diff --git a/Manuscript/seniority.tex b/Manuscript/seniority.tex index 996aef7..d354ef2 100644 --- a/Manuscript/seniority.tex +++ b/Manuscript/seniority.tex @@ -102,6 +102,7 @@ In this context, one accounts for all determinants generated by exciting up to $ In this way, the excitation degree $e$ defines the following sequence of models: CI with single excitations (CIS), CI with single and double excitations (CISD), CI with single, double, and triple excitations (CISDT), and so on. Excitation-based CI manages to quickly recover weak (dynamic) correlation effects, but struggles in strong (static) correlation regimes. +\titou{It also famously lacks size-consistency which explains issues at dissociations?} 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 $\Nbas^{2e}$. %This means that the contribution of higher excitations become progressively smaller. @@ -111,8 +112,9 @@ By truncating at the seniority zero ($s = 0$) sector (sCI0), one obtains the wel which has been shown to be particularly effective at catching static correlation, while higher sectors tend to contribute progressively less. \cite{Bytautas_2011,Bytautas_2015,Alcoba_2014b,Alcoba_2014} However, already at the sCI0 level, $\Ndet$ scales exponentially with $\Nbas$, since excitations of all excitation degrees are included. -Therefore, despite the encouraging successes of seniority-based CI methods, their unfavorable computational scaling restricts applications to very small systems. -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} +\titou{Is it therefore size-consistent?} +Therefore, despite the encouraging successes of seniority-based CI methods, their unfavorable computational scaling restricts applications to very small systems. \cite{Shepherd_2016} +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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -143,7 +145,7 @@ Figure \ref{fig:allCI} shows how the Hilbert space is populated in excitation-ba \includegraphics[width=0.3\linewidth]{table_sen_full} \caption{Partitioning of the Hilbert space into blocks of specific excitation degree $e$ (with respect to a closed-shell determinant) and seniority number $s$. This $e$-$s$ map is truncated differently in excitation-based CI (left), seniority-based CI (right), and hierarchy-based CI (center). -The color tones represent the determinants that are included at a given level of CI.} +The color tones represent the determinants that are included at a given CI level.} \label{fig:allCI} \end{figure*} %%% %%% %%% @@ -178,7 +180,7 @@ In the latter case, the scaling of $\Ndet$ would be dominated by the rightmost b 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. In comparison to previous approaches, our hybrid hCI scheme has two key advantages. First, it is defined by a single parameter that unifies excitation degree and seniority number [see Eq.~\eqref{eq:h}]. -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. +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. Each level of excitation-based CI has a hCI counterpart with the same scaling of $\Ndet$ with respect to $\Nbas$. For example, $\Ndet = \order*{\Nbas^4}$ in both hCI2 and CISD, whereas $\Ndet = \order*{\Nbas^6}$ in hCI3 and CISDT, and so on.