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Manuscript/Ec.bib
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Manuscript/Ec.tex
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\QP}{\textsc{quantum package}}
\newcommand{\Ndet}{N_\text{det}}
\newcommand{\Norb}{N_\text{orb}}
\newcommand{\br}{\boldsymbol{r}}
\newcommand{\bc}{\boldsymbol{c}}
\newcommand{\bC}{\boldsymbol{C}}
\usepackage[
colorlinks=true,
citecolor=blue,
@ -106,6 +114,7 @@ The CCSD(T) method, \cite{Raghavachari_1989} known as the gold-standard of quant
Motivated by the recent blind test of Eriksen \textit{et al.}\cite{Eriksen_2020}~reporting the performance of a large panel of emerging electronic structure methods [the many-body expansion FCI (MBE-FCI), \cite{Eriksen_2017,Eriksen_2018,Eriksen_2019a,Eriksen_2019b} adaptive sampling CI (ASCI), \cite{Tubman_2016,Tubman_2018,Tubman_2020} iterative CI (iCI), \cite{Liu_2014,Liu_2016,Lei_2017,Zhang_2020} semistochastic heat-bath CI (SHCI), \cite{Holmes_2016,Holmes_2017,Sharma_2017} the full coupled-cluster reduction (FCCR), \cite{Xu_2018,Xu_2020} density-matrix renormalization group (DMRG), \cite{White_1992,White_1993,Chan_2011} adaptive-shift FCI quantum Monte Carlo (AS-FCIQMC), \cite{Booth_2009,Cleland_2010,Ghanem_2019} and cluster-analysis-driven FCIQMC (CAD-FCIQMC) \cite{Deustua_2017,Deustua_2018}] on the non-relativistic frozen-core correlation energy of the benzene molecule in the standard correlation-consistent double-$\zeta$ Dunning basis set (cc-pVDZ), some of us have recently investigated the performance of the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) method \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2018,Garniron_2019} on the very same system \cite{Loos_2020e} [see also Ref.~\onlinecite{Lee_2020} for a study of the performance of phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) \cite{Motta_2018}].
In the continuity of this recent work, we report here a significant extension by estimating the (frozen-core) FCI/cc-pVDZ correlation energy of twelve cyclic molecules (cyclopentadiene, furan, imidazole, pyrrole, thiophene, benzene, pyrazine, pyridazine, pyridine, pyrimidine, tetrazine, and triazine) with the help of CIPSI employing energetically-optimized orbitals at the same level of theory. \cite{Yao_2020,Yao_2021}
These systems are depicted in Fig.~\ref{fig:mol}.
This set of molecular systems corresponds to Hilbert spaces with sizes ranging from $10^{28}$ (for thiophene) to $10^{36}$ (for benzene).
In addition to CIPSI, the performance and convergence properties of several series of methods are investigated.
In particular, we study i) the MP perturbation series up to fifth-order (MP2, MP3, MP4, and MP5), ii) the CC2, CC3, and CC4 approximate series, and ii) the ``complete'' CC series up to quadruples (\ie, CCSD, CCSDT, and CCSDTQ).
@ -120,16 +129,17 @@ The performance of the ground-state gold standard CCSD(T) is also investigated.
\end{figure*}
%%% FIG 1 %%%
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\section{Computational details}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The geometries of the twelve systems considered in the present study have been all obtained at the CC3/aug-cc-pVTZ level of theory and have been extracted from a previous study. \cite{Loos_2020a}
Note that, for the sake of consistency, the geometry of benzene considered here is different from one of Ref.~\onlinecite{Loos_2020e} which has been computed at a lower level of theory [MP2/6-31G(d)]. \cite{Schreiber_2008}
The MP2, MP3, MP4, CC2, CC3, CC4, CCSD, CCSDT, and CCSDTQ calculations have been performed with CFOUR, \cite{Matthews_2020} while the CCSD(T) and MP5 calculations have been computed with Gaussian 09. \cite{g09}
The CIPSI calculations have been performed with {\QP}. \cite{Garniron_2019}
The particularity of the current implementation is that the selection step and the PT2 correction are computed \textit{simultaneously} via a hybrid semistochastic algorithm \cite{Garniron_2017,Garniron_2019} (which explains the statistical error associated with the PT2 correction in the following).
\titou{The particularity of the current implementation is that the selection step and the PT2 correction are computed \textit{simultaneously} via a hybrid semistochastic algorithm \cite{Garniron_2017,Garniron_2019} (which explains the statistical error associated with the PT2 correction in the following).
Moreover, a renormalized version of the PT2 correction (dubbed rPT2 below) has been recently implemented and tested for a more efficient extrapolation to the FCI limit thanks to a partial resummation of the higher-order of perturbation. \cite{Garniron_2019}
We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the rPT2 correction and the CIPSI algorithm.
We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the rPT2 correction and the CIPSI algorithm.}
For all these calculations, we consider Dunning's correlation-consistent double-$\zeta$ basis (cc-pVDZ) which consists of Hilbert space sizes ranging from $10^{20}$ (for thiophene) to $10^{36}$ (for benzene).
Although the FCI energy has the enjoyable property of being independent of the set of one-electron orbitals used to construct the many-electron Slater determinants, as a truncated CI method, the convergence properties of CIPSI strongly dependent on this orbital choice.
@ -143,14 +153,51 @@ Using these localized orbitals as starting point, we also perform successive orb
When convergence is achieved in terms of orbital optimization, as our ``production'' run, we perform a new CIPSI calculation from scratch using this set of optimized orbitals.
As we shall see below, employing optimized orbitals has the advantage to produce a smoother and faster convergence of the SCI energy toward the FCI limit.
The total SCI energy is defined as the sum of the variational energy $E_\text{var.}$ (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction $E_\text{(r)PT2}$ which takes into account the external determinants, \ie, the determinants which do not belong to the variational space but are linked to the reference space via a nonzero matrix element.
The magnitude of $E_\text{(r)PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit.
We then linearly extrapolate the total SCI energy to $E_\text{(r)PT2} = 0$ (which effectively corresponds to the FCI limit).
%%%%%%%%%%%%%%%%%%%%%%%%%
\section{CIPSI with optimized orbitals}
%%%%%%%%%%%%%%%%%%%%%%%%%
Here, we provide key details about the CIPSI method.
Note that we focus on the ground state but the present discussion can be easily extended to excited states. \cite{Scemama_2019,Veril_2021}
At each iteration $k$, the total CIPSI energy $E_\text{CIPSI}^{(k)}$ is defined as the sum of the variational energy
\begin{equation}
E_\text{var}^{(k)} = \frac{\mel*{\Psi_\text{var}^{(k)}}{\Hat{H}}{\Psi_\text{var}^{(k)}}}{\braket*{\Psi_\text{var}^{(k)}}{\Psi_\text{var}^{(k)}}}
\end{equation}
and a second-order perturbative correction
\begin{equation}
E_\text{PT2}^{(k)}
= \sum_{\alpha \in \mathcal{A}_k} e_{\alpha}
= \sum_{\alpha \in \mathcal{A}_k} \frac{\mel*{\Psi_\text{var}^{(k)}}{\Hat{H}}{\alpha}}{E_\text{var}^{(k)} - \mel*{\alpha}{\Hat{H}}{\alpha}}
\end{equation}
where $\Psi_\text{var}^{(k)} = \sum_{I \in \mathcal{I}_k} c_I^{(k)} \ket*{I}$ is the variational wave function, $\mathcal{I}_k$ is the set of internal determinants $\ket*{I}$ and $\mathcal{A}_k$ is the set of external determinants $\ket*{\alpha}$ which do not belong to the variational space but are linked to it via a nonzero matrix element, \ie, $\mel*{\Psi_\text{var}^{(k)}}{\Hat{H}}{\alpha} \neq 0$.
The sets $\mathcal{I}_k$ and $\mathcal{A}_k$ define, at the $k$th iteration, the internal and external spaces, respectively.
In practice, $E_\text{var}^{(k)}$ is computed by diagonalizing the CI matrix in the reference space and the magnitude of $E_\text{PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit.
We then linearly extrapolate, using large variational space, the CIPSI energy to $E_\text{PT2} = 0$ (which effectively corresponds to the FCI limit).
Note that, unlike excited-state calculations where it is important to enforce that the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, \cite{Chilkuri_2021} the present wave functions do not fulfil this property as we aim for the lowest possible energy of a singlet state.
We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero for each system.
From a more general point of view, the variational energy $E_\text{var}^{(k)}$ depends on both the coefficient $\{ c_I \}_{1 \le I \le \Ndet^{(k)}}$ but also on the orbital coefficients $\{C_{\mu p}\}_{1 \le \mu,p \le \Norb,1 \le \mu,p \le \Norb}$ such that the $p$th orbital is
\begin{equation}
\phi_p(\br) = \sum_{\mu} C_{\mu p} \chi_{\mu}(\br)
\end{equation}
where $\chi_{\mu}(\br)$ is a basis function.
The diagonalization of the CI matrix ensures that
\begin{equation}
\pdv{E_\text{var}(\bc,\bC)}{c_I} = 0,
\end{equation}
but, a priori, we have
\begin{equation}
\pdv{E_\text{var}(\bc,\bC)}{C_{\mu p}} \neq 0,
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and discussion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table*}
\caption{Total energy $E$ (in \Eh) and correlation energy $\Ec$ (in \mEh) for the frozen-core ground state of five-membered rings in the cc-pVDZ basis set.
@ -219,16 +266,24 @@ We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero for
\end{squeezetable}
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\section{Conclusion}
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\begin{acknowledgements}
This work was performed using HPC resources from GENCI-TGCC (2020-gen1738) and from CALMIP (Toulouse) under allocation 2021-18005.
PFL, AS, and MC have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).
\end{acknowledgements}
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\section*{Data availability statement}
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The data that support the findings of this study are openly available in Zenodo at \href{http://doi.org/XX.XXXX/zenodo.XXXXXXX}{http://doi.org/XX.XXXX/zenodo.XXXXXXX}.
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\bibliography{Ec}
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\end{document}