saving work

Manuscript/seniority.bib

@@ -140,3 +140,27 @@ volume = {141},
 year = {2014}
 }
 
+@book{Ring_1980,
+	address = {{Berlin Heidelberg}},
+	author = {Ring, Peter and Schuck, Peter},
+	file = {/home/antoinem/Zotero/storage/R89CB5M7/9783540212065.html},
+	isbn = {978-3-540-21206-5},
+	publisher = {{Springer-Verlag}},
+	series = {Theoretical and {{Mathematical Physics}}, {{The Nuclear Many}}-{{Body Problem}}},
+	title = {The {{Nuclear Many}}-{{Body Problem}}},
+	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.},
+author = {Bytautas, Laimutis and Scuseria, Gustavo E. and Ruedenberg, Klaus},
+doi = {10.1063/1.4929904},
+file = {:home/fabris/Downloads/1.4929904.pdf:pdf},
+issn = {00219606},
+journal = {Journal of Chemical Physics},
+number = {9},
+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},
+volume = {143},
+year = {2015}
+}
+

Manuscript/seniority.tex

@@ -104,8 +104,8 @@ Importantly, the number of determinants $N_{det}$ (which control the computation
 
 %In turn, seniority-based CI is specially targeted to describe static correlation.
 \fk{Still have to work in this paragraph.}
-Alternatively, CI methods based on the seniority number \cite{} have been proposed \cite{Bytautas_2011}.
-In short, the seniority number $\Omega$ is the number of unpaired electrons in a given determinant \cite{}.
+Alternatively, CI methods based on the seniority number \cite{Ring_1980} have been proposed \cite{Bytautas_2011,Bytautas_2015}.
+In short, the seniority number $\Omega$ is the number of unpaired electrons in a given determinant.
 The seniority zero ($\Omega = 0$) sector has been shown to be the most important for static correlation, while higher sectors tend to contribute progressively less ~\cite{}.
 % scaling
 However, already at the CI$\Omega$0 level the number of determinants scale exponentially with $N$, since excitations of all excitation degrees $d$ are included.
@@ -209,11 +209,17 @@ Besides a physical or computational perspective, the question of what makes for
 Does our CIo class of methods perform better than excitation-based or seniority-based CI,
 in the sense of recovering most of the correlation energy with the least computational effort?
 
-A hybrid approach based on both excitation degree and seniority number has been proposed before \cite{Alcoba_2014,Alcoba_2018}.
-In their approach, the authors established separate thresholds for each of the two variables,
+\fk{Still have to work in this paragraph.}
+A hybrid approach based on both excitation degree and seniority number has been proposed by Alcoba et al.\cite{Alcoba_2014,Alcoba_2018}.
+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,
 meaning that the Hilbert space would be filled rectangle-wise in our excitation-seniority map.
 Therefore, the scaling with the number of determinants would be dominated by the rightmost bottom block.
-In contrast, a single combined parameter (eq.\ref{eq:o}) defines our CIo method, which allows for all classes of determinants sharing a common scaling with system size, as discussed before.
+%Different in spirit, but with the same exponential scaling,
+Bytautas et al.\cite{Bytautas_2015} explored a different hybrid scheme combining determinants from a complete active space and with a maximum seniority number.
+In contrast to previous approaches, our hybrid CIo scheme has two key advantages.
+First, it is defined by a single parameter that unifies excitation degree and seniority number (eq.\ref{eq:o}).
+And second, 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.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Computational details}