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\title{AB INITIO PREDICTION OF THE VIBRATION-ROTATION SPECTRA AND EFFECTIVE HAMILTONIANS USING HIGH-ORDER OPERATOR PERTURBATION THEORY}

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

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% \address{Saint-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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\category {1,3}
\begin{document}

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Nowadays, two concurrent approaches are being used for prediction and interpretation of high-resolution vibration-rotation spectra. 
The direct problem can be accurately solved by matrix diagonalization with exact KEO/\textit{ab initio} PES, followed by their small adjustments to fit the observed spectra. 
On the contrary, more traditional and precise techniques of fitted effective Hamiltonians are being used for solving the inverse problem. 
The need of building the bridge between these direct and inverse problems is evident, while only a few attempts in this direction were made.\footnote{
\href{https://doi.org/10.1080/00268976.2022.2096140}{doi:10.1080/00268976.2022.2096140}, 
V. Tyuterev, S. Tashkun, M. Rey, A. Nikitin, \textit{Mol. Phys.}, \textbf{120}, e2096140 (2022).}

The operator version of the vibration-rotation canonical Van Vleck perturbation theory (CVPT($n$)) is a suitable tool for solving the direct problem.\footnote{
D. Papou{\v s}ek, M. R. Aliev, \textit{Molecular Vibrational-rotational Spectra: Theory and Applications of High Resolution Infrared, Microwave and Raman Spectroscopy of Polyatomic Molecules}, Elsevier, 323 pp (1982).} 
The CVPT($n$) is based on a series of vibrational operator unitary transformations of the full Watson Hamiltonian followed by rotational transformations (reduction) that allow to derive numerical values of observable quartic ($\Delta, \delta$), sextic ($H,h$) and even octic ($L,l$) spectroscopic constants (in 6th order), as well as obtain constants of effective resonance operators. 
Also, the derived vibrationally transformed effective Hamiltonian can be composed in the matrix form for a chosen value of $J_{max}$ with accounting of Fermi, Darling-Dennison and Coriolis resonances. 
Its diagonalization produces vibration-rotation energy levels as well as wave functions that can be used for evaluation of intensities, provided that the dipole moment operator is also unitary transformed.

The traditional analytic form of CVPT is rather simple in the second order, while going to higher orders requires evaluation of complex rotational commutators that can reliably be done with the aid of analytic computations software. 
However, even for triatomic molecules such computations become rather heavy in orders higher than second. 
For a suitable solution of this problem and extension of applicability of CVPT($n$) to larger molecules %(4,5 atoms)
it is necessary to employ high-performance programming languages and solve certain theoretical problems for efficient evaluation of rotational commutators with possible participation of Wigner $D^1_{0,\varepsilon}$ functions.

We created a suite of high-performance routines written in Fortran90 that systematically performs CVPT($n$),$n=2,4,6$ calculations on a desk-top computer with a minimum manual control. 
This approach is largely based on evaluation of vibrational and rotational commutators using the normal ordering of ladder operators of angular momentum ($J_z, \, J_{\pm} = J_x \mp i J_y$):\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).}
\begin{equation}
\label{eq:AngularMoment:NormOrder}
\begin{array}{l}
: J_z^a J_+^b J_-^c J_z^d J_+^e J_-^f : 
= \sum\limits_{k = 0}^d {d \choose k} {(c-b)}^k 
\sum\limits_{m = 0}^{min(c,e)} {e \choose m} \frac{c!}{(c-m)!} \sum\limits_{l = 0}^{m} s(m,l) \times \\
\times \sum\limits_{j = 0}^{l} (-2)^j {l \choose j} 
\left(2b - c + e \right)^{l-j} J_z^{a + d - k + j} J_+^{b + e - m} J_-^{c + f - m}.
\end{array}
\end{equation}
% After a series of even number of unitary transformations on vibrational operators the original Hamiltonian takes a (quasi-)diagonal form.
The reduction can also be accomplished with the aid of such normal ordering. 
Similarly, unitary transformations of the dipole moment operator can be made using the normal ordering with the Wigner $D_{0,\varepsilon}^1$-functions ($\varepsilon=-1,0,+1$):
\begin{equation}
\label{eq:WignerNormOrderProblem}
    : J_z^a J_+^b J_-^c D_{0,\varepsilon}^1 : \;
    = \sum\limits_{\tau}^{} {
    D_{0,\tau}^1 \sum\limits_j c_{\tau j} J_z^{k_{\tau j}} J_+^{l_{\tau j}} J_-^{m_{\tau j}} } .
\end{equation}

The obtained effective Hamiltonian and dipole moment operators can be further used for calculation of simulated IR spectra for a chosen spectral range and vibrational transitions. 
The spectroscopic constants $\Delta,\delta, H,h,L,l$ can also be calculated after the a series of rotational unitary transformations. 
The effectiveness of this procedure for the ${}^{32}$S${}^{16}$O${}_{2}$ molecule is illustrated below, where the \textit{ab initio} values are compared with empirical fittings:
\footnote{
\href{https://doi.org/10.1006/jmsp.1993.1174}{doi:10.1006/jmsp.1993.1174}, J.M. Flaud, A. Perrin, L.M. Salah, W.J. Lafferty, G. Guelachvili, \textit{J. Mol. Spectrosc.}, \textbf{160}, 272--278 (1993).}
\footnote{
\href{https://doi.org/10.1016/j.jqsrt.2013.04.011}{doi:10.1016/j.jqsrt.2013.04.011}, O.N. Ulenikov, G.A. Onopenko, O.V. Gromova, E.S. Bekhtereva, V.-M. Horneman, \textit{J. Quant. Spectrosc. Radiat. Transfer}, \textbf{130}, 220--232 (2013).}

%For instance, sextic constants $H,h$ can be obtained after evaluation of the following expression:
%\begin{equation}
%\label{eq:Reduction}
%\begin{array}{l}
%\widetilde{H}_{06} = H_{02} + H_{04} + \big[ H_{02}, S(3) \big] + H_{06} + i \big[ H_{04}, S(3) \big] \\
%- 1/2 \big[ \big[ H_{02}, S(3) \big], S(3) \big] + i \big[ H_{02}, S(5) \big].
%\end{array}
%\end{equation}

\begin{table}[h]
\centering
  \caption{ Comparison of the effective sextic spectroscopic constants, cm${}^{-1}$ }
  \begin{tabular}{ p{1.5cm} p{1.25cm} p{1.75cm} p{1.75cm} p{2cm} p{2cm} }
  \hline
  Constant & & CVPT(4) & CVPT(6) & Observed\textcolor{blue}{${}^4$} & Observed\textcolor{blue}{${}^5$} \\ 
  \hline 
  $H_{J}$  & $\times 10^{12}$ & \multicolumn{1}{c}{0.3803} & \multicolumn{1}{c}{0.3859} & \multicolumn{1}{c}{0.3821} & \multicolumn{1}{c}{0.3746}  \\
  $H_{JK}$ & $\times 10^{12}$ & \multicolumn{1}{c}{-0.1247} & \multicolumn{1}{c}{1.3860} & \multicolumn{1}{c}{1.3250} & \multicolumn{1}{c}{1.1603} \\
  $H_{KJ}$ & $\times 10^{12}$ & \multicolumn{1}{c}{-598.3840} & \multicolumn{1}{c}{-647.5335} & \multicolumn{1}{c}{-649.4890} & \multicolumn{1}{c}{-649.6069} \\
  $H_{K}$  & $\times 10^{12}$ & \multicolumn{1}{c}{10773.5277} & \multicolumn{1}{c}{11986.1209} & \multicolumn{1}{c}{12412.9000} & \multicolumn{1}{c}{12360.4280} \\
  $h_{J}$  & $\times 10^{12}$ & \multicolumn{1}{c}{0.1850} & \multicolumn{1}{c}{0.1875} & \multicolumn{1}{c}{0.1831} & \multicolumn{1}{c}{0.1830} \\
  $h_{JK}$ & $\times 10^{12}$ & \multicolumn{1}{c}{-0.4181} & \multicolumn{1}{c}{-0.2406} & \multicolumn{1}{c}{-0.3787} & \multicolumn{1}{c}{-0.2430} \\
  $h_{K}$  & $\times 10^{12}$ & \multicolumn{1}{c}{480.5198} & \multicolumn{1}{c}{556.6921} & \multicolumn{1}{c}{557.9130} & \multicolumn{1}{c}{567.9872} \\
\hline
\end{tabular}
\end{table}

\end{document}
