up to fig1

Manuscript/seniority.bib

@@ -1,3 +1,83 @@
+%% 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 
+
+
+%% Saved with string encoding Unicode (UTF-8) 
+
+
+
+@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},
+	date-modified = {2022-03-06 15:23:54 +0100},
+	doi = {10.1063/1.5044503},
+	journal = {J. Chem. Phys.},
+	pages = {064103},
+	title = {Selected Configuration Interaction Dressed by Perturbation},
+	volume = {149},
+	year = {2018},
+	bdsk-url-1 = {https://doi.org/10.1063/1.5044503}}
+
+@article{Knowles_1989,
+	author = {Knowles, Peter J. and Handy, Nicholas C.},
+	date-added = {2022-03-06 15:20:23 +0100},
+	date-modified = {2022-03-06 15:20:23 +0100},
+	file = {/Users/loos/Zotero/storage/XWJJMYAA/Knowles_1989.pdf},
+	journal = {Comput. Phys. Commun.},
+	number = {1},
+	pages = {75--83},
+	title = {A Determinant Based Full Configuration Interaction Program},
+	volume = {54},
+	year = {1989}}
+
+@article{Knowles_1984,
+	author = {P. J. Knowles and N. C. Handy},
+	date-added = {2022-03-06 15:20:23 +0100},
+	date-modified = {2022-03-06 15:20:23 +0100},
+	journal = {Chem. Phys. Lett.},
+	keywords = {correlation},
+	pages = {315--321},
+	title = {A new determinant-based full configuration interaction method},
+	volume = {111},
+	year = {1984}}
+
+@article{Booth_2009,
+	author = {Booth, George H. and Thom, Alex J. W. and Alavi, Ali},
+	date-added = {2022-03-06 15:19:42 +0100},
+	date-modified = {2022-03-06 15:19:54 +0100},
+	doi = {10.1063/1.3193710},
+	journal = {J. Chem. Phys.},
+	month = aug,
+	number = {5},
+	pages = {054106},
+	title = {Fermion {Monte} {Carlo} without fixed nodes: {A} game of life, death, and annihilation in {Slater} determinant space},
+	volume = {131},
+	year = {2009},
+	bdsk-url-1 = {http://aip.scitation.org/doi/full/10.1063/1.3193710},
+	bdsk-url-2 = {http://dx.doi.org/10.1063/1.3193710}}
+
+@book{SzaboBook,
+	address = {New York},
+	author = {A. Szabo and N. S. Ostlund},
+	date-added = {2022-03-06 15:16:04 +0100},
+	date-modified = {2022-03-06 15:16:04 +0100},
+	keywords = {qmech},
+	publisher = {McGraw-Hill},
+	title = {Modern quantum chemistry},
+	year = {1989}}
+
+@book{Helgakerbook,
+	author = {T. Helgaker and P. J{\o}rgensen and J. Olsen},
+	date-added = {2022-03-06 15:15:23 +0100},
+	date-modified = {2022-03-06 15:15:23 +0100},
+	owner = {joshua},
+	publisher = {John Wiley \& Sons, Inc.},
+	timestamp = {2014.11.24},
+	title = {Molecular Electronic-Structure Theory},
+	year = {2013}}
+
 @article{Huron_1973,
 	author = {Huron, B. and Malrieu, J. P. and Rancurel, P.},
 	date-added = {2021-01-06 09:31:37 +0100},
@@ -13,7 +93,7 @@
 	url = {http://dx.doi.org/10.1063/1.1679199},
 	volume = {58},
 	year = {1973},
-        Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.1679199}}
+	bdsk-url-1 = {http://dx.doi.org/10.1063/1.1679199}}
 
 @article{Giner_2013,
 	author = {Giner, Emmanuel and Scemama, Anthony and Caffarel, Michel},
@@ -30,7 +110,7 @@
 	url = {http://dx.doi.org/10.1139/cjc-2013-0017},
 	volume = {91},
 	year = {2013},
-        Bdsk-Url-1 = {http://dx.doi.org/10.1139/cjc-2013-0017}}
+	bdsk-url-1 = {http://dx.doi.org/10.1139/cjc-2013-0017}}
 
 @article{Giner_2015,
 	author = {Emmanuel Giner and Anthony Scemama and Michel Caffarel},
@@ -47,7 +127,7 @@
 	url = {http://dx.doi.org/10.1063/1.4905528},
 	volume = {142},
 	year = {2015},
-        Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4905528}}
+	bdsk-url-1 = {http://dx.doi.org/10.1063/1.4905528}}
 
 @article{Garniron_2019,
 	author = {Y. Garniron and K. Gasperich and T. Applencourt and A. Benali and A. Fert{\'e} and J. Paquier and B. Pradines and R. Assaraf and P. Reinhardt and J. Toulouse and P. Barbaresco and N. Renon and G. David and J. P. Malrieu and M. V{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama},
@@ -59,7 +139,7 @@
 	title = {Quantum Package 2.0: a open-source determinant-driven suite of programs},
 	volume = {15},
 	year = {2019},
-        Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00176}}
+	bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.9b00176}}
 
 @article{Bytautas_2011,
 	author = {Bytautas, Laimutis and Henderson, Thomas M. and {Jim{\'e}nez-Hoyos}, Carlos A. and Ellis, Jason K. and Scuseria, Gustavo E.},
@@ -73,103 +153,94 @@
 	title = {Seniority and Orbital Symmetry as Tools for Establishing a Full Configuration Interaction Hierarchy},
 	volume = {135},
 	year = {2011},
-        Bdsk-Url-1 = {https://doi.org/10.1063/1.3613706}}
+	bdsk-url-1 = {https://doi.org/10.1063/1.3613706}}
 
 @article{Damour_2021,
-author = {Damour, Yann and V{\'{e}}ril, Micka{\"{e}}l and Kossoski, F{\'{a}}bris and Caffarel, Michel and Jacquemin, Denis and Scemama, Anthony and Loos, Pierre-Fran{\c{c}}ois},
+	author = {Damour, Yann and V{\'{e}}ril, Micka{\"{e}}l and Kossoski, F{\'{a}}bris and Caffarel, Michel and Jacquemin, Denis and Scemama, Anthony and Loos, Pierre-Fran{\c{c}}ois},
-doi = {10.1063/5.0065314},
+	doi = {10.1063/5.0065314},
-file = {:home/fabris/Downloads/5.0065314.pdf:pdf},
+	file = {:home/fabris/Downloads/5.0065314.pdf:pdf},
-issn = {0021-9606},
+	issn = {0021-9606},
-journal = {The Journal of Chemical Physics},
+	journal = {The Journal of Chemical Physics},
-number = {13},
+	number = {13},
-pages = {134104},
+	pages = {134104},
-publisher = {AIP Publishing, LLC},
+	publisher = {AIP Publishing, LLC},
-title = {{Accurate full configuration interaction correlation energy estimates for five- and six-membered rings}},
+	title = {{Accurate full configuration interaction correlation energy estimates for five- and six-membered rings}},
-url = {https://doi.org/10.1063/5.0065314},
+	url = {https://doi.org/10.1063/5.0065314},
-volume = {155},
+	volume = {155},
-year = {2021}
+	year = {2021},
-}
+	bdsk-url-1 = {https://doi.org/10.1063/5.0065314}}
-
 
 @article{Hollett_2022,
-author = {Elayan,Ismael A  and Gupta,Rishabh  and Hollett,Joshua Wallace },
+	author = {Elayan,Ismael A and Gupta,Rishabh and Hollett,Joshua Wallace},
-title = {ΔNO and the complexities of electron correlation in simple hydrogen clusters},
+	doi = {10.1063/5.0073227},
-journal = {The Journal of Chemical Physics},
+	eprint = {https://doi.org/10.1063/5.0073227},
-volume = {0},
+	journal = {The Journal of Chemical Physics},
-number = {ja},
+	number = {ja},
-pages = {null},
+	pages = {null},
-year = {0},
+	title = {ΔNO and the complexities of electron correlation in simple hydrogen clusters},
-doi = {10.1063/5.0073227},
+	url = {https://doi.org/10.1063/5.0073227},
-URL = {
+	volume = {0},
-        https://doi.org/10.1063/5.0073227
+	year = {0},
-},
+	bdsk-url-1 = {https://doi.org/10.1063/5.0073227}}
-eprint = {
-        https://doi.org/10.1063/5.0073227
-}
-}
 
 @article{Alcoba_2014b,
-abstract = {This work deals with the configuration interaction method when an N-electron Hamiltonian is projected on Slater determinants which are classified according to their seniority number values. We study the spin features of the wave functions and the size of the matrices required to formulate states of any spin symmetry within this treatment. Correlation energies associated with the wave functions arising from the seniority-based configuration interaction procedure are determined for three types of molecular orbital basis: canonical molecular orbitals, natural orbitals, and the orbitals resulting from minimizing the expectation value of the N-electron seniority number operator. The performance of these bases is analyzed by means of numerical results obtained from selected N-electron systems of several spin symmetries. The comparison of the results highlights the efficiency of the molecular orbital basis which minimizes the mean value of the seniority number for a state, yielding energy values closer to those provided by the full configuration interaction procedure. {\textcopyright} 2014 AIP Publishing LLC.},
+	abstract = {This work deals with the configuration interaction method when an N-electron Hamiltonian is projected on Slater determinants which are classified according to their seniority number values. We study the spin features of the wave functions and the size of the matrices required to formulate states of any spin symmetry within this treatment. Correlation energies associated with the wave functions arising from the seniority-based configuration interaction procedure are determined for three types of molecular orbital basis: canonical molecular orbitals, natural orbitals, and the orbitals resulting from minimizing the expectation value of the N-electron seniority number operator. The performance of these bases is analyzed by means of numerical results obtained from selected N-electron systems of several spin symmetries. The comparison of the results highlights the efficiency of the molecular orbital basis which minimizes the mean value of the seniority number for a state, yielding energy values closer to those provided by the full configuration interaction procedure. {\textcopyright} 2014 AIP Publishing LLC.},
-author = {Alcoba, Diego R. and Torre, Alicia and Lain, Luis and Massaccesi, Gustavo E. and O{\~{n}}a, Ofelia B.},
+	author = {Alcoba, Diego R. and Torre, Alicia and Lain, Luis and Massaccesi, Gustavo E. and O{\~{n}}a, Ofelia B.},
-doi = {10.1063/1.4882881},
+	doi = {10.1063/1.4882881},
-file = {:home/fabris/Downloads/1.4882881.pdf:pdf},
+	file = {:home/fabris/Downloads/1.4882881.pdf:pdf},
-issn = {00219606},
+	issn = {00219606},
-journal = {Journal of Chemical Physics},
+	journal = {Journal of Chemical Physics},
-number = {23},
+	number = {23},
-title = {{Configuration interaction wave functions: A seniority number approach}},
+	title = {{Configuration interaction wave functions: A seniority number approach}},
-url = {http://dx.doi.org/10.1063/1.4882881},
+	url = {http://dx.doi.org/10.1063/1.4882881},
-volume = {140},
+	volume = {140},
-year = {2014}
+	year = {2014},
-}
+	bdsk-url-1 = {http://dx.doi.org/10.1063/1.4882881}}
 
 @article{Raemdonck_2015,
-author = {Van Raemdonck,Mario  and Alcoba,Diego R.  and Poelmans,Ward  and De Baerdemacker,Stijn  and Torre,Alicia  and Lain,Luis  and Massaccesi,Gustavo E.  and Van Neck,Dimitri  and Bultinck,Patrick },
+	author = {Van Raemdonck,Mario and Alcoba,Diego R. and Poelmans,Ward and De Baerdemacker,Stijn and Torre,Alicia and Lain,Luis and Massaccesi,Gustavo E. and Van Neck,Dimitri and Bultinck,Patrick},
-title = {Polynomial scaling approximations and dynamic correlation corrections to doubly occupied configuration interaction wave functions},
+	doi = {10.1063/1.4930260},
-journal = {The Journal of Chemical Physics},
+	eprint = {https://doi.org/10.1063/1.4930260},
-volume = {143},
+	journal = {The Journal of Chemical Physics},
-number = {10},
+	number = {10},
-pages = {104106},
+	pages = {104106},
-year = {2015},
+	title = {Polynomial scaling approximations and dynamic correlation corrections to doubly occupied configuration interaction wave functions},
-doi = {10.1063/1.4930260},
+	url = {https://doi.org/10.1063/1.4930260},
-URL = {
+	volume = {143},
-        https://doi.org/10.1063/1.4930260
+	year = {2015},
-},
+	bdsk-url-1 = {https://doi.org/10.1063/1.4930260}}
-eprint = {
-        https://doi.org/10.1063/1.4930260
-}
-}
 
 @book{Alcoba_2018,
-abstract = {In this work we project the Hamiltonian of an N-electron system onto a set of N-electron determinants cataloged by their seniority numbers and their excitation levels with respect to a reference determinant. We show that, in open-shell systems, the diagonalization of the N-electron Hamiltonian matrix leads to eigenstates of the operator Ŝ2 when the excitation levels are counted in terms of spatial orbitals instead of spin-orbitals. Our proposal is based on the commutation relations between the N-electron operators seniority number and spatial excitation level, as well as between these operators and the spin operators Ŝ2 and Ŝz. Energy and 〈Ŝ2〉 expectation values of molecular systems obtained from our procedure are compared with those arising from the standard hybrid configuration interaction methods based on seniority numbers and spin-orbital-excitation levels. We analyze the behavior of these methods, evaluating their computational costs and establishing their usefulness.},
+	abstract = {In this work we project the Hamiltonian of an N-electron system onto a set of N-electron determinants cataloged by their seniority numbers and their excitation levels with respect to a reference determinant. We show that, in open-shell systems, the diagonalization of the N-electron Hamiltonian matrix leads to eigenstates of the operator {\^S}2 when the excitation levels are counted in terms of spatial orbitals instead of spin-orbitals. Our proposal is based on the commutation relations between the N-electron operators seniority number and spatial excitation level, as well as between these operators and the spin operators {\^S}2 and {\^S}z. Energy and 〈{\^S}2〉 expectation values of molecular systems obtained from our procedure are compared with those arising from the standard hybrid configuration interaction methods based on seniority numbers and spin-orbital-excitation levels. We analyze the behavior of these methods, evaluating their computational costs and establishing their usefulness.},
-author = {Alcoba, Diego R. and Torre, Alicia and Lain, Luis and O{\~{n}}a, Ofelia B. and Massaccesi, Gustavo E. and Capuzzi, Pablo},
+	author = {Alcoba, Diego R. and Torre, Alicia and Lain, Luis and O{\~{n}}a, Ofelia B. and Massaccesi, Gustavo E. and Capuzzi, Pablo},
-booktitle = {Advances in Quantum Chemistry},
+	booktitle = {Advances in Quantum Chemistry},
-doi = {10.1016/bs.aiq.2017.05.003},
+	doi = {10.1016/bs.aiq.2017.05.003},
-edition = {1},
+	edition = {1},
-file = {:home/fabris/Downloads/alcoba2017.pdf:pdf},
+	file = {:home/fabris/Downloads/alcoba2017.pdf:pdf},
-issn = {00653276},
+	issn = {00653276},
-keywords = {Configuration interaction methodology,Excitation level operators,Excitation levels in N-electron determinants,Hybrid methods in CI treatments,Seniority number of N-electron determinants,Seniority number operators,Spin contamination of wave functions},
+	keywords = {Configuration interaction methodology,Excitation level operators,Excitation levels in N-electron determinants,Hybrid methods in CI treatments,Seniority number of N-electron determinants,Seniority number operators,Spin contamination of wave functions},
-pages = {315--332},
+	pages = {315--332},
-publisher = {Elsevier Inc.},
+	publisher = {Elsevier Inc.},
-title = {{Hybrid Treatments Based on Determinant Seniority Numbers and Spatial Excitation Levels in the Configuration Interaction Framework}},
+	title = {{Hybrid Treatments Based on Determinant Seniority Numbers and Spatial Excitation Levels in the Configuration Interaction Framework}},
-url = {http://dx.doi.org/10.1016/bs.aiq.2017.05.003},
+	url = {http://dx.doi.org/10.1016/bs.aiq.2017.05.003},
-volume = {76},
+	volume = {76},
-year = {2018}
+	year = {2018},
-}
+	bdsk-url-1 = {http://dx.doi.org/10.1016/bs.aiq.2017.05.003}}
 
 @article{Alcoba_2014,
-abstract = {We present a configuration interaction method in which the Hamiltonian of an N-electron system is projected on Slater determinants selected according to the seniority-number criterion along with the traditional excitation-based procedure. This proposed method is especially useful to describe systems which exhibit dynamic (weak) correlation at determined geometric arrangements (where the excitation-based procedure is more suitable) but show static (strong) correlation at other arrangements (where the seniority-number technique is preferred). The hybrid method amends the shortcomings of both individual determinant selection procedures, yielding correct shapes of potential energy curves with results closer to those provided by the full configuration interaction method.},
+	abstract = {We present a configuration interaction method in which the Hamiltonian of an N-electron system is projected on Slater determinants selected according to the seniority-number criterion along with the traditional excitation-based procedure. This proposed method is especially useful to describe systems which exhibit dynamic (weak) correlation at determined geometric arrangements (where the excitation-based procedure is more suitable) but show static (strong) correlation at other arrangements (where the seniority-number technique is preferred). The hybrid method amends the shortcomings of both individual determinant selection procedures, yielding correct shapes of potential energy curves with results closer to those provided by the full configuration interaction method.},
-author = {Alcoba, Diego R. and Torre, Alicia and Lain, Luis and O{\~{n}}a, Ofelia B. and Capuzzi, Pablo and {Van Raemdonck}, Mario and Bultinck, Patrick and {Van Neck}, Dimitri},
+	author = {Alcoba, Diego R. and Torre, Alicia and Lain, Luis and O{\~{n}}a, Ofelia B. and Capuzzi, Pablo and {Van Raemdonck}, Mario and Bultinck, Patrick and {Van Neck}, Dimitri},
-doi = {10.1063/1.4904755},
+	doi = {10.1063/1.4904755},
-file = {:home/fabris/Downloads/1.4904755.pdf:pdf},
+	file = {:home/fabris/Downloads/1.4904755.pdf:pdf},
-issn = {0021-9606},
+	issn = {0021-9606},
-journal = {The Journal of Chemical Physics},
+	journal = {The Journal of Chemical Physics},
-number = {24},
+	number = {24},
-pages = {244118},
+	pages = {244118},
-title = {{A hybrid configuration interaction treatment based on seniority number and excitation schemes}},
+	title = {{A hybrid configuration interaction treatment based on seniority number and excitation schemes}},
-url = {http://dx.doi.org/10.1063/1.4904755},
+	url = {http://dx.doi.org/10.1063/1.4904755},
-volume = {141},
+	volume = {141},
-year = {2014}
+	year = {2014},
-}
+	bdsk-url-1 = {http://dx.doi.org/10.1063/1.4904755}}
 
 @book{Ring_1980,
 	address = {{Berlin Heidelberg}},
@@ -182,69 +253,66 @@ year = {2014}
 	year = {1980}}
 
 @article{Bytautas_2015,
-abstract = {The present study further explores the concept of the seniority number ($\Omega$) by examining different configuration interaction (CI) truncation strategies in generating compact wave functions in a systematic way. While the role of $\Omega$ in addressing static (strong) correlation problem has been addressed in numerous previous studies, the usefulness of seniority number in describing weak (dynamic) correlation has not been investigated in a systematic way. Thus, the overall objective in the present work is to investigate the role of $\Omega$ in addressing also dynamic electron correlation in addition to the static correlation. Two systematic CI truncation strategies are compared beyond minimal basis sets and full valence active spaces. One approach is based on the seniority number (defined as the total number of singly occupied orbitals in a determinant) and another is based on an excitation-level limitation. In addition, molecular orbitals are energy-optimized using multiconfigurational-self-consistent-field procedure for all these wave functions. The test cases include the symmetric dissociation of water (6-31G), N2 (6-31G), C2 (6-31G), and Be2 (cc-pVTZ). We find that the potential energy profile for H2O dissociation can be reasonably well described using only the $\Omega$ = 0 sector of the CI wave function. For the Be2 case, we show that the full CI potential energy curve (cc-pVTZ) is almost exactly reproduced using either $\Omega$-based (including configurations having up to $\Omega$ = 2 in the virtual-orbital-space) or excitation-based (up to single-plus-double-substitutions) selection methods, both out of a full-valence-reference function. Finally, in dissociation cases of N2 and C2, we shall also consider novel hybrid wave functions obtained by a union of a set of CI configurations representing the full valence space and a set of CI configurations where seniority-number restriction is imposed for a complete set (full-valence-space and virtual) of correlated molecular orbitals, simultaneously. We discuss the usefulness of the seniority number concept in addressing both static and dynamic electron correlation problems along dissociation paths.},
+	abstract = {The present study further explores the concept of the seniority number ($\Omega$) by examining different configuration interaction (CI) truncation strategies in generating compact wave functions in a systematic way. While the role of $\Omega$ in addressing static (strong) correlation problem has been addressed in numerous previous studies, the usefulness of seniority number in describing weak (dynamic) correlation has not been investigated in a systematic way. Thus, the overall objective in the present work is to investigate the role of $\Omega$ in addressing also dynamic electron correlation in addition to the static correlation. Two systematic CI truncation strategies are compared beyond minimal basis sets and full valence active spaces. One approach is based on the seniority number (defined as the total number of singly occupied orbitals in a determinant) and another is based on an excitation-level limitation. In addition, molecular orbitals are energy-optimized using multiconfigurational-self-consistent-field procedure for all these wave functions. The test cases include the symmetric dissociation of water (6-31G), N2 (6-31G), C2 (6-31G), and Be2 (cc-pVTZ). We find that the potential energy profile for H2O dissociation can be reasonably well described using only the $\Omega$ = 0 sector of the CI wave function. For the Be2 case, we show that the full CI potential energy curve (cc-pVTZ) is almost exactly reproduced using either $\Omega$-based (including configurations having up to $\Omega$ = 2 in the virtual-orbital-space) or excitation-based (up to single-plus-double-substitutions) selection methods, both out of a full-valence-reference function. Finally, in dissociation cases of N2 and C2, we shall also consider novel hybrid wave functions obtained by a union of a set of CI configurations representing the full valence space and a set of CI configurations where seniority-number restriction is imposed for a complete set (full-valence-space and virtual) of correlated molecular orbitals, simultaneously. We discuss the usefulness of the seniority number concept in addressing both static and dynamic electron correlation problems along dissociation paths.},
-author = {Bytautas, Laimutis and Scuseria, Gustavo E. and Ruedenberg, Klaus},
+	author = {Bytautas, Laimutis and Scuseria, Gustavo E. and Ruedenberg, Klaus},
-doi = {10.1063/1.4929904},
+	doi = {10.1063/1.4929904},
-file = {:home/fabris/Downloads/1.4929904.pdf:pdf},
+	file = {:home/fabris/Downloads/1.4929904.pdf:pdf},
-issn = {00219606},
+	issn = {00219606},
-journal = {Journal of Chemical Physics},
+	journal = {Journal of Chemical Physics},
-number = {9},
+	number = {9},
-title = {{Seniority number description of potential energy surfaces: Symmetric dissociation of water, N2, C2, and Be2}},
+	title = {{Seniority number description of potential energy surfaces: Symmetric dissociation of water, N2, C2, and Be2}},
-url = {http://dx.doi.org/10.1063/1.4929904},
+	url = {http://dx.doi.org/10.1063/1.4929904},
-volume = {143},
+	volume = {143},
-year = {2015}
+	year = {2015},
-}
+	bdsk-url-1 = {http://dx.doi.org/10.1063/1.4929904}}
 
 @article{Chen_2015,
-author = {Chen, Zhenhua and Zhou, Chen and Wu, Wei},
+	author = {Chen, Zhenhua and Zhou, Chen and Wu, Wei},
-title = {Seniority Number in Valence Bond Theory},
+	doi = {10.1021/acs.jctc.5b00416},
-journal = {Journal of Chemical Theory and Computation},
+	eprint = {https://doi.org/10.1021/acs.jctc.5b00416},
-volume = {11},
+	journal = {Journal of Chemical Theory and Computation},
-number = {9},
+	note = {PMID: 26575906},
-pages = {4102-4108},
+	number = {9},
-year = {2015},
+	pages = {4102-4108},
-doi = {10.1021/acs.jctc.5b00416},
+	title = {Seniority Number in Valence Bond Theory},
-    note ={PMID: 26575906},
+	url = {https://doi.org/10.1021/acs.jctc.5b00416},
-URL = {
+	volume = {11},
-        https://doi.org/10.1021/acs.jctc.5b00416
+	year = {2015},
-},
+	bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.5b00416}}
-eprint = {
-        https://doi.org/10.1021/acs.jctc.5b00416
-}
-}
 
 @article{Bytautas_2018,
-title = {Seniority based energy renormalization group (Ω-ERG) approach in quantum chemistry: Initial formulation and application to potential energy surfaces},
+	abstract = {This investigation combines the concept of the seniority number Ω (defined as the total number of singly occupied orbitals in a determinant) with the energy renormalization group (ERG) approach to obtain the lowest-energy electronic states on molecular potential energy surfaces. The proposed Ω-ERG method uses Slater determinants that are ordered according to seniority number Ω in ascending order. In the Ω-ERG procedure, the active system consists of M (N-electron) states and K additional complement (N-electron) states (complement-system). Among the M states in the active system the lowest-energy m states represent target states of interest (target-states), thus m ≤ M. The environment consists of Full Configuration Interaction (FCI) determinants that represent a reservoir from which the complement-states K are being selected. The goal of the Ω-ERG procedure is to obtain lowest-energy target states m of FCI quality in an iterative way at a reduced computational cost. In general, the convergence rate of Ω-ERG energies towards FCI values depends on m and M, thus, the notation Ω-ERG(m, M) is used. It is found that the Ω-ERG(m, M) method can be very effective for calculating lowest-energy m (ground and excited) target states when a sufficiently large number of sweeps is used. We find that the fastest convergence is observed when M > m. The performance of the Ω-ERG(m, M) procedure in describing strongly correlated molecular systems has been illustrated by examining bond-breaking processes in N2, H8, H2O and C2. The present, proof-of-principle study yields encouraging results for calculating multiple electronic states on potential energy surfaces with near Full CI quality.},
-journal = {Computational and Theoretical Chemistry},
+	author = {Laimutis Bytautas and Jorge Dukelsky},
-volume = {1141},
+	doi = {https://doi.org/10.1016/j.comptc.2018.08.011},
-pages = {74-88},
+	issn = {2210-271X},
-year = {2018},
+	journal = {Computational and Theoretical Chemistry},
-issn = {2210-271X},
+	pages = {74-88},
-doi = {https://doi.org/10.1016/j.comptc.2018.08.011},
+	title = {Seniority based energy renormalization group (Ω-ERG) approach in quantum chemistry: Initial formulation and application to potential energy surfaces},
-url = {https://www.sciencedirect.com/science/article/pii/S2210271X18304651},
+	url = {https://www.sciencedirect.com/science/article/pii/S2210271X18304651},
-author = {Laimutis Bytautas and Jorge Dukelsky},
+	volume = {1141},
-abstract = {This investigation combines the concept of the seniority number Ω (defined as the total number of singly occupied orbitals in a determinant) with the energy renormalization group (ERG) approach to obtain the lowest-energy electronic states on molecular potential energy surfaces. The proposed Ω-ERG method uses Slater determinants that are ordered according to seniority number Ω in ascending order. In the Ω-ERG procedure, the active system consists of M (N-electron) states and K additional complement (N-electron) states (complement-system). Among the M states in the active system the lowest-energy m states represent target states of interest (target-states), thus m ≤ M. The environment consists of Full Configuration Interaction (FCI) determinants that represent a reservoir from which the complement-states K are being selected. The goal of the Ω-ERG procedure is to obtain lowest-energy target states m of FCI quality in an iterative way at a reduced computational cost. In general, the convergence rate of Ω-ERG energies towards FCI values depends on m and M, thus, the notation Ω-ERG(m, M) is used. It is found that the Ω-ERG(m, M) method can be very effective for calculating lowest-energy m (ground and excited) target states when a sufficiently large number of sweeps is used. We find that the fastest convergence is observed when M > m. The performance of the Ω-ERG(m, M) procedure in describing strongly correlated molecular systems has been illustrated by examining bond-breaking processes in N2, H8, H2O and C2. The present, proof-of-principle study yields encouraging results for calculating multiple electronic states on potential energy surfaces with near Full CI quality.}
+	year = {2018},
-}
+	bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S2210271X18304651},
+	bdsk-url-2 = {https://doi.org/10.1016/j.comptc.2018.08.011}}
 
 @article{Henderson_2014,
-abstract = {Doubly occupied configuration interaction (DOCI) with optimized orbitals often accurately describes strong correlations while working in a Hilbert space much smaller than that needed for full configuration interaction. However, the scaling of such calculations remains combinatorial with system size. Pair coupled cluster doubles (pCCD) is very successful in reproducing DOCI energetically, but can do so with low polynomial scaling (N3, disregarding the two-electron integral transformation from atomic to molecular orbitals). We show here several examples illustrating the success of pCCD in reproducing both the DOCI energy and wave function and show how this success frequently comes about. What DOCI and pCCD lack are an effective treatment of dynamic correlations, which we here add by including higher-seniority cluster amplitudes which are excluded from pCCD. This frozen pair coupled cluster approach is comparable in cost to traditional closed-shell coupled cluster methods with results that are competitive for weakly correlated systems and often superior for the description of strongly correlated systems.},
+	abstract = {Doubly occupied configuration interaction (DOCI) with optimized orbitals often accurately describes strong correlations while working in a Hilbert space much smaller than that needed for full configuration interaction. However, the scaling of such calculations remains combinatorial with system size. Pair coupled cluster doubles (pCCD) is very successful in reproducing DOCI energetically, but can do so with low polynomial scaling (N3, disregarding the two-electron integral transformation from atomic to molecular orbitals). We show here several examples illustrating the success of pCCD in reproducing both the DOCI energy and wave function and show how this success frequently comes about. What DOCI and pCCD lack are an effective treatment of dynamic correlations, which we here add by including higher-seniority cluster amplitudes which are excluded from pCCD. This frozen pair coupled cluster approach is comparable in cost to traditional closed-shell coupled cluster methods with results that are competitive for weakly correlated systems and often superior for the description of strongly correlated systems.},
-archivePrefix = {arXiv},
+	archiveprefix = {arXiv},
-arxivId = {1410.6529},
+	arxivid = {1410.6529},
-author = {Henderson, Thomas M. and Bulik, Ireneusz W. and Stein, Tamar and Scuseria, Gustavo E.},
+	author = {Henderson, Thomas M. and Bulik, Ireneusz W. and Stein, Tamar and Scuseria, Gustavo E.},
-doi = {10.1063/1.4904384},
+	doi = {10.1063/1.4904384},
-eprint = {1410.6529},
+	eprint = {1410.6529},
-file = {:home/fabris/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Henderson et al. - 2014 - Seniority-based coupled cluster theory.pdf:pdf},
+	file = {:home/fabris/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Henderson et al. - 2014 - Seniority-based coupled cluster theory.pdf:pdf},
-issn = {0021-9606},
+	issn = {0021-9606},
-journal = {The Journal of Chemical Physics},
+	journal = {The Journal of Chemical Physics},
-month = {dec},
+	month = {dec},
-number = {24},
+	number = {24},
-pages = {244104},
+	pages = {244104},
-title = {{Seniority-based coupled cluster theory}},
+	title = {{Seniority-based coupled cluster theory}},
-url = {http://aip.scitation.org/doi/10.1063/1.4904384},
+	url = {http://aip.scitation.org/doi/10.1063/1.4904384},
-volume = {141},
+	volume = {141},
-year = {2014}
+	year = {2014},
-}
+	bdsk-url-1 = {http://aip.scitation.org/doi/10.1063/1.4904384},
-
+	bdsk-url-2 = {https://doi.org/10.1063/1.4904384}}
 
 @article{Allen_1962,
 	author = {Allen, Thomas L. and Shull, Harrison},
@@ -255,8 +323,7 @@ year = {2014}
 	title = {Electron {{Pairs}} in the {{Beryllium Atom}}},
 	volume = {66},
 	year = {1962},
-	Bdsk-Url-1 = {https://doi.org/10.1021/j100818a001}}
+	bdsk-url-1 = {https://doi.org/10.1021/j100818a001}}
-
 
 @article{Smith_1965,
 	author = {Smith, Darwin W. and Fogel, Sidney J.},
@@ -267,8 +334,7 @@ year = {2014}
 	title = {Natural {{Orbitals}} and {{Geminals}} of the {{Beryllium Atom}}},
 	volume = {43},
 	year = {1965},
-	Bdsk-Url-1 = {https://doi.org/10.1063/1.1701519}}
+	bdsk-url-1 = {https://doi.org/10.1063/1.1701519}}
-
 
 @article{Veillard_1967,
 	author = {Veillard, A. and Clementi, E.},
@@ -279,6 +345,4 @@ year = {2014}
 	title = {Complete Multi-Configuration Self-Consistent Field Theory},
 	volume = {7},
 	year = {1967},
-	Bdsk-Url-1 = {https://doi.org/10.1007/BF01151915}}
+	bdsk-url-1 = {https://doi.org/10.1007/BF01151915}}
-
-

Manuscript/seniority.tex

@@ -38,6 +38,9 @@
 \newcommand{\EVCC}{E_\text{VCC}}
 \newcommand{\EpCCD}{E_\text{pCCD}}
 
+\newcommand{\Ndet}{N_\text{det}}
+\newcommand{\Nb}{N}
+
 \newcommand{\si}{\sigma}
 
 \usepackage[
@@ -97,29 +100,31 @@ at a very modest computational cost.
 %\label{sec:intro}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-In electronic structure theory, configuration interaction (CI) methods allow for a systematic way to obtain approximate or exact solutions of the electronic Hamiltonian,
+In electronic structure theory, configuration interaction (CI) methods allow for a systematic way to obtain approximate and exact solutions of the electronic Hamiltonian,
-by expanding the wave function as a linear combination of Slater determinants (or configuration state functions).
+by expanding the wave function as a linear combination of Slater determinants (or configuration state functions). \cite{SzaboBook,Helgakerbook}
-At the full CI (FCI) level, the complete Hilbert space is spanned in the wave function expansion, leading to the exact solution for a given one-particle basis set.
+At the full CI (FCI) level, the complete Hilbert space is spanned in the wave function expansion, leading to the exact solution for a given one-electron basis set.
-Except for very small systems, the FCI limit is unattainable, and in practice the expansion of the CI wave function must be truncated.
+Except for very small systems, \cite{Knowles_1984,Knowles_1989} the FCI limit is unattainable, and in practice the expansion of the CI wave function must be truncated.
 The question is then how to construct an effective and computationally tractable hierarchy of truncated CI methods
-that quickly recover the correlation energy, understood as the energy difference between the FCI and the mean-field restricted Hartree-Fock (HF) solutions.
+that quickly recover the correlation energy, understood as the energy difference between the FCI and the mean-field \titou{restricted?} Hartree-Fock (HF) solutions.
 
-The most well-known and popular class of CI methods is excitation-based,
+Excitation-based CI is surely the most well-known and popular class of CI methods.
-where one accounts for all determinants generated by exciting up to $e$ electrons from a given close-shell reference, which is usually the restricted HF solution, but does not have to.
+In this context, one accounts for all determinants generated by exciting up to $e$ electrons from a given \titou{closed-shell?} reference, which is usually the \titou{restricted?} HF solution, but does not have to.
-In this way, the excitation degree $e$ parameter defines the sequence
+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 at strong (static) correlation regimes.
+Excitation-based CI manages to quickly recover weak (dynamic) correlation effects, but struggles in strong (static) correlation regimes.
-Importantly, the number of determinants $N_{det}$ (which is the key parameter governing the computational cost) scales polynomially with the number of electrons $N$ as $N^{2d}$.
+Importantly, the number of determinants $\Ndet$ (which is the key parameter governing the computational cost) scales polynomially with the number of \titou{basis functions} $\Nb$ as $N^{2e}$.
 %This means that the contribution of higher excitations become progressively smaller.
  
-Alternatively, CI methods based on the seniority number \cite{Ring_1980} have been proposed \cite{Bytautas_2011}.
+Alternatively, seniority-based CI methods (sCI) have been proposed in both nuclear \cite{Ring_1980} and electronic \cite{Bytautas_2011} structure calculations.
 In short, the seniority number $s$ is the number of unpaired electrons in a given determinant.
-By truncating at the seniority zero ($s = 0$) sector, one obtains the 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 the most important for 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, $N_{det}$ scales exponentially with $N$, since excitations of all excitation degrees $e$ are included.
+However, already at the sCI0 level, $\Ndet$ scales exponentially with $\Nb$, since excitations of all excitation degrees are included.
-Therefore, despite the encouraging successes of seniority-based CI methods, their unfavourable computational scaling restricts applications to very small systems.
+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.}
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %\section{Hierarchy configuration interaction}
@@ -127,35 +132,37 @@ Besides CI, other methods that exploit the concept of seniority number have been
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 At this point, we notice the current dicothomy.
-When targeting static correlation, seniority-based CI methods tend to have a better performance than excitation-based CI, despite the higher computational cost.
+When targeting static correlation, seniority-based CI methods tend to have a better performance than excitation-based CI, despite their higher computational cost.
 The latter class of methods, in contrast, are well-suited for recovering dynamic correlation, and only at polynomial cost with system size.
 Ideally, we aim for a method that captures most of both static and dynamic correlation, with as few determinants as possible.
-With this goal in mind, we propose a new partitioning of the Hilbert space, named hierarchy configuration interaction (hCI).
+With this goal in mind, we propose a new partitioning of the Hilbert space, named \textit{hierarchy} CI (hCI).
-It combines both the excitation degree $e$ and the seniority number $s$ into one single hierarchy parameter $h$,
+It combines both the excitation degree $e$ and the seniority number $s$ into one single hierarchy parameter
 \begin{equation}
 	\label{eq:h}
 	h = \frac{e+s/2}{2},
 \end{equation}
 which assumes half-integer values.
 % open-shell
-Fig.~\ref{fig:allCI} shows how the Hilbert space is populated in excitation-based CI, seniority-based CI, and our hybrid hCI methods.
+Figure \ref{fig:allCI} shows how the Hilbert space is populated in excitation-based CI, seniority-based CI, and our hybrid hCI methods.
 
 %%% FIG 1 %%%
-\begin{figure}[h!]
+\begin{figure*}%[h!]
-\includegraphics[width=0.48\linewidth]{table_exc_full}
+\includegraphics[width=0.3\linewidth]{table_exc_full}
-\includegraphics[width=0.48\linewidth]{table_sen_full}
+\hspace{0.02\linewidth}
-\includegraphics[width=0.48\linewidth]{table_hCI}
+\includegraphics[width=0.3\linewidth]{table_hCI}
-\caption{Partitioning of the full Hilbert space into blocks of specific excitation degree $e$ (with respect to a closed-shell determinant) and seniority number $s$.
+\hspace{0.02\linewidth}
-This $e$-$s$ map is truncated differently in excitation-based CI (top left), seniority-based CI (top right), and hierarchy-CI (bottom).
+\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.}
 \label{fig:allCI}
-\end{figure}
+\end{figure*}
 %%% %%% %%%
 
 We have three key justifications for this new CI hierarchy.
 The first one is physical.
 We know that the lower degrees of excitations and lower seniority sectors, when looked at individually, often carry the most important contribution to the FCI expansion.
-By combining $e$ and $s$ as is eq.~\ref{eq:h}, we ensure that both directions in the excitation-seniority map (see Fig.~\ref{fig:allCI}) will be contemplated.
+By combining $e$ and $s$ as is Eq.~\eqref{eq:h}, we ensure that both directions in the excitation-seniority map (see Fig.~\ref{fig:allCI}) will be contemplated.
 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).
@@ -176,40 +183,40 @@ in the sense of recovering most of the correlation energy with the least computa
 Hybrid approaches based on both excitation degree and seniority number have been proposed before. \cite{Alcoba_2014,Raemdonck_2015,Alcoba_2018}
 In these works, the authors established separate maximum values for the excitation and the seniority,
 and either the union or the intersection between the two sets of determinants have been considered.
-For the union case, $N_{det}$ grows exponentially with $N$,
+For the union case, $\Ndet$ grows exponentially with $N$,
 while in the intersection approach the Hilbert space is filled rectangle-wise in our excitation-seniority map.
-In the latter case, the scaling of $N_{det}$ would be dominated by the rightmost bottom block.
+In the latter case, the scaling of $\Ndet$ would be dominated by the rightmost bottom block.
 Bytautas 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 (eq.\ref{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 keeping the method at a polynomial scaling.
 
-Each level of excitation-based CI has a hCI counterpart with the same scaling of $N_{det}$ with respect to $N$.
+Each level of excitation-based CI has a hCI counterpart with the same scaling of $\Ndet$ with respect to $N$.
-For example, $N_{det} \sim N^4$ in both hCI2 and CISD, whereas $N_{det} \sim N^6$ in hCI3 and CISDT, and so on.
+For example, $\Ndet \sim N^4$ in both hCI2 and CISD, whereas $\Ndet \sim N^6$ in hCI3 and CISDT, and so on.
 From this computational perspective, hCI can be seen as a more natural choice than the traditional excitation-based CI,
 because if one can afford for, say, a CISDT calculation, than one could probably afford a hCI3 calculation, which has the same computational scaling.
 Of course, in practice an integer-$h$ hCI method will have more determinants than its excitation-based counterpart (despite the same scaling),
 and thus one should first ensure whether including the lower-triangular blocks (going from CISDT to hCI3 in our example)
 is a better strategy than adding the next column (going from CISDT to CISDTQ).
-Therefore, here we decided to discuss the results in terms of $N_{det}$, rather than the formal scaling of $N_{det}$,
+Therefore, here we decided to discuss the results in terms of $\Ndet$, rather than the formal scaling of $\Ndet$,
 which could make the comparison somewhat biased toward hCI.
 It is interesting to compare the lowest levels of hCI (hCI1) and excitation-based CI (CIS).
 Since single excitations do not connect with the reference (at least for HF orbitals), CIS provides the same energy as HF.
 In contrast, the paired double excitations of hCI1 do connect with the reference (and the singles contribute indirectly via the doubles).
 Therefore, while CIS based on HF orbitals does not improve with respect to the mean-field HF wave function,
-the hCI1 counterpart already represents a minimally correlated model, with the same and favourable $N_{det} \sim N^2$ scaling.
+the hCI1 counterpart already represents a minimally correlated model, with the same and favourable $\Ndet \sim N^2$ scaling.
 hCI also allows for half-integer values of $h$, with no parallel in excitation-based CI.
 This gives extra flexibility in terms of choice of method.
 For a particular application with excitation-based CI, CISD might be too inaccurate, for example, while the improved accuracy of CISDT might be too expensive.
 hCI2.5 could represent an alternative, being more accurate than hCI2 and less expensive than hCI3.
 
 Our main goal here is to assess the performance of hCI against excitation-based and seniority-based CI.
-To do so, we evaluated how fast different observables converge to the FCI limit as a function of $N_{det}$.
+To do so, we evaluated how fast different observables converge to the FCI limit as a function of $\Ndet$.
 We have calculated the potential energy curves (PECs) for the dissociation of six systems, 
 \ce{HF}, \ce{F2}, \ce{N2}, ethylene, \ce{H4}, and \ce{H8},
 which display a variable number of bond breaking.
 For the latter two, we considered linearly arranged and equally spaced hydrogen atoms, and computed PECs along the symmetric dissociation coordinate.
-For ethylene, we considered the C$=$C double bond breaking, while freezing the remaining internal coordinates.
+For ethylene, we considered the \ce{C=C} double bond breaking, while freezing the remaining internal coordinates.
 Due to the (multiple) bond breaking, these are challenging systems for electronic structure methods,
 being often considered when assessing novel methodologies.
 We evaluated the convergence of four observables: the non-parallelity error (NPE), the distance error, the vibrational frequencies, and the equilibrium geometries.
@@ -266,16 +273,16 @@ 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, $N_{det}$.
+as functions of the number of determinants, $\Ndet$.
 The corresponding PECs and the energy differences with respect to the FCI results can be found in the \SI.
 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 $N_{det}$ is slower in the latter, more challenging cases, irrespective of the class of CI methods, as would be expected.
+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}).
 
 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 \SI).
-While a larger $N_{det}$ is required to achieve the same level of convergence, as expected,
+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.
 
@@ -288,17 +295,17 @@ 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 $N_{det}$ scales as $N^4$.
+For all systems (specially ethylene and \ce{N2}), hCI2 is better than CISD, two methods where $\Ndet$ scales as $N^4$.
 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 \SI) 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, 
-which are accounted for in hCI but not in excitation-based CI (for a given scaling with $N_{det}$).
+which are accounted for in hCI but not in excitation-based CI (for a given scaling with $\Ndet$).
  
 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 $N_{det}$ (hCI2.5 and CISDT).
+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 \SI, which shows that augmenting the basis set leads to a much steeper increase of $N_{det}$ for seniority-based CI.
+This can be seen in Fig.Sx of the \SI, 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}.
@@ -318,7 +325,7 @@ and the performance of seniority-based CI is much poorer (due to the slow recove
 %\subsection{Equilibrium geometries and vibrational frequencies}
 
 In Fig.~\ref{fig:xe} and \ref{fig:freq}, we present the convergence of the equilibrium geometries and vibrational frequencies, respectively,
-as functions of $N_{det}$, for the three classes of CI methods.
+as functions of $\Ndet$, for the three classes of CI methods.
 For the equilibrium geometries, hCI performs slightly better overall than excitation-based CI.
 A more significant advantage of hCI can be seen for the vibrational frequencies.
 For both observables, hCI and excitation-based CI largely outperform seniority-based CI.
@@ -353,7 +360,7 @@ Up to this point, all results and discussions have been based on CI calculations
 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,
 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 \SI.
+The PECs and analogous results to those of Figs.~\ref{fig:plot_stat}, \ref{fig:xe}, and \ref{fig:freq} are shown in the \SI.
 
 At a given CI level, orbital optimization will lead to lower energies than with HF orbitals. 
 However, even though the energy is lowered (thus improved) at each geometry, such improvement may vary largely along the PEC, which may or may not decrease the NPE.
@@ -413,10 +420,10 @@ by comparing PECs and derived quantities (non-parallelity errors, distance error
 for six systems, ranging from single to multiple bond breaking.
 
 Our key finding is that the overall performance of hCI either surpasses or equals that of excitation-based CI,
-in the sense of convergence with respect to $N_{det}$.
+in the sense of convergence with respect to $\Ndet$.
 The superiority of hCI methods is more noticeable for the non-parallelity and distance errors, but also observed to a lesser extent for the vibrational frequencies and equilibrium geometries.
 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 $N_{det}$, but it falls short in describing the other properties investigated here.
+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.
@@ -450,8 +457,8 @@ This project has received funding from the European Research Council (ERC) under
 \section*{Supporting information available}
 \label{sec:SI}
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-PECs, energy differences with respect to FCI results, NPE, distance errors, vibrational frequencies, and equilibrium geometries,
+Potential energy curves, energy differences with respect to FCI results, non-parallelity errors, distance errors, vibrational frequencies, and equilibrium geometries,
-according to the three classes of CI methods (excitation-based CI, seniority-based CI, and hierarchy-CI),
+according to the three classes of CI methods (excitation-based CI, seniority-based CI, and hierarchy-based CI),
 computed for \ce{HF}, \ce{F2}, ethylene, \ce{N2}, \ce{H4}, and \ce{H8}.
 
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