From e4878ea9c1ff683d9f6fc1e627b4bf5f5d9da831 Mon Sep 17 00:00:00 2001 From: kossoski Date: Wed, 23 Feb 2022 20:58:37 +0100 Subject: [PATCH] saving work --- Manuscript/comp.sh | 7 ++++ Manuscript/seniority.bib | 81 ++++++++++++++++++++++++++++++++++++++++ Manuscript/seniority.tex | 48 ++++++++++++------------ 3 files changed, 111 insertions(+), 25 deletions(-) create mode 100755 Manuscript/comp.sh diff --git a/Manuscript/comp.sh b/Manuscript/comp.sh new file mode 100755 index 0000000..7fd9ced --- /dev/null +++ b/Manuscript/comp.sh @@ -0,0 +1,7 @@ +#!/bin/bash + +pdflatex seniority +bibtex seniority +pdflatex seniority +pdflatex seniority +okular seniority.pdf diff --git a/Manuscript/seniority.bib b/Manuscript/seniority.bib index fe0b210..7fcf774 100644 --- a/Manuscript/seniority.bib +++ b/Manuscript/seniority.bib @@ -108,6 +108,37 @@ eprint = { } } +@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.}, +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}, +file = {:home/fabris/Downloads/1.4882881.pdf:pdf}, +issn = {00219606}, +journal = {Journal of Chemical Physics}, +number = {23}, +title = {{Configuration interaction wave functions: A seniority number approach}}, +url = {http://dx.doi.org/10.1063/1.4882881}, +volume = {140}, +year = {2014} +} + +@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 }, +title = {Polynomial scaling approximations and dynamic correlation corrections to doubly occupied configuration interaction wave functions}, +journal = {The Journal of Chemical Physics}, +volume = {143}, +number = {10}, +pages = {104106}, +year = {2015}, +doi = {10.1063/1.4930260}, +URL = { + 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.}, author = {Alcoba, Diego R. and Torre, Alicia and Lain, Luis and O{\~{n}}a, Ofelia B. and Massaccesi, Gustavo E. and Capuzzi, Pablo}, @@ -164,3 +195,53 @@ volume = {143}, year = {2015} } +@article{Chen_2015, +author = {Chen, Zhenhua and Zhou, Chen and Wu, Wei}, +title = {Seniority Number in Valence Bond Theory}, +journal = {Journal of Chemical Theory and Computation}, +volume = {11}, +number = {9}, +pages = {4102-4108}, +year = {2015}, +doi = {10.1021/acs.jctc.5b00416}, + note ={PMID: 26575906}, +URL = { + 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}, +journal = {Computational and Theoretical Chemistry}, +volume = {1141}, +pages = {74-88}, +year = {2018}, +issn = {2210-271X}, +doi = {https://doi.org/10.1016/j.comptc.2018.08.011}, +url = {https://www.sciencedirect.com/science/article/pii/S2210271X18304651}, +author = {Laimutis Bytautas and Jorge Dukelsky}, +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.} +} + +@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.}, +archivePrefix = {arXiv}, +arxivId = {1410.6529}, +author = {Henderson, Thomas M. and Bulik, Ireneusz W. and Stein, Tamar and Scuseria, Gustavo E.}, +doi = {10.1063/1.4904384}, +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}, +issn = {0021-9606}, +journal = {The Journal of Chemical Physics}, +month = {dec}, +number = {24}, +pages = {244104}, +title = {{Seniority-based coupled cluster theory}}, +url = {http://aip.scitation.org/doi/10.1063/1.4904384}, +volume = {141}, +year = {2014} +} + diff --git a/Manuscript/seniority.tex b/Manuscript/seniority.tex index 424ba1d..6af9aa3 100644 --- a/Manuscript/seniority.tex +++ b/Manuscript/seniority.tex @@ -103,22 +103,20 @@ Importantly, the number of determinants $N_{det}$ (which control the computation %This means that the contribution of higher excitations become progressively smaller. %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{Ring_1980} have been proposed \cite{Bytautas_2011,Bytautas_2015}. +%\fk{Still have to work in this paragraph.} +Alternatively, CI methods based on the seniority number \cite{Ring_1980} have been proposed \cite{Bytautas_2011}. 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{}. +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{Bytautas_2011,Bytautas_2015,Alcoba_2014b,Alcoba_2014}. % 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. Therefore, despite the encouraging successes of seniority-based CI methods, their unfourable computational scaling restricts applications to very small systems. - -%as the associated %doubly-occupied CI (DOCI) - +Besides CI, other methods that exploit the concpet of seniority number have been pursued. \cite{Henderson_2014,Chen_2015,Bytautas_2018} % Seniority Number in Valence Bond Theory %https://doi.org/10.1021/acs.jctc.5b00416 - % Seniority based energy renormalization group (Ω-ERG) approach in quantum chemistry: Initial formulation and application to potential energy surfaces %https://doi.org/10.1016/j.comptc.2018.08.011 + 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. The latter class of methods, in contrast, are well-suited for recovering dynamic correlation, and only at polynomial cost with system size. @@ -154,13 +152,13 @@ Fig.~\ref{fig:allCI} shows how the Hilbert space is populated in excitation-base \caption{Hybrid excitation-seniority CI.} \label{fig:CIo} \end{subfigure} - \caption{Partionining of the full Hilbert space into blocks of specific excitation degree $d$ and seniority number $\Omega$ (with respect to a closed-shell determinant). - Each of three CI methods truncate this $d$-$\Omega$ map in a different way.} + \caption{Partionining of the full Hilbert space into blocks of specific excitation degree $d$ (with respect to a closed-shell determinant) and seniority number $\Omega$. + Each of three classes of CI methods truncate this $d$-$\Omega$ map differently, and each color tone represents the added determinants at a given CI level.} \label{fig:allCI} \end{figure} %%% %%% %%% -There are three justifications for this new CI hierarchy. +We have three key justifications for this new CI hierarchy. The first one is physical. We know that low degree excitations and low seniority sectors, when looked at individually, often have the most important contribution to the FCI expansion. %carry the most important weights. @@ -177,22 +175,23 @@ The key realization of the CIo hierarchy is that the number of additional determ %to $O$ and $V$, for all excitation-seniority sectors of a given order $o$. %This computational realization represents the second justification for the introduction of the CIo method. This further justifies the parameter $o$ as being the simple average between $d$ and $\Omega/2$. + Each level of excitation-based CI has a CIo counterpart with the same scaling of $N_{det}$ with respect to $N$. -However, CIo counts with additional half-integer levels of theory, with no parallel in excitation-based CI. -For example, in both CIo2 and CISD we have $N_{det} \sim O^2V^2 \sim N^4$, whereas in CIo3 and CISDT scale, $N_{det} \sim N^6$, and so on. +%However, CIo counts with additional half-integer levels of theory, with no parallel in excitation-based CI. +For example, in both CIo2 and CISD we have $N_{det} \sim N^4$, whereas in CIo3 and CISDT, $N_{det} \sim N^6$, and so on. %the number of determinants of CIo2 and CISD scale as $O^2V^2$, those of CIo3 and CISDT scale as $O^3V^3$, and so on. From this computational perspective, the CIo hierarchy can be seen as a more natural choice than the traditional excitation-based CI, because if one can afford for, say, the $N_{det} \sim N^6$ cost of a CISDT calculation, than one can probably afford a CIo3 calculation, which has the same computational scaling. -Of course, in practice an integer-CIo method will have more determinants than its excitation-based counterpart (though the same scaling), -and thus one has to first ensure whether including the lower-triangular blocks (going from CISDT to CIo3 in our example) +Of course, in practice an integer-$o$ CIo method will have more determinants than its excitation-based counterpart (though the same scaling), +and thus one should first ensure whether including the lower-triangular blocks (going from CISDT to CIo3 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 the number of determinants, rather than the computational scaling, which could make the comparison somewhat biased toward the CIo hierarchy. - +% The lowest level in the CIo hierarchy (CIo1) parallels CIS of excitation-based CI. However, the single excitations do not connect with the reference, at least for HF orbitals, and therefore CIS provides the same energy as HF. In contrast, the paired doubles excitations of CIo1 do connect with the reference (as well as the singles, indireclty via the doubles). -Therefore, while the lowest level of excitation-based CI (CIS) does not improve with respect to the mean-field HF wave function, +Therefore, while the HF-based lowest level of excitation-based CI (CIS) does not improve with respect to the mean-field HF wave function, the CIo1 counterpart already represents a minimally correlated model, with the very favourable $N_{det} \sim N^2$ scaling. %number of determinants scaling only as $OV$. % @@ -209,15 +208,14 @@ 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? -\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. -%Different in spirit, but with the same exponential scaling, +A hybrid approach based on both excitation degree and seniority number has been proposed. \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, the number of determinants 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. 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. +In comparison 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. @@ -321,7 +319,7 @@ Inspection of the PECs (see SI) reveal that the lower NPE in the CIo results ste This result demonstrates the importance of higher-order excitations with low seniority number in this strong correlation regime, which are accounted for in CIo but not in excitation-based CI (for a given scaling with the number of determinants). %The situation at the Franck-Condon region will be discussed later. % -Meanwhile, the first level of seniority-based CI (CI$\Omega$0) tends to offer a rather low NPE when compare to the other CI methods with a similar number of determinants (CIo2.5 and CISDT). +Meanwhile, the first level of seniority-based CI (CI$\Omega$0, which is the same as doubly-occupied CI) tends to offer a rather low NPE when compare to the other CI methods with a similar number of determinants (CIo2.5 and CISDT). However, convergence is clearly slower for the next levels in this hierarchy (CI$\Omega$2 and CI$\Omega$4), while excitation-based CI and specially CIo methods converge faster. % For the symmetric dissociation of linear \ce{H4} and \ce{H8} the performance of CIo and excitation-based CI are similar, both being superior to seniority-based CI.