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\begin{document}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\title{Hierarchy Configuration Interaction: Combining Seniority Number and Excitation Degree}
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\author{F\'abris Kossoski}
\email{fkossoski@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
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\author{Yann Damour}
\affiliation{\LCPQ}
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\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
% Abstract
\begin{abstract}
%Here comes the abstract.
%\bigskip
%\begin{center}
% \boxed{\includegraphics[width=0.4\linewidth]{TOC}}
%\end{center}
%\bigskip
\end{abstract}
% Title
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{\ce{Computational details}}
%\label{sec:comp_details}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{20cm}
%x
%\newpage
%x
%\newpage
%Start.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{oo-CI}
%\label{sec:oo-CI}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{plot_stat_opt}
\caption{Non-parallelity errors as function of the number of determinants, for the three classes of CI methods: seniority-based CI (blue), excitation-based CI (red), and our proposed hCI (green),
with orbitals optimized at each CI level.
}
\label{fig:plot_stat_opt}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{plot_distance_opt}
\caption{Distance errors as function of the number of determinants, for the three classes of CI methods: seniority-based CI (blue), excitation-based CI (red), and our proposed hCI (green),
with orbitals optimized at each CI level.
}
\label{fig:plot_distance_opt}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{xe_opt}
\caption{Equilibrium geometries as function of the number of determinants, for the three classes of CI methods: seniority-based CI (blue), excitation-based CI (red), and our proposed hCI (green),
with orbitals optimized at each CI level.
}
\label{fig:xe_opt}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{freq_opt}
\caption{Vibrational frequencies (or force constants) as function of the number of determinants, for the three classes of CI methods: seniority-based CI (blue), excitation-based CI (red), and our proposed hCI (green),
with orbitals optimized at each CI level.
}
\label{fig:freq_opt}
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\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{\ce{HF}, different basis sets}
%\label{sec:HF_basis}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{plot_pes_HF}
\caption{Potential energy curves (top) and energy differences with respect to FCI (bottom), for dissociation of \ce{HF},
according to the three classes of CI methods: seniority-based CI (blue), excitation-based CI (red), and our proposed hybrid hCI (green),
with Hartree-Fock orbitals,
and for the cc-pVDZ (left), cc-pVTZ (center), and cc-pVQZ (right) basis sets.
}
\label{fig:plot_pes_HF}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{freq_HF}
\caption{Nonparallelity error (left), vibrational frequencies (center), and equilibrium geometries (right) for \ce{HF},
as function of the number of determinants,
for the three classes of CI methods: seniority-based CI (blue), excitation-based CI (red), and our proposed hybrid hCI (green),
with Hartree-Fock orbitals,
and for the cc-pVDZ (left), cc-pVTZ (center), and cc-pVQZ (right) basis sets.}
\label{fig:freq_HF}
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\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{\ce{HF}}
%\label{sec:HF}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{HF_pes}
\caption{Potential energy curves for \ce{HF},
computed with different CI methods and the cc-pVDZ basis set,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
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\label{fig:HF_pes}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{HF_pes_error}
\caption{Energy difference to the FCI results for \ce{HF},
computed with different CI methods and the cc-pVDZ basis set,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
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\label{fig:HF_pes_error}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{HF_npe}
\caption{Nonparallelity error for \ce{HF},
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as function of the computational scaling of excitation-based CI (red) and CIo (green) methods,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
\label{fig:HF_npe}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{HF_distance}
\caption{Nonparallelity error for \ce{HF},
as function of the computational scaling of excitation-based CI (red) and CIo (green) methods,
with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
\label{fig:HF_distance}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=\linewidth]{HF_freq}
\caption{Vibrational frequency of \ce{HF},
as function of the computational scaling of excitation-based CI (red) and CIo (green) methods,
with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
\label{fig:HF_freq}
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\end{figure}
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\begin{figure}[h!]
\includegraphics[width=\linewidth]{HF_xe}
\caption{Equilibrium bond length of \ce{HF},
as function of the computational scaling of excitation-based CI (red) and CIo (green) methods,
with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
\label{fig:HF_xe}
\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{\ce{F2}}
%\label{sec:F2}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{F2_pes}
\caption{Potential energy curves for \ce{F2},
computed with different CI methods and the cc-pVDZ basis set,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
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\label{fig:F2_pes}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{F2_pes_error}
\caption{Energy difference to the FCI results for \ce{F2},
computed with different CI methods and the cc-pVDZ basis set,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
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\label{fig:F2_pes_error}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{F2_npe}
\caption{Nonparallelity error for \ce{F2},
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as function of the computational scaling of excitation-based CI (red) and CIo (green) methods,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
\label{fig:F2_npe}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{F2_xe}
\caption{Equilibrium bond length of \ce{F2},
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as function of the computational scaling of excitation-based CI (red) and CIo (green) methods,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
\label{fig:F2_xe}
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\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{\ce{Ethylene}}
%\label{sec:ethylene}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h!]
\includegraphics[width=\linewidth]{ethylene_pes}
\caption{Potential energy curves for ethylene, as function of the C$=$C distance,
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computed with different CI methods and the cc-pVDZ basis set,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
\label{fig:ethylene_pes}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=\linewidth]{ethylene_pes_error}
\caption{Energy difference to the FCI results for ethylene, as function of the C$=$C distance,
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computed with different CI methods and the cc-pVDZ basis set,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
\label{fig:ethylene_pes_error}
\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{ethylene_npe}
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\caption{Nonparallelity error for ethylene,
as function of the computational scaling of excitation-based CI (red) and CIo (green) methods,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
\label{fig:ethylene_npe}
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\end{figure}
\begin{figure}[h!]
\includegraphics[width=\linewidth]{ethylene_xe}
\caption{C$=$C equilibrium bond length of ethylene,
as function of the computational scaling of excitation-based CI (red) and CIo (green) methods,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
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\label{fig:ethylene_xe}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{\ce{N2}}
%\label{sec:N2}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{N2_pes}
\caption{Potential energy curves for \ce{N2},
computed with different CI methods and the cc-pVDZ basis set,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
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\label{fig:N2_pes}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{N2_pes_error}
\caption{Energy difference to the FCI results for \ce{N2},
computed with different CI methods and the cc-pVDZ basis set,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
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\label{fig:N2_pes_error}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{N2_npe}
\caption{Nonparallelity error for \ce{N2},
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as function of the computational scaling of excitation-based CI (red) and CIo (green) methods,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
\label{fig:N2_npe}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{N2_xe}
\caption{Equilibrium bond length of \ce{N2},
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as function of the computational scaling of excitation-based CI (red) and CIo (green) methods,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
\label{fig:N2_xe}
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\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{\ce{H4}}
%\label{sec:H4}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{H4_pes}
\caption{Potential energy curves for linear \ce{H4}, as function of the symmetric stretching coordinate,
computed with different CI methods and the cc-pVDZ basis set,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
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\label{fig:H4_pes}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{H4_pes_error}
\caption{Energy difference to the FCI results for linear \ce{H4}, as function of the symmetric stretching coordinate,
computed with different CI methods and the cc-pVDZ basis set,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
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\label{fig:H4_pes_error}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{H4_npe}
\caption{Nonparallelity error for linear \ce{H4},
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as function of the computational scaling of excitation-based CI (red) and CIo (green) methods,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
\label{fig:H4_npe}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{H4_xe}
\caption{Equilibrium bond length of linear \ce{H4},
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as function of the computational scaling of excitation-based CI (red) and CIo (green) methods,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
\label{fig:H4_xe}
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{\ce{H8}}
%\label{sec:H8}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{H8_pes}
\caption{Potential energy curves for linear \ce{H8}, as function of the symmetric stretching coordinate,
computed with different CI methods and the cc-pVDZ basis set,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
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\label{fig:H8_pes}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{H8_pes_error}
\caption{Energy difference to the FCI results for linear \ce{H8}, as function of the symmetric stretching coordinate,
computed with different CI methods and the cc-pVDZ basis set,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
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\label{fig:H8_pes_error}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{H8_npe}
\caption{Nonparallelity error for linear \ce{H8},
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as function of the computational scaling of excitation-based CI (red) and CIo (green) methods,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
\label{fig:H8_npe}
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\end{figure}
\begin{figure}[h!]
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\includegraphics[width=\linewidth]{H8_xe}
\caption{Equilibrium bond length of linear \ce{H8},
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as function of the computational scaling of excitation-based CI (red) and CIo (green) methods,
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with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
\label{fig:H8_xe}
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\end{figure}
%End.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibliography{seniority}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}