Construction and analysis of dimensionally reduced potential energy surface of S$_{\mathrm{4}}$ and its vibrational states

Electronic structure calculations are carried out for S$_{\mathrm{4}}$ molecule at the CCSD(T)-F12a/VTZ-F12 level of theory to map out its potential energy surface that possesses a double-well shape with low-energy barrier. Two degrees of freedom are considered, the distance $R$ and the angle $\alpha $ between two weakly-perturbed S$_{\mathrm{2}}$ dimers that form S$_{\mathrm{4}}$. The resultant PES was computed near the bottom of the covalent well exposing transition state barrier energy of about 690 cm$^{\mathrm{-1}}$. Vibrational states are computed on this 2D-surface and assigned quantum numbers based on their energies and shapes of wavefunctions. Two progressions of vibrational states are identified: a long progression that develop nodes along the ``channels'' on the surface, and a shorter progression of states that develop nodes across the ``channels''. Normal mode analysis indicates that these two modes in S$_{\mathrm{4}}$ represent a significant mixture of conventional bending and stretching motions. Frequencies of the modes, approximately 180 cm$^{\mathrm{-1}}$ and 420 cm$^{\mathrm{-1}}$, are in qualitative agreement with earlier \textit{ab initio} studies of S$_{\mathrm{4}}$, and with sparse experimental data.