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Manuscript/seniority.bib
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@ -1,13 +1,45 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/ %% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-03-09 10:25:47 +0100 %% Created for Pierre-Francois Loos at 2022-03-09 11:15:21 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Motta_2020,
author = {Motta, Mario and Genovese, Claudio and Ma, Fengjie and Cui, Zhi-Hao and Sawaya, Randy and Chan, Garnet Kin-Lic and Chepiga, Natalia and Helms, Phillip and Jim\'enez-Hoyos, Carlos and Millis, Andrew J. and Ray, Ushnish and Ronca, Enrico and Shi, Hao and Sorella, Sandro and Stoudenmire, Edwin M. and White, Steven R. and Zhang, Shiwei},
collaboration = {Simons Collaboration on the Many-Electron Problem},
date-added = {2022-03-09 10:53:47 +0100},
date-modified = {2022-03-09 10:53:47 +0100},
doi = {10.1103/PhysRevX.10.031058},
issue = {3},
journal = {Phys. Rev. X},
month = {Sep},
numpages = {9},
pages = {031058},
publisher = {American Physical Society},
title = {Ground-State Properties of the Hydrogen Chain: Dimerization, Insulator-to-Metal Transition, and Magnetic Phases},
url = {https://link.aps.org/doi/10.1103/PhysRevX.10.031058},
volume = {10},
year = {2020},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevX.10.031058},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevX.10.031058}}
@article{Motta_2017,
author = {Motta, Mario and Ceperley, David M and Chan, Garnet Kin-Lic and Gomez, John A and Gull, Emanuel and Guo, Sheng and Jim{\'e}nez-Hoyos, Carlos A and Lan, Tran Nguyen and Li, Jia and Ma, Fengjie and others},
date-added = {2022-03-09 10:53:17 +0100},
date-modified = {2022-03-09 10:53:17 +0100},
doi = {10.1103/PhysRevX.7.031059},
journal = {Phys. Rev. X},
number = {3},
pages = {031059},
title = {Towards the solution of the many-electron problem in real materials: Equation of state of the hydrogen chain with state-of-the-art many-body methods},
volume = {7},
year = {2017},
bdsk-url-1 = {https://doi.org/10.1103/PhysRevX.7.031059}}
@article{Henderson_2014b, @article{Henderson_2014b,
author = {Henderson, Thomas M. and Scuseria, Gustavo E. and Dukelsky, Jorge and Signoracci, Angelo and Duguet, Thomas}, author = {Henderson, Thomas M. and Scuseria, Gustavo E. and Dukelsky, Jorge and Signoracci, Angelo and Duguet, Thomas},
date-added = {2022-03-09 10:25:38 +0100}, date-added = {2022-03-09 10:25:38 +0100},
@ -43,7 +75,8 @@
eprint = {2202.12402}, eprint = {2202.12402},
primaryclass = {physics.chem-ph}, primaryclass = {physics.chem-ph},
title = {Near-exact treatment of seniority-zero ground and excited states with a Richardson-Gaudin mean-field}, title = {Near-exact treatment of seniority-zero ground and excited states with a Richardson-Gaudin mean-field},
year = {2022}} year = {2022},
bdsk-url-1 = {https://doi.org/10.48550/arXiv.2202.12402}}
@article{Davidson_1975, @article{Davidson_1975,
author = {E. R. Davidson}, author = {E. R. Davidson},

16
Manuscript/seniority.tex
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@ -101,7 +101,7 @@ In short, the seniority number $s$ is the number of unpaired electrons in a give
By truncating at the seniority zero ($s = 0$) sector (sCI0), one obtains the well-known doubly-occupied CI (DOCI) method, \cite{Bytautas_2011,Allen_1962,Smith_1965,Veillard_1967} By truncating at the seniority zero ($s = 0$) sector (sCI0), one obtains the well-known doubly-occupied CI (DOCI) method, \cite{Bytautas_2011,Allen_1962,Smith_1965,Veillard_1967}
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}
\titou{In addition, sCI0 is size-consistent, a property that is not shared by higher orders of seniority-based CI.} In addition, sCI0 is size-consistent, a property that is not shared by higher orders of seniority-based CI.
However, already at the sCI0 level, $\Ndet$ scales exponentially with $\Nbas$, since excitations of all degrees are included. However, already at the sCI0 level, $\Ndet$ scales exponentially with $\Nbas$, since excitations of all degrees are included.
Therefore, despite the encouraging successes of seniority-based CI methods, their unfavorable computational scaling restricts applications to very small systems. \cite{Shepherd_2016} 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,Johnson_2022,Fecteau_2022} 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,Johnson_2022,Fecteau_2022}
@ -262,7 +262,7 @@ The corresponding PECs and the energy differences with respect to FCI can be fou
The main result contained in Fig.~\ref{fig:plot_stat} concerns the overall faster convergence of hCI when compared to excitation-based and seniority-based CI. The main result contained in Fig.~\ref{fig:plot_stat} concerns the overall faster convergence of hCI when compared to excitation-based and seniority-based CI.
This is observed for single bond breaking (\ce{HF} and \ce{F2}) as well as the more challenging double (ethylene), triple (\ce{N2}), and quadruple (\ce{H4}) bond breaking. This is observed for single bond breaking (\ce{HF} and \ce{F2}) as well as the more challenging double (ethylene), triple (\ce{N2}), and quadruple (\ce{H4}) bond breaking.
For \ce{H8}, hCI and excitation-based CI perform similarly. For \ce{H8}, hCI and excitation-based CI perform similarly.
The convergence with respect to $\Ndet$ is slower in the latter, more challenging cases, irrespective of the class of CI methods, as expected. The convergence with respect to $\Ndet$ is slower in the latter, more challenging cases, irrespective of the class of CI methods, as expected. \cite{Motta_2017,Motta_2020}
But more importantly, the superiority of hCI appears to be highlighted in the multiple bond break systems (compare ethylene and \ce{N2} with \ce{HF} and \ce{F2} in Fig.~\ref{fig:plot_stat}). But more importantly, the superiority of hCI appears to be highlighted in the multiple bond break systems (compare ethylene and \ce{N2} with \ce{HF} and \ce{F2} in Fig.~\ref{fig:plot_stat}).
%%% FIG 2 %%% %%% FIG 2 %%%
@ -279,7 +279,7 @@ hCI2.5 is better than CISDT (except for \ce{H8}), despite its lower computationa
Inspection of the PECs (see \SupInf) reveals that the lower NPEs observed for hCI stem mostly from the contribution of the dissociation region. Inspection of the PECs (see \SupInf) reveals that the lower NPEs observed for hCI stem mostly from the contribution of the dissociation region.
This result demonstrates the importance of higher-order excitations with low seniority number in this strong correlation regime, This result demonstrates the importance of higher-order excitations with low seniority number in this strong correlation regime,
which are accounted for in hCI but not in excitation-based CI (for a given scaling of $\Ndet$). which are accounted for in hCI but not in excitation-based CI (for a given scaling of $\Ndet$).
\fk{These determinants are responsible for alleviating the size-consistency problem when going from excitation-based CI to hCI.} These determinants are responsible for alleviating the size-consistency problem when going from excitation-based CI to hCI.
Meanwhile, the first level of seniority-based CI (sCI0, which is the same as DOCI) Meanwhile, the first level of seniority-based CI (sCI0, which is the same as DOCI)
tends to offer a rather low NPE when compared to the other CI methods with a similar $\Ndet$ scaling (hCI2.5 and CISDT). tends to offer a rather low NPE when compared to the other CI methods with a similar $\Ndet$ scaling (hCI2.5 and CISDT).
@ -297,8 +297,8 @@ become less apparent as progressively more bonds are being broken (compare, for
This reflects the fact that higher-order excitations are needed to properly describe multiple bond breaking, This reflects the fact that higher-order excitations are needed to properly describe multiple bond breaking,
and also hints at some cancelation of errors in low-order hCI methods for single bond breaking. and also hints at some cancelation of errors in low-order hCI methods for single bond breaking.
In Fig.~Sx of the \SupInf, we present the distance error, which is also found to decrease faster with the hCI methods. In Fig.~Sx of the \SupInf, we present the distance error, which is also found to decrease faster with hCI.
Most of observations discussed for the NPE also hold for the distance error, with two main differences. Most of the observations discussed for the NPE also hold for the distance error, with two main differences.
The convergence is always monotonic for the latter observable (which is expected from the definition of the observable), The convergence is always monotonic for the latter observable (which is expected from the definition of the observable),
and the performance of seniority-based CI is much poorer (due to the slow recovery of dynamic correlation). and the performance of seniority-based CI is much poorer (due to the slow recovery of dynamic correlation).
@ -342,10 +342,10 @@ We thus believe that the main findings discussed here for the other systems woul
%\subsection{Orbital optimized configuration interaction} %\subsection{Orbital optimized configuration interaction}
Up to this point, all results and discussions have been based on CI calculations with HF orbitals. Up to this point, all results and discussions have been based on CI calculations with HF orbitals.
\fk{We recall that seniority-based CI (in contrast to excitation-based CI) is not invariant with respect to orbital rotations within the occupied and virtual subspaces, \cite{Bytautas_2011} We recall that seniority-based CI (in contrast to excitation-based CI) is not invariant with respect to orbital rotations within the occupied and virtual subspaces, \cite{Bytautas_2011}
and for this reason it is customary to optimize the corresponding wave function by performing such rotations. and for this reason it is customary to optimize the corresponding wave function by performing such rotations.
Similarly, hCI wave functions are not invariant under orbital rotations within each subspace. Similarly, hCI wave functions are not invariant under orbital rotations within each subspace.
Thus, we decided to further assess the role of orbital optimization (occupied-virtual rotations included) for each class of CI methods.} Thus, we decided to further assess the role of orbital optimization (occupied-virtual rotations included) for each class of CI methods.
Due to the significantly higher computational cost and numerical difficulties associated with orbital optimization at higher CI levels, Due to the significantly higher computational cost and numerical difficulties associated with orbital optimization at higher CI levels,
such calculations were typically limited up to oo-CISD (for excitation-based), oo-DOCI (for seniority-based), and oo-hCI2 (for hCI). such calculations were typically limited up to oo-CISD (for excitation-based), oo-DOCI (for seniority-based), and oo-hCI2 (for hCI).
The PECs and analogous results to those of Figs.~\ref{fig:plot_stat}, \ref{fig:xe}, and \ref{fig:freq} are shown in the \SupInf. The PECs and analogous results to those of Figs.~\ref{fig:plot_stat}, \ref{fig:xe}, and \ref{fig:freq} are shown in the \SupInf.
@ -358,7 +358,7 @@ similar NPEs for ethylene, and smaller NPEs for \ce{N2}, \ce{H4}, and \ce{H8}.
% oo-hCI2 % oo-hCI2
Following the same trend, oo-CISD presents smaller NPEs than HF-CISD for the multiple bond breaking systems, but very similar ones for the single bond breaking cases. Following the same trend, oo-CISD presents smaller NPEs than HF-CISD for the multiple bond breaking systems, but very similar ones for the single bond breaking cases.
oo-CIS has significantly smaller NPEs than HF-CIS, being comparable to oo-hCI1 for all systems except for \ce{H4} and \ce{H8}, where the latter method performs better. oo-CIS has significantly smaller NPEs than HF-CIS, being comparable to oo-hCI1 for all systems except for \ce{H4} and \ce{H8}, where the latter method performs better.
We will come back to oo-CIS latter. (We will come back to oo-CIS later.)
Based on the present oo-CI results, hCI still has the upper hand when compared with excitation-based CI, though by a much smaller margin. Based on the present oo-CI results, hCI still has the upper hand when compared with excitation-based CI, though by a much smaller margin.
Orbital optimization usually reduces the NPE for seniority-based CI (in this case we only considered oo-DOCI) as well. Orbital optimization usually reduces the NPE for seniority-based CI (in this case we only considered oo-DOCI) as well.