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\title{HIGH-ORDER VIBRATIONAL RESONANCES OF CARBONYL SULFIDE REVEALED BY THE RESUMMATION TECHNIQUE}

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\author{\FirstName E.~O. \LastName DOBROLYUBOV}
\address{Emanuel Institute of Biochemical Physics of the Russian Academy of Sciences, Kosygina ul., 4, Moscow, 119334, Russia}

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

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\author{\FirstName R.~V. \LastName PALVELEV}
\address{Dept. of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskiye Gory 1-1, 119991 Moscow, Russia}

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\fancyhead[C]{{\footnotesize \textit{HRMS, 28th Colloquium, Dijon, France,
  Aug. 28 - Sept. 1, 2023}}}
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%EGOR
The high-order Rayleigh-Schr{\"o}dinger perturbation theory (RSPT), being a classical quantum-mechanical technique for solving many problems with a known analytic solution for the zero-order approximation, is rarely used in the context of molecular vibrations since such RSPT series usually diverge. 
This divergence is caused by resonance couplings that mix vibrational states and produce abnormally small denominators in RSPT corrections. 
%EGOR

%The Rayleigh-Schr{\"o}dinger perturbation theory (RSPT), being a classical quantum-mechanical technique for solving many problems with a known analytic solution for the zero-order approximation, is rarely used in the context of molecular vibrations for the reason that such RSPT series almost universally diverge. 
%This divergence is usually caused by resonance couplings that mix wave functions and produce abnormally large matrix elements in RSPT summations. 

%EGOR
However, divergent series still contain information about coupled vibrational states that can be properly extracted and described in terms of specially designed multi-valued functions, called approximants. 
Among them the quartic Pad{\'e}-Hermite approximant (PHA) is one of the most efficient:
\[
P(\lambda) + Q(\lambda) E(\lambda) + R(\lambda) E^2 (\lambda) + S(z) E^3 (\lambda) + T(\lambda) E^4 (\lambda) = 0.
\]
The resummation procedure transforms a large portion (up to $\approx$ 200${}^{\rm{th}}$ order) of corrections of a divergent vibrational RSPT series into PHA with algebraic coefficients via solving a linear system of equations or in an incremental way.\footnote{
\href{https://doi.org/10.1016/j.cpc.2004.02.002}{doi:10.1016/j.cpc.2004.02.002}, 
T. M. Feil, H. H. H. Homeier, \textit{Comput. Phys. Commun.}, \textbf{158}, 124--135 (2004).}
%EGOR

%However, such divergent series still contain information about vibrational manifolds (polyads) that can be properly extracted and described in terms of specially designed multi-valued functions, called approximants. 
%Among such algebraic functions, an especially powerful option is the quartic Pad{\'e}-Hermite approximant (PHA). 
%The resummation procedure equates a large portion (up to $\approx$200${}^{\rm{th}}$ order) of a divergent vibrational RSPT series to PHA with algebraic coefficients and defines their values by solving a linear system of equations or in an incremental way. 
%EGOR
As a result, one of several PHA values accurately reproduces the exact variational solution of the same problem, while the others usually coincide with nearby resonant energy levels. 
It is very interesting that singularities of the RSPT/PHA solutions can be described in terms of the complex-valued perturbation parameter $\bar \lambda = \Re (\bar \lambda) + \Im (\bar \lambda)$, while the resonant states can be characterized by coinciding complex singularities with the condition $|\bar \lambda| \le 1$.\footnote{
\href{https://doi.org/10.1021/jp211400w}{doi:10.1021/jp211400w}, 
X. Chang, E. O. Dobrolyubov, S. V.  Krasnoshchekov, \textit{Phys. Chem. Chem. Phys.}, \textbf{24}, 6655--6675 (2022).}
%EGOR

%As a result, one of several PHA values reproduce the exact matrix solution of the same problem, while the others usually coincide with nearby resonant energy levels. 
%It is very interesting that singularities (bifurcations, etc.) of the RSPT/PHA solutions can be describles in terms of the complex-valued RSPT parameter $\bar \lambda = Re(\lambda) + Im(\lambda)$, while the resonant states can be characterized by coinciding complex singularities with the condition $|\bar \lambda| \le 1$.

%EGOR
This RSPT resummation technique was applied to vibrational states of the series of OCS isotopologues. 
The Watson Hamiltonians were calculated using the \textit{ab initio} equilibrium geometry and quartic/sextic PES at CCSD(T)/cc-pV(Q+d)Z level. The analysis of the resulting PHA approximants revealed most of experimentally observed high-order resonances including those which break down the known polyad formula $P = 2v_1 + v_2 + 4v_3$.
%EGOR

%This RSPT resummation technique was applied to the vibrational states of the series of OCS isotopologues. 
%The vibrational Watson Hamiltonians were calculated using the \textit{ab initio} equilibrium geometry and quartic/sextic PES at CCSD(T)/cc-pV(Q+d)Z level. 
%The Table below illustrates some predicted resonances which were earlier detected experimentally. 

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