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Pierre-Francois Loos 2022-03-07 22:39:54 +01:00
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Manuscript/seniority.tex
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@ -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: 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. 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. 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 $\Nbas^{2e}$. 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. %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, 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} 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. 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. \titou{Is it therefore size-consistent?}
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} 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}
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@ -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} \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 $s$. \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). 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} \label{fig:allCI}
\end{figure*} \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. 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. 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}]. 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 $\Nbas$. 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. For example, $\Ndet = \order*{\Nbas^4}$ in both hCI2 and CISD, whereas $\Ndet = \order*{\Nbas^6}$ in hCI3 and CISDT, and so on.