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\title{QUADRATIC VIBRATION-ROTATION CONSTANTS $\gamma^{\beta}_{r,s}$ AND SEMI-EXPERIMENTAL ${}^{32}$S${}^{16}$O${}_2$ EQUILIBRIUM GEOMETRY}

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\author{\underline{\FirstName S.~V. \LastName KRASNOSHCHEKOV}, \FirstName E.~O. \LastName DOBROLYUBOV, \\
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\FirstName I.~V. \LastName POLYAKOV}
\address{Dept. of Chemistry, Lomonosov Moscow State University, Leninskiye Gory 1-3, 119991 Moscow, Russia}

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\author{\FirstName A.~V. \LastName BELYAKOV}
\address{St. Petersburg State Technological Institute, 190013, Russia}

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\fancyhead[C]{{\footnotesize \textit{HRMS, 28th Colloquium, Dijon, France,
  Aug. 28 - Sept. 1, 2023}}}
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Probably the most precise geometrical nuclear configuration of a semi-rigid molecule can be obtained through the wide-spread technique combining experimental values of rotational constants $B^\beta_v (\beta=x,y,z)$ (obtainable from high-resolution MW spectra) and non-empirical values of linear and quadratic vibration-rotation constants $\alpha^{\beta}_{r}$ and $\gamma^{\beta}_{r,s}$ plus a small electronic correction obtained through
the rotational $\textbf{g}$-tensor ($\Delta B^\beta_{g} = (m_e/m_p)g^{\alpha \alpha} B_e^\alpha$):\footnote{
\href{https://doi.org/10.1021/acs.jctc.7b00279}{doi:10.1021/acs.jctc.7b00279}, 
M. Mendolicchio, E. Penocchio, D. Licari, N. Tasinato, V. Barone, \textit{J. Chem. Theory Comput.}, \textbf{13}, 3060--3075 (2017).}
\begin{equation}
{B^\beta_v} = {B^\beta_e} - \sum\limits_r {\alpha^{\beta}_{r}\left( v_r + \frac{1}{2} \right)} + \sum\limits_{r \ge s} {\gamma^{\beta}_{r,s}\left( v_r + \frac{1}{2} \right)\left( v_s + \frac{1}{2} \right)} + \dots 
\end{equation}

While linear constants $\alpha^{\beta}_{r}$ can be trivially calculated using the second-order perturbation theory (VPT2), the evaluation of the quadratic constants $\gamma^{\beta}_{rs}$ requires a number of complex rotational commutators and this problem was never solved systematically. 
Only a few experimental studies for the ${}^{32}$S${}^{16}$O${}_2$ molecule included evaluation of $\gamma^{\beta}_{rs}$.
%\footnote{\href{https://doi.org/10.1006/jmsp.1993.1245}{doi:10.1006/jmsp.1993.1245}, S. Saito, \textit{J. Mol. Spectrosc.}, \textbf{30}, 1--16 (1966).} 
\footnote{
\href{https://doi.org/10.1006/jmsp.1993.1245}{doi:10.1006/jmsp.1993.1245}, J.-M. Flaud, W. J. Lafferty, \textit{J. Mol. Spectrosc.}, \textbf{161}, 396--402 (1993).} 
\footnote{
\href{https://doi.org/10.1006/jmsp.1993.1245}{doi:10.1006/jmsp.1993.1245}, Y. Morino, M. Tanimoto, \textit{J. Mol. Spectrosc.}, \textbf{166}, 310--320 (1994).}
The theoretical research include an early work by Brown\footnote{
\href{https://doi.org/10.1016/0022-2852(71)90050-6}{doi:10.1016/0022-2852(71)90050-6}, J.~M. Brown, \textit{J. Mol. Spectrosc.}, \textbf{37}, 179--195 (1971).} and a recent review by Demaison \textit{et al}.\footnote{
\href{https://doi.org/10.1080/00268976.2021.1950857}{doi:10.1080/00268976.2021.1950857}, J. Demaison, J. Li{\'e}vin, \textit{Mol. Phys.}, \textbf{120}, e1950857(13pp) (2022).}

We propose the systematic procedure for evaluation of $\gamma^{\beta}_{r,s}$ using the fourth-order canonical Van Vleck perturbation theory (CVPT4). This approach is largely based on evaluation of vibrational and rotational commutators using the normal ordering of ladder operators of angular momentum.\footnote{
\href{https://doi.org/10.1063/5.0142809}{doi:10.1063/5.0142809}, X. Chang, D.~V. Millionshchikov, I.~M. Efremov, S.~V. Krasnoshchekov, \textit{J. Chem. Phys.}, \textbf{158}, 104802(1--12) (2023).}
The values of $\gamma^{\beta}_{r,s}$ can be obtained by comparing the operator form of the vibrationally transformed Hamiltonian $\hat H^{(4)}$ with the corresponding theoretical terms,
%\begin{equation}
%H_{vib-rot} = {\sum\limits_\beta^{xyz} B_e^\beta J_\beta^2} - \sum\limits_\beta^{xyz} {\sum\limits_{r} {\left( \alpha^{\beta}_{r} J_\beta^2 a_r^\dagger a_r^{} \right)}} + \sum\limits_\beta^{xyz} {\sum\limits_{r \ge s} {\left( \gamma^{\beta}_{rs} J_\beta^2 a_r^\dagger a_r^{} a_s^\dagger a_s^{} \right)}}.
%\end{equation}
\begin{equation}
H_{vib-rot} = \sum\limits_\beta^{xyz} { J_\beta^2 \bigg( B_e^\beta - \sum\limits_{r} { \alpha^{\beta}_{r} a_r^\dagger a_r^{} } + \sum\limits_{r \ge s} { \gamma^{\beta}_{rs} a_r^\dagger a_r^{} a_s^\dagger a_s^{} + \dots } \bigg) }.
\end{equation}

% Numerical example for the ${}^{32}$S${}^{16}$O${}_2$ molecule is considered.


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