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\begin{document}

\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}

\title{Hierarchy Configuration Interaction: Combining Seniority Number and Excitation Degree}

\author{F\'abris Kossoski}
\email{fkossoski@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Yann Damour}
\affiliation{\LCPQ}
\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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{oo-CI}
%\label{sec:oo-CI}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[h!]
\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}
\end{figure}

\begin{figure}[h!]
\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}
\end{figure}

\begin{figure}[h!]
        \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}
\end{figure}

\begin{figure}[h!]
\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}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{\ce{HF}, different basis sets}
%\label{sec:HF_basis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[h!]
\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}
\end{figure}

\begin{figure}[h!]
\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}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{\ce{HF}}
%\label{sec:HF}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[h!]
\includegraphics[width=\linewidth]{HF_pes}
	\caption{Potential energy curves for \ce{HF},
	computed with different CI methods and the cc-pVDZ basis set, 
	with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
        \label{fig:HF_pes}
\end{figure}

\begin{figure}[h!]
\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, 
	with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
        \label{fig:HF_pes_error}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=\linewidth]{HF_npe}
	\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_npe}
\end{figure}

\begin{figure}[h!]
\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}
\end{figure}

\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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{\ce{F2}}
%\label{sec:F2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[h!]
\includegraphics[width=\linewidth]{F2_pes}
	\caption{Potential energy curves for \ce{F2},
	computed with different CI methods and the cc-pVDZ basis set, 
	with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
        \label{fig:F2_pes}
\end{figure}

\begin{figure}[h!]
\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, 
	with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
        \label{fig:F2_pes_error}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=\linewidth]{F2_npe}
	\caption{Nonparallelity error for \ce{F2},
	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:F2_npe}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=\linewidth]{F2_xe}
	\caption{Equilibrium bond length of \ce{F2},
	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:F2_xe}
\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,
	computed with different CI methods and the cc-pVDZ basis set, 
	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,
	computed with different CI methods and the cc-pVDZ basis set, 
	with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
        \label{fig:ethylene_pes_error}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=\linewidth]{ethylene_npe}
	\caption{Nonparallelity error for ethylene,
	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:ethylene_npe}
\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,
	with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right), for the cc-pVDZ basis set.}
        \label{fig:ethylene_xe}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{\ce{N2}}
%\label{sec:N2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[h!]
\includegraphics[width=\linewidth]{N2_pes}
	\caption{Potential energy curves for \ce{N2},
	computed with different CI methods and the cc-pVDZ basis set, 
	with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
        \label{fig:N2_pes}
\end{figure}

\begin{figure}[h!]
\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, 
	with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
        \label{fig:N2_pes_error}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=\linewidth]{N2_npe}
	\caption{Nonparallelity error for \ce{N2},
	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:N2_npe}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=\linewidth]{N2_xe}
	\caption{Equilibrium bond length of \ce{N2},
	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:N2_xe}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{\ce{H4}}
%\label{sec:H4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[h!]
\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, 
	with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
        \label{fig:H4_pes}
\end{figure}

\begin{figure}[h!]
\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, 
	with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
        \label{fig:H4_pes_error}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=\linewidth]{H4_npe}
	\caption{Nonparallelity error for linear \ce{H4},
	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:H4_npe}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=\linewidth]{H4_xe}
	\caption{Equilibrium bond length of linear \ce{H4},
	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:H4_xe}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{\ce{H8}}
%\label{sec:H8}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[h!]
\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, 
	with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
        \label{fig:H8_pes}
\end{figure}

\begin{figure}[h!]
\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, 
	with Hartree-Fock orbitals (left) and orbitals optimized for a given CI method (right).}
        \label{fig:H8_pes_error}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=\linewidth]{H8_npe}
	\caption{Nonparallelity error for linear \ce{H8},
	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:H8_npe}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=\linewidth]{H8_xe}
	\caption{Equilibrium bond length of linear \ce{H8},
	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:H8_xe}
\end{figure}

%End.

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%\bibliography{seniority}
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\end{document}