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\title{A COMPLETE CONSIDERATION OF NON-ADIABATIC EFFECTS (UP TO ORDER M$^{-2}$, M=REDUCED NUCLEAR MASS) IN A TRIATOMIC MOLECULE:
INFLUENCE ON THE ROVIBRATIONAL ENERGIES OF H$_3^+$.}

\author{\underline{\FirstName RALPH \LastName JAQUET}}
\address{Theoretical Chemistry, University Siegen, 57068 Siegen, Germany}

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\fancyhead[C]{{\footnotesize \textit{HRMS, 28th Colloquium, Dijon, France,
  Aug. 28 - Sept. 1, 2023}}}
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%
%It is generally accepted that radiative association of ions with molecules plays an important role in the
%  molecular synthesis in the super-cold, low-density environment of the interstellar medium.
%One of the crucial steps in the investigation  of cross-sections for \textcolor{black}{\bf radiative association}
% (and the infrared \textcolor{black}{\bf predissociation} spectrum [1] of H$_3^+$, with ca. \textcolor{black}{\bf 100 transitions per wavenumber})
%is the calculation of the ro-vibrational bound states and resonances  near the dissociation limit.
%%Because H$_3^+$ has a binding energy of $\approx$ 35000 cm$^{-1}$, the accurate  determination of the highest bound energy levels can be demanding.
% For the description of radiative association of  a strongly bound molecule,
% we need a good potential energy and dipole surface, an efficient reactive scattering implementation
%and  an efficient way of calculating high-lying bound states and resonances.
%But, \textcolor{black}{\bf it is already problematic to calculate with spectroscopy accuracy all rovibrational transitions $<$ 16000 cm$^{-1}$.}
For a triatomic molecule (like H$_3^+$) it is already problematic to calculate with spectroscopy accuracy {\it all} rovibrational transitions $<$ 16000 cm$^{-1}$.

%\bigskip
Our main focus   is the investigation of the influence of non-adiabaticity on the rovibrational bound states of H$_3^+$.
For this purpose a full configuration interaction (FCI) treatment using Gaussian basis functions
is applied to calculate the energies of the electronic states as well as all couplings between them caused by the nuclear motion.

%/bigskip
These \textcolor{black}{\it 'derivative couplings'} were evaluated up to second order by means of a perturbation treatment
using the 'some-over-states'-strategy. While this has been possible already for diatomics, e.g. H$_2^+$,   H$_2$, etc.,
%nothing equivalent has been available for H$_3^+$.  The present work [2] is an extension of the investigation
nothing equivalent has been available for H$_3^+$.  The present work \footnote{R. Jaquet, J. Mol. Spectr., 384, 111585 (2022).} is an extension of the investigation
of earlier non-adiabatic investigations based on first derivative couplings of electronic states
that led to the \textcolor{black}{\it concept of geometry-dependent effective nuclear masses} \footnote{R.M. Herman, A. Ashgarian, J. Mol. Spectr. 19, 305 (1966).}
  needed for only \textcolor{black}{\it one single potential energy surface} 
\footnote{R. Jaquet, M. Khoma, J. Phys. Chem. A, 121, 7016 (2017).}$^,$
\footnote{M. Khoma, R. Jaquet, J. Chem. Phys., 147, 114106 (2018).}$^,$
\footnote{R. Jaquet, M. Khoma, J. Mol. Phys., 116, 3507 (2018).}$^,$
\footnote{M. Khoma, R. Jaquet, J. Math. Chem., 57, 701 (2019).}.
Our new implementation allows to include for the first time also for  H$_3^+$ all non-adiabatic effects (\textcolor{black}{\it coupling of second-derivatives,
 and second-derivatives with first-derivatives})  up to the order of $\mu^{-2}$,  $\mu$ being the reduced nuclear mass.

%\bigskip
These new inclusions of all nonadiabatic effects could reduce also for  H$_3^+$
the  deviations to experimental data for most rovibrational levels ($<$ 16000 cm$^{-1}$) to less than 0.1 cm$^{-1}$
without any empirically adjustable parameters. For H$_3^+$, the accuracy is slightly improved
when also \textcolor{black}{\it relativistic and QED effects} are taken into account \footnote{R. Jaquet, M. Lesiuk, J. Chem. Phys., 152, 104109 (2020).}.
For some questionable assignments of observed transitions in H$_3^+$ we propose a new labeling.
In future investigations we are going to  analyse  non-adiabaticity for triatomics with more than 2 electrons.
%\large

%\bigskip
%[1]  Carrington, R. A. Kennedy, J. Chem. Phys., 81, 91 (1984)

%[2] Jaquet, J. Mol. Spectr., 384, 111585 (2022)

%[3] R.M. Herman, A. Ashgarian, J. Mol. Spectr. 19, 305 (1966)

%[4] Jaquet, M. Khoma, J. Phys. Chem. A, 121, 7016 (2017); J. Chem. Phys., 147, 114106 (2018); Mol. Phys., 116, 3507 (2018); J. Math. Chem., 57, 701 (2019)

%[5] Jaquet, M. Lesiuk, J. Chem. Phys., 152, 104109 (2020)




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