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@ -102,6 +102,7 @@ In this context, one accounts for all determinants generated by exciting up to $
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In this way, the excitation degree $e$ defines the following sequence of models:
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CI with single excitations (CIS), CI with single and double excitations (CISD), CI with single, double, and triple excitations (CISDT), and so on.
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Excitation-based CI manages to quickly recover weak (dynamic) correlation effects, but struggles in strong (static) correlation regimes.
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\titou{It also famously lacks size-consistency which explains issues at dissociations?}
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Importantly, the number of determinants $\Ndet$ (which is the key parameter governing the computational cost) scales polynomially with the number of \titou{basis functions} $\Nbas$ as $\Nbas^{2e}$.
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%This means that the contribution of higher excitations become progressively smaller.
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@ -111,8 +112,9 @@ By truncating at the seniority zero ($s = 0$) sector (sCI0), one obtains the wel
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which has been shown to be particularly effective at catching static correlation,
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while higher sectors tend to contribute progressively less. \cite{Bytautas_2011,Bytautas_2015,Alcoba_2014b,Alcoba_2014}
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However, already at the sCI0 level, $\Ndet$ scales exponentially with $\Nbas$, since excitations of all excitation degrees are included.
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Therefore, despite the encouraging successes of seniority-based CI methods, their unfavorable computational scaling restricts applications to very small systems.
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Besides CI, other methods that exploit the concept of seniority number have been pursued. \cite{Limacher_2013,Limacher_2014,Tecmer_2014,Boguslawski_2014a,Boguslawski_2015,Boguslawski_2014b,Boguslawski_2014c,Johnson_2017,Fecteau_2020,Johnson_2020,Henderson_2014,Stein_2014,Henderson_2015,Chen_2015,Shepherd_2016,Bytautas_2018}
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\titou{Is it therefore size-consistent?}
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Therefore, despite the encouraging successes of seniority-based CI methods, their unfavorable computational scaling restricts applications to very small systems. \cite{Shepherd_2016}
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Besides CI, other methods that exploit the concept of seniority number have been pursued. \cite{Limacher_2013,Limacher_2014,Tecmer_2014,Boguslawski_2014a,Boguslawski_2015,Boguslawski_2014b,Boguslawski_2014c,Johnson_2017,Fecteau_2020,Johnson_2020,Henderson_2014,Stein_2014,Henderson_2015,Chen_2015,Bytautas_2018}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -143,7 +145,7 @@ Figure \ref{fig:allCI} shows how the Hilbert space is populated in excitation-ba
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\includegraphics[width=0.3\linewidth]{table_sen_full}
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\caption{Partitioning of the Hilbert space into blocks of specific excitation degree $e$ (with respect to a closed-shell determinant) and seniority number $s$.
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This $e$-$s$ map is truncated differently in excitation-based CI (left), seniority-based CI (right), and hierarchy-based CI (center).
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The color tones represent the determinants that are included at a given level of CI.}
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The color tones represent the determinants that are included at a given CI level.}
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\label{fig:allCI}
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\end{figure*}
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%%% %%% %%%
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@ -178,7 +180,7 @@ In the latter case, the scaling of $\Ndet$ would be dominated by the rightmost b
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Bytautas \textit{et al.}\cite{Bytautas_2015} explored a different hybrid scheme combining determinants having a maximum seniority number and those from a complete active space.
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In comparison to previous approaches, our hybrid hCI scheme has two key advantages.
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First, it is defined by a single parameter that unifies excitation degree and seniority number [see Eq.~\eqref{eq:h}].
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Second and most importantly, each next level includes all classes of determinants sharing the same scaling with system size, as discussed before, thus preserving the polynomial cost of the method.
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Second and most importantly, each next level includes all classes of determinants \titou{sharing the same scaling with system size}, as discussed before, thus preserving the polynomial cost of the method.
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Each level of excitation-based CI has a hCI counterpart with the same scaling of $\Ndet$ with respect to $\Nbas$.
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For example, $\Ndet = \order*{\Nbas^4}$ in both hCI2 and CISD, whereas $\Ndet = \order*{\Nbas^6}$ in hCI3 and CISDT, and so on.
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