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39
Manuscript/seniority.bib
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@ -1,13 +1,50 @@
%% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-03-09 10:00:09 +0100
%% Created for Pierre-Francois Loos at 2022-03-09 10:25:47 +0100
%% Saved with string encoding Unicode (UTF-8)
@article{Henderson_2014b,
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-modified = {2022-03-09 10:25:38 +0100},
doi = {10.1103/PhysRevC.89.054305},
file = {/home/antoinem/Zotero/storage/NBW3DPNI/Henderson et al. - 2014 - Quasiparticle coupled cluster theory for pairing i.pdf;/home/antoinem/Zotero/storage/URGVY3VT/PhysRevC.89.html},
journal = {Phys. Rev. C},
pages = {054305},
publisher = {{American Physical Society}},
title = {Quasiparticle Coupled Cluster Theory for Pairing Interactions},
volume = {89},
year = {2014},
bdsk-url-1 = {https://doi.org/10.1103/PhysRevC.89.054305}}
@misc{Johnson_2022,
archiveprefix = {arXiv},
author = {Paul A. Johnson and Paul W. Ayers and Stijn De Baerdemacker and Peter A. Limacher and Dimitri Van Neck},
date-added = {2022-03-09 10:22:06 +0100},
date-modified = {2022-03-09 10:22:06 +0100},
doi = {10.48550/arXiv.2203.02624},
eprint = {2203.02624},
primaryclass = {physics.chem-ph},
title = {Bivariational Principle for an Antisymmetrized Product of Nonorthogonal Geminals Appropriate for Strong Electron Correlation},
year = {2022},
bdsk-url-1 = {https://doi.org/10.48550/arXiv.2203.02624}}
@misc{Fecteau_2022,
archiveprefix = {arXiv},
author = {Charles-{\'E}mile Fecteau and Samuel Cloutier and Jean-David Moisset, J{\'e}r{\'e}my Boulay and Patrick Bultinck and Alexandre Faribault and Paul A. Johnson},
date-added = {2022-03-09 10:19:23 +0100},
date-modified = {2022-03-09 10:23:18 +0100},
doi = {10.48550/arXiv.2202.12402},
eprint = {2202.12402},
primaryclass = {physics.chem-ph},
title = {Near-exact treatment of seniority-zero ground and excited states with a Richardson-Gaudin mean-field},
year = {2022}}
@article{Davidson_1975,
author = {E. R. Davidson},
date-added = {2022-03-09 10:00:08 +0100},

19
Manuscript/seniority.tex
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@ -104,7 +104,7 @@ while higher sectors tend to contribute progressively less. \cite{Bytautas_2011,
\titou{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.
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}
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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Hierarchy configuration interaction}
@ -146,7 +146,6 @@ By combining $e$ and $s$ as is Eq.~\eqref{eq:h}, we ensure that both directions
Rather than filling the map top-bottom (as in excitation-based CI) or left-right (as in seniority-based CI), the hCI methods fills it diagonally.
In this sense, we hope to recover dynamic correlation by moving right in the map (increasing the excitation degree while keeping a low seniority number),
at the same time as static correlation, by moving down (increasing the seniority number while keeping a low excitation degree).
%dynamic correlation is recovered with traditional CI.
The second justification is computational.
In the hCI class of methods, each level of theory accommodates additional determinants from different excitation-seniority sectors (each block of same color tone in Fig.~\ref{fig:allCI}).
@ -231,20 +230,19 @@ We employed the algorithm described elsewhere \cite{Damour_2021} and also implem
In order to avoid converging to a saddle point solution, we employed a similar strategy as recently described in Ref.~\onlinecite{Elayan_2022}.
Namely, whenever the eigenvalue of the orbital rotation Hessian is negative and the corresponding gradient component $g_i$ lies below a given threshold $g_0$,
then this gradient component is replaced by $g_0 \abs{g_i}/g_i$.
\fk{Here we took $g_0 = $ \SI{1}{\micro\hartree}, and considered the orbitals to be converged when the maximum orbital rotation gradient lies below \SI{0.1}{\milli\hartree}.}
Here we took $g_0 = $ \SI{1}{\micro\hartree}, and considered the orbitals to be converged when the maximum orbital rotation gradient lies below \SI{0.1}{\milli\hartree}.
While we cannot ensure that the obtained solutions are global minima in the orbital parameter space, we verified that in all stationary solutions surveyed here
correspond to real minima (rather than maxima or saddle points).
All CI calculations were performed for the cc-pVDZ basis set and with frozen core orbitals.
For the \ce{HF} molecule we have also tested basis set effects, by considered the cc-pVTZ and cc-pVQZ basis sets.
All CI calculations were performed with the cc-pVDZ basis set and within the frozen core approximation.
For the \ce{HF} molecule we have also tested basis set effects, by considered the larger cc-pVTZ and cc-pVQZ basis sets.
It is worth mentioning that obtaining smooth PECs for the orbital optimized calculations proved to be far from trivial.
First, the orbital optimization started from the HF orbitals of each geometry.
This usually lead to discontinuous PECs, meaning that distinct solutions of the orbital optimization have been found with our algorithm.
First, the orbital optimization was started from the HF orbitals of each geometry.
This usually led to discontinuous PECs, meaning that distinct solutions were found by our algorithm.
Then, at some geometries that seem to present the lowest lying solution,
the optimized orbitals were employed as the guess orbitals for the neighboring geometries, and so on, until a new PEC is obtained.
This protocol is repeated until the PEC built from the lowest lying oo-CI solution becomes continuous.
This protocol was repeated until the PEC built from the lowest lying oo-CI solution becomes continuous.
%While we cannot guarantee that the presented solutions represent the global minima, we believe that in most cases the above protocol provides at least close enough solutions.
%Multiple solutions for the orbital optimization are usually found, meaning several local minimal in the orbital parameter landscape.
We recall that saddle point solutions were purposely avoided in our orbital optimization algorithm. If that was not the case, then even more stationary solutions would have been found.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -362,10 +360,9 @@ Following the same trend, oo-CISD presents smaller NPEs than HF-CISD for the mul
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.
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.
%It does, however, lead to a more monotonic convergence in the case of hCI, which is not necessarily an advantage.
Orbital optimization usually reduces the NPE for seniority-based CI (in this case we only considered oo-DOCI) as well.
The gain is specially noticeable for \ce{H4} and \ce{H8} (where the orbitals become symmetry-broken \cite{}),
The gain is specially noticeable for \ce{H4} and \ce{H8} (where the orbitals become symmetry-broken \cite{Henderson_2014}),
and much less so for \ce{HF}, ethylene, and \ce{N2} (where the orbitals remain symmetry-preserved).
This is in line with what has been observed before for \ce{N2}. \cite{Bytautas_2011}
For \ce{F2}, we found that orbital optimization actually increases the NPE (though by a small amount),