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Manuscript/seniority.bib
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@ -1,13 +1,354 @@
%% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-03-06 15:23:57 +0100
%% Created for Pierre-Francois Loos at 2022-03-06 22:50:10 +0100
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@article{Magoulas_2021,
author = {Magoulas, Ilias and Gururangan, Karthik and Piecuch, Piotr and Deustua, J. Emiliano and Shen, Jun},
date-added = {2022-03-06 22:49:34 +0100},
date-modified = {2022-03-06 22:49:34 +0100},
doi = {10.1021/acs.jctc.1c00181},
journal = {J. Chem. Theory Comput.},
number = {7},
pages = {4006-4027},
title = {Is Externally Corrected Coupled Cluster Always Better Than the Underlying Truncated Configuration Interaction?},
volume = {17},
year = {2021},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.1c00181}}
@article{Lee_2021,
author = {Lee, Seunghoon and Zhai, Huanchen and Sharma, Sandeep and Umrigar, C. J. and Chan, Garnet Kin-Lic},
date-added = {2022-03-06 22:47:21 +0100},
date-modified = {2022-03-06 22:47:21 +0100},
doi = {10.1021/acs.jctc.1c00205},
journal = {J. Chem. Theory Comput.},
number = {6},
pages = {3414-3425},
title = {Externally Corrected CCSD with Renormalized Perturbative Triples (R-ecCCSD(T)) and the Density Matrix Renormalization Group and Selected Configuration Interaction External Sources},
volume = {17},
year = {2021},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.1c00205}}
@article{Aroeira_2021,
author = {Aroeira, Gustavo J. R. and Davis, Madeline M. and Turney, Justin M. and Schaefer, Henry F.},
date-added = {2022-03-06 22:47:03 +0100},
date-modified = {2022-03-06 22:47:03 +0100},
doi = {10.1021/acs.jctc.0c00888},
journal = {J. Chem. Theory Comput.},
number = {1},
pages = {182-190},
title = {Coupled Cluster Externally Corrected by Adaptive Configuration Interaction},
volume = {17},
year = {2021},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.0c00888}}
@article{Dash_2019,
author = {Dash, Monika and Feldt, Jonas and Moroni, Saverio and Scemama, Anthony and Filippi, Claudia},
date-added = {2022-03-06 22:42:28 +0100},
date-modified = {2022-03-06 22:43:48 +0100},
doi = {10.1021/acs.jctc.9b00476},
journal = {J. Chem. Theory Comput.},
number = {9},
pages = {4896-4906},
title = {Excited States with Selected Configuration Interaction-Quantum Monte Carlo: Chemically Accurate Excitation Energies and Geometries},
volume = {15},
year = {2019},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.9b00476}}
@article{Cuzzocrea_2022,
author = {Cuzzocrea, Alice and Moroni, Saverio and Scemama, Anthony and Filippi, Claudia},
date-added = {2022-03-06 22:42:28 +0100},
date-modified = {2022-03-06 22:43:15 +0100},
doi = {10.1021/acs.jctc.1c01162},
journal = {J. Chem. Theory Comput.},
number = {2},
pages = {1089-1095},
title = {Reference Excitation Energies of Increasingly Large Molecules: A QMC Study of Cyanine Dyes},
volume = {18},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.1c01162}}
@article{Dash_2021,
author = {Dash, Monika and Moroni, Saverio and Filippi, Claudia and Scemama, Anthony},
date-added = {2022-03-06 22:42:28 +0100},
date-modified = {2022-03-06 22:42:57 +0100},
doi = {10.1021/acs.jctc.1c00212},
journal = {J. Chem. Theory Comput.},
number = {6},
pages = {3426-3434},
title = {Tailoring CIPSI Expansions for QMC Calculations of Electronic Excitations: The Case Study of Thiophene},
volume = {17},
year = {2021},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.1c00212}}
@article{Fecteau_2020,
author = {Fecteau,Charles-{\'E}mile and Fortin,Hubert and Cloutier,Samuel and Johnson,Paul A.},
date-added = {2022-03-06 20:16:57 +0100},
date-modified = {2022-03-06 20:16:57 +0100},
doi = {10.1063/5.0027393},
journal = {J. Chem. Phys.},
number = {16},
pages = {164117},
title = {Reduced density matrices of Richardson--Gaudin states in the Gaudin algebra basis},
volume = {153},
year = {2020},
bdsk-url-1 = {https://doi.org/10.1063/5.0027393}}
@article{Johnson_2020,
author = {Johnson,Paul A. and Fecteau,Charles-{\'E}mile and Berthiaume,Fr{\'e}d{\'e}ric and Cloutier,Samuel and Carrier,Laurie and Gratton,Marianne and Bultinck,Patrick and De Baerdemacker,Stijn and Van Neck,Dimitri and Limacher,Peter and Ayers,Paul W.},
date-added = {2022-03-06 20:16:57 +0100},
date-modified = {2022-03-06 20:16:57 +0100},
doi = {10.1063/5.0022189},
journal = {J. Chem. Phys.},
number = {10},
pages = {104110},
title = {Richardson--Gaudin mean-field for strong correlation in quantum chemistry},
volume = {153},
year = {2020},
bdsk-url-1 = {https://doi.org/10.1063/5.0022189}}
@article{Boguslawski_2016a,
author = {Boguslawski, Katharina and Tecmer, Pawe{\l} and Legeza, {\"O}rs},
date-added = {2022-03-06 20:16:11 +0100},
date-modified = {2022-03-06 20:16:11 +0100},
doi = {10.1103/PhysRevB.94.155126},
file = {/home/antoinem/Zotero/storage/XLC79SPJ/Boguslawski et al. - 2016 - Analysis of two-orbital correlations in wave funct.pdf;/home/antoinem/Zotero/storage/5BTASXE6/PhysRevB.94.html;/home/antoinem/Zotero/storage/9IQSWNFY/PhysRevB.94.html},
journal = {Phys. Rev. B},
pages = {155126},
publisher = {{American Physical Society}},
title = {Analysis of Two-Orbital Correlations in Wave Functions Restricted to Electron-Pair States},
volume = {94},
year = {2016},
bdsk-url-1 = {https://doi.org/10.1103/PhysRevB.94.155126}}
@article{Tecmer_2014,
author = {Tecmer, Pawe{\l} and Boguslawski, Katharina and Johnson, Paul A. and Limacher, Peter A. and Chan, Matthew and Verstraelen, Toon and Ayers, Paul W.},
date-added = {2022-03-06 20:16:11 +0100},
date-modified = {2022-03-06 20:16:11 +0100},
doi = {10.1021/jp502127v},
file = {/home/antoinem/Zotero/storage/Q2XDCYBY/Tecmer et al. - 2014 - Assessing the Accuracy of New Geminal-Based Approa.pdf;/home/antoinem/Zotero/storage/GNPWKYBS/jp502127v.html},
journal = {J. Phys. Chem. A},
pages = {9058--9068},
publisher = {{American Chemical Society}},
title = {Assessing the {{Accuracy}} of {{New Geminal}}-{{Based Approaches}}},
volume = {118},
year = {2014},
bdsk-url-1 = {https://doi.org/10.1021/jp502127v}}
@article{Boguslawski_2017a,
author = {Boguslawski, Katharina and Tecmer, Pawe{\l}},
date-added = {2022-03-06 20:16:11 +0100},
date-modified = {2022-03-06 20:16:11 +0100},
doi = {10.1021/acs.jctc.6b01134},
file = {/home/antoinem/Zotero/storage/QQLX55VV/Boguslawski and Tecmer - 2017 - Benchmark of Dynamic Electron Correlation Models f.pdf;/home/antoinem/Zotero/storage/FIW7UJXS/acs.jctc.html},
journal = {J. Chem. Theory Comput.},
pages = {5966--5983},
publisher = {{American Chemical Society}},
title = {Benchmark of {{Dynamic Electron Correlation Models}} for {{Seniority}}-{{Zero Wave Functions}} and {{Their Application}} to {{Thermochemistry}}},
volume = {13},
year = {2017},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.6b01134}}
@article{Boguslawski_2014a,
author = {Boguslawski, Katharina and Tecmer, Pawe{\l} and Ayers, Paul W. and Bultinck, Patrick and De Baerdemacker, Stijn and Van Neck, Dimitri},
date-added = {2022-03-06 20:16:11 +0100},
date-modified = {2022-03-06 20:16:11 +0100},
doi = {10.1103/PhysRevB.89.201106},
file = {/home/antoinem/Zotero/storage/MR7E5X7Y/Boguslawski et al. - 2014 - Efficient description of strongly correlated elect.pdf;/home/antoinem/Zotero/storage/FPAQQW6N/PhysRevB.89.html},
journal = {Phys. Rev. B},
pages = {201106},
publisher = {{American Physical Society}},
title = {Efficient Description of Strongly Correlated Electrons with Mean-Field Cost},
volume = {89},
year = {2014},
bdsk-url-1 = {https://doi.org/10.1103/PhysRevB.89.201106}}
@article{Boguslawski_2017b,
author = {Boguslawski, Katharina},
date-added = {2022-03-06 20:16:11 +0100},
date-modified = {2022-03-06 20:16:11 +0100},
doi = {10.1063/1.5006124},
file = {/home/antoinem/Zotero/storage/YLPA4GNY/Boguslawski - 2017 - Erratum ``Targeting excited states in all-trans po.pdf;/home/antoinem/Zotero/storage/7MJG68RQ/1.html},
journal = {J. Chem. Phys.},
pages = {139901},
publisher = {{American Institute of Physics}},
title = {Erratum: ``{{Targeting}} Excited States in All-Trans Polyenes with Electron-Pair States'' [{{J}}. {{Chem}}. {{Phys}}. 145, 234105 (2016)]},
volume = {147},
year = {2017},
bdsk-url-1 = {https://doi.org/10.1063/1.5006124}}
@article{Boguslawski_2014b,
author = {Boguslawski, Katharina and Tecmer, Pawe{\l} and Bultinck, Patrick and De Baerdemacker, Stijn and Van Neck, Dimitri and Ayers, Paul W.},
date-added = {2022-03-06 20:16:11 +0100},
date-modified = {2022-03-06 20:16:11 +0100},
doi = {10.1021/ct500759q},
file = {/home/antoinem/Zotero/storage/S2ZFWFVC/Boguslawski et al. - 2014 - Nonvariational Orbital Optimization Techniques for.pdf;/home/antoinem/Zotero/storage/Z26HYNRT/ct500759q.html},
journal = {J. Chem. Theory Comput.},
pages = {4873--4882},
publisher = {{American Chemical Society}},
title = {Nonvariational {{Orbital Optimization Techniques}} for the {{AP1roG Wave Function}}},
volume = {10},
year = {2014},
bdsk-url-1 = {https://doi.org/10.1021/ct500759q}}
@article{Boguslawski_2019,
author = {Boguslawski, Katharina},
date-added = {2022-03-06 20:16:11 +0100},
date-modified = {2022-03-06 20:16:11 +0100},
doi = {10.1021/acs.jctc.8b01053},
file = {/home/antoinem/Zotero/storage/9P2GLQNQ/Boguslawski - 2019 - Targeting Doubly Excited States with Equation of M.pdf;/home/antoinem/Zotero/storage/Q7RN4NWG/acs.jctc.html},
journal = {J. Chem. Theory Comput.},
pages = {18--24},
publisher = {{American Chemical Society}},
title = {Targeting {{Doubly Excited States}} with {{Equation}} of {{Motion Coupled Cluster Theory Restricted}} to {{Double Excitations}}},
volume = {15},
year = {2019},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.8b01053}}
@article{Boguslawski_2016b,
author = {Boguslawski, Katharina},
date-added = {2022-03-06 20:16:11 +0100},
date-modified = {2022-03-06 20:16:11 +0100},
doi = {10.1063/1.4972053},
file = {/home/antoinem/Zotero/storage/KHBWEYJD/Boguslawski - 2016 - Targeting excited states in all-trans polyenes wit.pdf;/home/antoinem/Zotero/storage/3APDNSK4/1.html},
journal = {J. Chem. Phys.},
pages = {234105},
publisher = {{American Institute of Physics}},
title = {Targeting Excited States in All-Trans Polyenes with Electron-Pair States},
volume = {145},
year = {2016},
bdsk-url-1 = {https://doi.org/10.1063/1.4972053}}
@article{Limacher_2013,
author = {Limacher, Peter A. and Ayers, Paul W. and Johnson, Paul A. and De Baerdemacker, Stijn and Van Neck, Dimitri and Bultinck, Patrick},
date-added = {2022-03-06 20:15:46 +0100},
date-modified = {2022-03-06 20:15:46 +0100},
doi = {10.1021/ct300902c},
file = {/home/antoinem/Zotero/storage/DC5HMNVA/Limacher et al. - 2013 - A New Mean-Field Method Suitable for Strongly Corr.pdf;/home/antoinem/Zotero/storage/GMVQZCGN/ct300902c.html},
journal = {J. Chem. Theory Comput.},
pages = {1394--1401},
publisher = {{American Chemical Society}},
title = {A {{New Mean}}-{{Field Method Suitable}} for {{Strongly Correlated Electrons}}: {{Computationally Facile Antisymmetric Products}} of {{Nonorthogonal Geminals}}},
volume = {9},
year = {2013},
bdsk-url-1 = {https://doi.org/10.1021/ct300902c}}
@article{Wouters_2014,
author = {Wouters, Sebastian and Poelmans, Ward and Ayers, Paul W. and Van Neck, Dimitri},
date-added = {2022-03-06 20:15:46 +0100},
date-modified = {2022-03-06 20:15:46 +0100},
doi = {10.1016/j.cpc.2014.01.019},
issn = {00104655},
journal = {Computer Physics Communications},
month = jun,
number = {6},
pages = {1501-1514},
shorttitle = {{{CheMPS2}}},
title = {{{CheMPS2}}: {{A}} Free Open-Source Spin-Adapted Implementation of the Density Matrix Renormalization Group for Ab Initio Quantum Chemistry},
volume = {185},
year = {2014},
bdsk-url-1 = {https://doi.org/10.1016/j.cpc.2014.01.019}}
@article{Boguslawski_2015,
author = {Boguslawski, Katharina and Ayers, Paul W.},
date-added = {2022-03-06 20:15:46 +0100},
date-modified = {2022-03-06 20:15:46 +0100},
doi = {10.1021/acs.jctc.5b00776},
file = {/home/antoinem/Zotero/storage/X9BZXVSJ/Boguslawski and Ayers - 2015 - Linearized Coupled Cluster Correction on the Antis.pdf;/home/antoinem/Zotero/storage/JYCLDH9E/acs.jctc.html},
journal = {J. Chem. Theory Comput.},
pages = {5252--5261},
publisher = {{American Chemical Society}},
title = {Linearized {{Coupled Cluster Correction}} on the {{Antisymmetric Product}} of 1-{{Reference Orbital Geminals}}},
volume = {11},
year = {2015},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.5b00776}}
@article{Boguslawski_2014c,
author = {Boguslawski, Katharina and Tecmer, Pawe{\l} and Limacher, Peter A. and Johnson, Paul A. and Ayers, Paul W. and Bultinck, Patrick and De Baerdemacker, Stijn and Van Neck, Dimitri},
date-added = {2022-03-06 20:15:46 +0100},
date-modified = {2022-03-06 20:15:46 +0100},
doi = {10.1063/1.4880820},
file = {/home/antoinem/Zotero/storage/BVAZILSS/Boguslawski et al. - 2014 - Projected seniority-two orbital optimization of th.pdf;/home/antoinem/Zotero/storage/LA6SD7AH/1.html},
journal = {J. Chem. Phys.},
pages = {214114},
publisher = {{American Institute of Physics}},
title = {Projected Seniority-Two Orbital Optimization of the Antisymmetric Product of One-Reference Orbital Geminal},
volume = {140},
year = {2014},
bdsk-url-1 = {https://doi.org/10.1063/1.4880820}}
@article{Limacher_2014a,
author = {Limacher, Peter and W. Ayers, Paul and A. Johnson, Paul and Baerdemacker, Stijn De and Van Neck, Dimitri and Bultinck, Patrick},
date-added = {2022-03-06 20:15:46 +0100},
date-modified = {2022-03-06 20:15:46 +0100},
doi = {10.1039/C3CP53301H},
file = {/home/antoinem/Zotero/storage/2KIWXTMG/A. Limacher et al. - 2014 - Simple and inexpensive perturbative correction sch.pdf;/home/antoinem/Zotero/storage/G2AM79JD/A. Limacher et al. - 2014 - Simple and inexpensive perturbative correction sch.pdf;/home/antoinem/Zotero/storage/2IG32UHQ/c3cp53301h.html;/home/antoinem/Zotero/storage/HWN8YQEX/c3cp53301h.html},
journal = {Phys. Chem. Chem. Phys.},
pages = {5061--5065},
publisher = {{Royal Society of Chemistry}},
title = {Simple and Inexpensive Perturbative Correction Schemes for Antisymmetric Products of Nonorthogonal Geminals},
volume = {16},
year = {2014},
bdsk-url-1 = {https://doi.org/10.1039/C3CP53301H}}
@article{Tecmer_2015,
author = {Tecmer, Pawe{\l} and Boguslawski, Katharina and Ayers, Paul W.},
date-added = {2022-03-06 20:15:46 +0100},
date-modified = {2022-03-06 20:15:46 +0100},
doi = {10.1039/C4CP05293E},
file = {/home/antoinem/Zotero/storage/XPJR6GIB/unauth.html},
journal = {Phys. Chem. Chem. Phys.},
pages = {14427--14436},
publisher = {{The Royal Society of Chemistry}},
title = {Singlet Ground State Actinide Chemistry with Geminals},
volume = {17},
year = {2015},
bdsk-url-1 = {https://doi.org/10.1039/C4CP05293E}}
@article{Johnson_2017,
abstract = {We discuss some strategies for extending recent geminal-based methods to open-shells by replacing the geminal-creation operators with more general composite boson creation operators, and even creation operators that mix fermionic and bosonic components. We also discuss the utility of symmetry-breaking and restoration, but using a projective (not a variational) approach. Both strategies---either together or separately---give a pathway for extending geminals-based methods to open shells, while retaining the computational efficiency and conceptual simplicity of existing geminal product wavefunctions.},
author = {Paul A. Johnson and Peter A. Limacher and Taewon D. Kim and Michael Richer and Ram{\'o}n Alain Miranda-Quintana and Farnaz Heidar-Zadeh and Paul W. Ayers and Patrick Bultinck and Stijn {De Baerdemacker} and Dimitri {Van Neck}},
date-added = {2022-03-06 20:15:46 +0100},
date-modified = {2022-03-06 20:15:46 +0100},
doi = {https://doi.org/10.1016/j.comptc.2017.05.010},
journal = {Comput. Theor. Chem.},
pages = {207-219},
title = {Strategies for extending geminal-based wavefunctions: Open shells and beyond},
volume = {1116},
year = {2017},
bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S2210271X17302359},
bdsk-url-2 = {https://doi.org/10.1016/j.comptc.2017.05.010}}
@article{Limacher_2014,
author = {Limacher, Peter A. and Kim, Taewon D. and Ayers, Paul W. and Johnson, Paul A. and Baerdemacker, Stijn De and Neck, Dimitri Van and Bultinck, Patrick},
date-added = {2022-03-06 20:15:46 +0100},
date-modified = {2022-03-06 20:15:46 +0100},
doi = {10.1080/00268976.2013.874600},
file = {/home/antoinem/Zotero/storage/QK9XACGM/Limacher et al. - 2014 - The influence of orbital rotation on the energy of.pdf;/home/antoinem/Zotero/storage/RB83UH42/00268976.2013.html},
journal = {Mol. Phys.},
pages = {853--862},
publisher = {{Taylor \& Francis}},
title = {The Influence of Orbital Rotation on the Energy of Closed-Shell Wavefunctions},
volume = {112},
year = {2014},
bdsk-url-1 = {https://doi.org/10.1080/00268976.2013.874600}}
@article{Ayers_2018,
author = {P. W. Ayers and M. Levy and \'A. Nagy},
date-added = {2022-03-06 20:15:46 +0100},
date-modified = {2022-03-06 20:15:46 +0100},
doi = {10.1007/s00214-018-2352-7},
journal = {Theor. Chem. Acc.},
pages = {137},
title = {Time‐independent density functional theory for degenerate excited states of Coulomb systems},
year = {2018},
bdsk-url-1 = {https://doi.org/10.1007/s00214-018-2352-7}}
@article{Garniron_2018,
author = {Y. Garniron and A. Scemama and E. Giner and M. Caffarel and P. F. Loos},
date-added = {2022-03-06 15:23:54 +0100},

49
Manuscript/seniority.tex
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@ -102,7 +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.
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 $N^{2e}$.
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.
Alternatively, seniority-based CI methods (sCI) have been proposed in both nuclear \cite{Ring_1980} and electronic \cite{Bytautas_2011} structure calculations.
@ -112,8 +112,7 @@ 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{Henderson_2014,Chen_2015,Bytautas_2018}
\titou{T2: I think we need to cite the papers of the Canadians here.}
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,Chen_2015,Bytautas_2018}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -214,8 +213,7 @@ The NPE is defined as the maximum minus the minimum differences between the PECs
We define the distance error as the maximum and the minimum differences between a given PEC and the FCI result.
Thus, while the NPE probes the similarity regarding the shape of the PECs, the distance error provides a measure of how their overall magnitudes compare.
From the PECs, we have also extracted the vibrational frequencies and equilibrium geometries (details can be found in the \SupInf).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Computational details}
%\label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -231,6 +229,13 @@ which are not related to the particular algorithmic choices of the CIPSI calcula
All CI calculations were performed for the cc-pVDZ basis set and with frozen core orbitals.
For \ce{HF} we have also tested basis set effects, by considered the cc-pVTZ and cc-pVQZ basis sets.
\titou{Geometries? SI?}
\titou{T2: I think it might be worth mentioning that the determinant-driven framework of {\QP} allows to include any arbitrary set of determinants.
This would also justify why we are focusing on the number of determinants instead of the actual scaling of the method.
I think this is a important point because the CISD Hilbert space has a size proportional to $N^4$ but the cost associated with solving the CISD equations scales as $N^6$... Actually, it follows the same rules as CC: CISD scales as $N^6$, CISDT as $N^8$, CISDTQ as $N^{10}$, etc.
We have to mention this somewhere.
Also, it is worth mentioning that one uses Davidson's iterative algorithm to seek for the ground-state energy which means that the computation and storage cost us $\order*{\Ndet^2}$ and $\order*{\Ndet}$, respectively.
This shows that the determinant-driven algorithm is definitely not optimal.
However, the selected nature of the CIPSI algorithm means that the actual number of determinants is quite small and therefore calculations are technically feasable.}
The CI calculations were performed with both canonical HF orbitals and optimized orbitals.
In the latter case, the energy is obtained variationally in the CI space and in the orbital parameter space, hence an orbital-optimized CI (oo-CI) method.
@ -244,14 +249,13 @@ correspond to real minima (rather than maxima or saddle points).
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.
Then, at some geometry or geometries that seem to present the lowest lying solution,
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.
%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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Results and discussion}
%\label{sec:res}
@ -264,15 +268,16 @@ We recall that saddle point solutions were purposely avoided in our orbital opti
%\subsection{Non-parallelity errors}
In Fig.~\ref{fig:plot_stat} we present the NPEs for the six systems studied, and for the three classes of CI methods,
as functions of the number of determinants, $\Ndet$.
as functions of $\Ndet$.
The corresponding PECs and the energy differences with respect to the FCI results can be found in the \SupInf.
The main result contained in Fig.~\ref{fig:plot_stat} concerns the overall faster convergence of the hCI methods when compared to excitation-based and seniority-based CI methods.
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.
The convergence with respect to $\Ndet$ is slower in the latter, more challenging cases, irrespective of the class of CI methods, as would be expected.
But more importantly, the superiority of the hCI methods 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}).
\titou{T2: Would it be a good idea to write the \ce{HF} molecule as \ce{FH}?}
For \ce{HF} we also evaluated the convergence is affected by increasing the basis sets, going from cc-pVDZ to cc-pVTZ and cc-pVQZ basis sets (see Fig.Sx in the \SupInf).
For \ce{HF} we have also evaluated how the convergence is affected by increasing the basis sets, going from cc-pVDZ to cc-pVTZ and cc-pVQZ (see Fig.~Sx in the \SupInf).
While a larger $\Ndet$ is required to achieve the same level of convergence, as expected,
the convergence profiles remain very similar for all basis sets.
We thus believe that the main findings discussed here for the other systems would be equally basis set independent.
@ -286,7 +291,7 @@ We thus believe that the main findings discussed here for the other systems woul
\end{figure}
%%% %%% %%%
For all systems (specially ethylene and \ce{N2}), hCI2 is better than CISD, two methods where $\Ndet$ scales as $N^4$.
For all systems (specially ethylene and \ce{N2}), hCI2 is better than CISD, two methods where $\Ndet$ scales as $\Nbas^4$.
hCI2.5 is better than CISDT (except for \ce{H8}), despite its lower computational cost, whereas hCI3 is much better than CISDT, and comparable in accuracy with CISDTQ (again for all systems).
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,
@ -296,7 +301,7 @@ Meanwhile, the first level of seniority-based CI (sCI0, which is the same as DOC
tends to offer a rather low NPE when compared to the other CI methods with a similar $\Ndet$ (hCI2.5 and CISDT).
However, convergence is clearly slower for the next levels (sCI2 and sCI4), whereas excitation-based CI and specially hCI methods converge faster.
Furthermore, seniority-based CI becomes less attractive for larger basis set in view of its exponential scaling.
This can be seen in Fig.Sx of the \SupInf, which shows that augmenting the basis set leads to a much steeper increase of $\Ndet$ for seniority-based CI.
This can be seen in Fig.~Sx of the \SupInf, which shows that augmenting the basis set leads to a much steeper increase of $\Ndet$ for seniority-based CI.
It is worth mentioning the surprisingly good performance of the hCI1 and hCI1.5 methods.
For \ce{HF}, \ce{F2}, and ethylene, they presented lower NPEs than the much more expensive CISDT method, being slightly higher in the case of \ce{N2}.
@ -306,9 +311,9 @@ Both findings are not observed for \ce{H4} and \ce{H8}.
It seems that both the relative worsening of hCI2 and the success of hCI1 and hCI1.5
become less apparent as progressively more bonds are being broken (compare for instance \ce{F2}, \ce{N2}, and \ce{H8} in Fig.~\ref{fig:plot_stat}).
This reflects the fact that higher-order excitations are needed to properly describe multiple bond breaking,
and also hints at some cancelation of erros 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 the hCI methods.
Most of 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),
and the performance of seniority-based CI is much poorer (due to the slow recovery of dynamic correlation).
@ -323,7 +328,7 @@ For both observables, hCI and excitation-based CI largely outperform seniority-b
Similarly to what we observed for the NPEs, the convergence of hCI was also found to be non-monotonic in some cases.
This oscillatory behavior is particularly evident for \ce{F2}, also noticeable for \ce{HF}, becoming less apparent for ethylene, virtually absent for \ce{N2},
and showing up again for \ce{H4} and \ce{H8}.
Results for \ce{HF} with larger basis sets (see Fig.Sx in the \SupInf) show very similar convergence behaviours, though with less oscillations for the hCI methods.
Results for \ce{HF} with larger basis sets (see Fig.Sx in the \SupInf) show very similar convergence behaviors, though with less oscillations for the hCI methods.
Interestingly, equilibrium geometries and vibrational frequencies of \ce{HF} and \ce{F2} (single bond breaking),
are rather accurate when evaluated at the hCI1.5 level, bearing in mind its relatively modest computational cost.
@ -347,6 +352,8 @@ are rather accurate when evaluated at the hCI1.5 level, bearing in mind its rela
%\subsection{Orbital optimized configuration interaction}
\titou{T2: Would it be a good idea to have mentioned that seniority-based schemes are not invariant with respect to orbital rotations?}
Up to this point, all results and discussions have been based on CI calculations with HF orbitals.
Now we discuss the role of further optimizing the orbitals at each given CI calculation.
Due to the significantly higher computational cost and numerical difficulties for optimizing the orbitals at higher levels of CI,
@ -374,12 +381,12 @@ due to the larger energy lowering at the Franck-Condon region than at dissociati
These results suggest that, when bond breaking involves one site, orbital optimization at the DOCI level does not have such an important role,
at least in the sense of decreasing the NPE.
Optimizing the orbitals at the CI level also tends to benefit the convergence of vibrational frequencies and equilibrium geometries (shown in Fig.Sx of the \SupInf).
Optimizing the orbitals at the CI level also tends to benefit the convergence of vibrational frequencies and equilibrium geometries (shown in Fig.~Sx of the \SupInf).
The impact is often somewhat larger for hCI than for excitation-based CI, by a small margin.
The large oscillations observed in the hCI convergence with HF orbitals (for \ce{HF} and \ce{F2}) are significantly suppressed upon orbital optimization.
We come back to the surprisingly good performance of oo-CIS, which is interesting due to its low computational cost.
The PECs are compared with those of HF and FCI in Fig.Sx of the \SupInf.
The PECs are compared with those of HF and FCI in Fig.~Sx of the \SupInf.
At this level, the orbital rotations provide an optimized reference (different from the HF solution), from which only single excitations are performed.
Since the reference is not the HF one, Brillouin's theorem no longer holds, and single excitations actually connect with the reference.
Thus, with only single excitations (and a reference that is optimized in the presence of these excitations), one obtains a minimally correlated model.
@ -405,7 +412,7 @@ Nevertheless, double (ethylene) and even triple (\ce{N2}) bond breaking still ap
In summary, here we have proposed a new scheme for truncating the Hilbert space in configuration interaction calculations, named hierarchy CI (hCI).
By merging the excitation degree and the seniority number into a single hierarchy parameter $h$,
the hCI method ensures that all classes of determinants sharing the same scaling with the number of electrons are included in each level of the hierarchy.
the hCI method ensures that all classes of determinants sharing \titou{the same scaling with the number of electrons} are included in each level of the hierarchy.
We evaluated the performance of hCI against the traditional excitation-based CI and seniority-based CI,
by comparing PECs and derived quantities (non-parallelity errors, distance errors, vibrational frequencies, and equilibrium geometries)
for six systems, ranging from single to multiple bond breaking.
@ -417,9 +424,9 @@ The comparison to seniority-based CI is less trivial.
DOCI (the first level of seniority-based CI) often provides even lower NPEs for a similar $\Ndet$, but it falls short in describing the other properties investigated here.
If higher accuracy is desired, than the convergence is faster with hCI (and also excitation-based CI) than seniority-based CI, at least for HF orbitals.
Finally, the exponential scaling of seniority-based CI in practice precludes this approach for larger systems and larger basis sets,
while the favourable polynomial scaling and encouraging performance of hCI as an alternative.
while the favorable polynomial scaling and encouraging performance of hCI as an alternative.
We found surprisingly good results for the first level of hCI (hCI1) and the orbital optimized version of CIS (oo-CIS), two methods with very favourable computational scaling.
We found surprisingly good results for the first level of hCI (hCI1) and the orbital optimized version of CIS (oo-CIS), two methods with very favorable computational scaling.
In particular, oo-CIS correctly describes single bond breaking.
We hope to report on generalizations to excited states in the future.
@ -434,8 +441,8 @@ One interesting possibility to explore is to first optimize the orbitals at a lo
The hCI pathway presented here offers several interesting possibilities to pursue.
One could generalize and adapt hCI for excited states and open-shell systems,
develop coupled cluster methods based on an analogous excitation-seniority truncation of the excitation operator,
and explore hCI wave functions for Quantum Monte Carlo simulations.
develop coupled-cluster methods based on an analogous excitation-seniority truncation of the excitation operator, \cite{Aroeira_2021,Magoulas_2021,Lee_2021}
and explore the accuracy of hCI trial wave functions for quantum Monte Carlo simulations. \cite{Dash_2019,Dash_2021,Cuzzocrea_2022}
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\begin{acknowledgements}