with all-electron basis set. The scalar relativistic corrections and core-valence correlations are also considered. In the CASSCF calculations, H(1s) and S(3s3p4s) shells are chosen as active space, and the rest orbitals S(1s2s2p) as closed-shell. In the MRCI+<i>Q</i> calculations, the S(1s2s2p) shells are used for the core-valence correlation. Spectroscopic parameters, Einstein spontaneous emission coefficient, Franck-Condon factors, and spontaneous radiative lifetimes are obtained by using Le Roy’s LEVEL8.0 program. The calculated spectroscopic parameters are in good agreement with available experimental data and theoretical values. Spin-orbit coupling (SOC) effects are evaluated with Breit-Pauli operators at the MRCI+<i>Q</i> level. Transition dipole moments (TDMs) for the <inline-formula><tex-math id="M16">\begin{document}${{\rm{A}}^1}{\Pi _1} \leftrightarrow {{\rm{X}}^1}\Sigma _{{0^ + }}^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M16.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M17">\begin{document}${{\rm{a}}^3}{\Pi _{{0^ + }}} \leftrightarrow {{\rm{X}}^1}\Sigma _{{0^ + }}^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M17.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M18">\begin{document}${{\rm{a}}^3}{\Pi _1} \leftrightarrow {{\rm{X}}^1}\Sigma _{{0^ + }}^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M18.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M19">\begin{document}${{\rm{A}}^1}{\Pi _1} \leftrightarrow {{\rm{a}}^3}{\Pi _{{0^ + }}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M19.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M20">\begin{document}${{\rm{A}}^1}{\Pi _1} \leftrightarrow {{\rm{a}}^3}{\Pi _1}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M20.png"/></alternatives></inline-formula> transitions are also calculated. The strength for the <inline-formula><tex-math id="Z-20190315031218-1">\begin{document}${{\rm{A}}^1}{\Pi _1} \leftrightarrow {{\rm{X}}^1}\Sigma _{{0^ + }}^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_Z-20190315031218-1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_Z-20190315031218-1.png"/></alternatives></inline-formula> is the strongest in these five transitions, the value of TDM at <i>R</i>_e is –1.3636 D. We find that the value of TDM for the <inline-formula><tex-math id="M21">\begin{document}${{\rm{a}}^3}{\Pi _1} \leftrightarrow {{\rm{X}}^1}\Sigma _{{0^ + }}^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M21.png"/></alternatives></inline-formula> transition at <i>R</i>_e is 0.5269 D. Therefore, this transition must be taken into account to build the scheme of laser-cooled SH^– anion. Highly diagonally distributed Franck-Condon factor <i>f</i>₀₀ for the <inline-formula><tex-math id="M22">\begin{document}${{\rm{a}}^3}{\Pi _1}(\nu ' = 0) \leftrightarrow {{\rm{X}}^1}\Sigma _{{0^ + }}^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M22.png"/></alternatives></inline-formula> <inline-formula><tex-math id="M22-1">\begin{document}$ (\nu '' = 0)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M22-1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M22-1.png"/></alternatives></inline-formula> transition is 0.9990 and the value for the <inline-formula><tex-math id="M23">\begin{document}${{\rm{A}}^1}{\Pi _1}(\nu ' = 0) \leftrightarrow {{\rm{X}}^1}\Sigma _{{0^ + }}^ + (\nu '' = 0)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M23.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M23.png"/></alternatives></inline-formula> transition is 0.9999. Spontaneous radiative lifetimes of <inline-formula><tex-math id="M24">\begin{document}$\tau \left( {{{\rm{a}}^3}{\Pi _1}} \right)= 1.472 \;{\text{μ}}{\rm{s}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M24.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M24.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M25">\begin{document}$\tau \left( {{{\rm{A}}^1}{\Pi _1}} \right)=0.188 \;{\text{μ}}{\rm{s}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M25.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M25.png"/></alternatives></inline-formula> are obtained, which can ensure that laser cools SH^– anion rapidly. To drive the <inline-formula><tex-math id="M26">\begin{document}${{\rm{a}}^3}{\Pi _1} \leftrightarrow {{\rm{X}}^1}\Sigma _{{0^ + }}^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M26.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M26.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M27">\begin{document}${{\rm{A}}^1}{\Pi _1} \leftrightarrow {{\rm{X}}^1}\Sigma _{{0^ + }}^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M27.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M27.png"/></alternatives></inline-formula> transitions, just one laser wavelength is required. The wavelengths are 492.27 nm and 478.57 nm for two transitions, respectively. Notably, the influences of the intervening states <inline-formula><tex-math id="M28">\begin{document}${{\rm{a}}^3}{\Pi _1}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M28.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M28.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M29">\begin{document}${{\rm{a}}^3}{\Pi _{{0^{\rm{ + }}}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M29.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M29.png"/></alternatives></inline-formula> on the <inline-formula><tex-math id="M30">\begin{document}${{\rm{A}}^1}{\Pi _1} \leftrightarrow {X^1}\Sigma _{{0^ + }}^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M30.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M30.png"/></alternatives></inline-formula> transition are small enough to implement a laser cooling project. A spin-forbidden transition and a three-electronic-level transition optical scheme of laser-cooled SH^– anion are constructed, respectively. In addition, the Doppler temperatures and recoil temperatures for the <inline-formula><tex-math id="M31">\begin{document}${{\rm{a}}^3}{\Pi _1} \leftrightarrow {{\rm{X}}^1}\Sigma _{{0^ + }}^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M31.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M31.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M32">\begin{document}${{\rm{A}}^1}{\Pi _1} \leftrightarrow {{\rm{X}}^1}\Sigma _{{0^ + }}^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M32.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182039_M32.png"/></alternatives></inline-formula> transitions of laser-cooled SH^– anion are also obtained, respectively.