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benzene.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-08-19 17:16:37 +0200
%% Created for Pierre-Francois Loos at 2020-08-20 10:17:19 +0200
%% Saved with string encoding Unicode (UTF-8)
@ -83,7 +83,8 @@
Pages = {4282},
Title = {The Coupled-Cluster Single, Double, Triple, and Quadruple Excitation Method},
Volume = {97},
Year = {1992}}
Year = {1992},
Bdsk-Url-1 = {https://doi.org/10.1063/1.463930}}
@article{Oliphant_1991,
Author = {Oliphant, N. and Adamowicz, L.},
@ -94,7 +95,8 @@
Pages = {6645},
Title = {Coupled-Cluster Method Truncated at Quadruples},
Volume = {95},
Year = {1991}}
Year = {1991},
Bdsk-Url-1 = {https://doi.org/10.1063/1.461534}}
@article{Motta_2018,
Author = {M. Motta and S. Zhang},
@ -117,7 +119,8 @@
Pages = {2863},
Title = {Density Matrix Formulation for Quantum Renormalization Groups},
Volume = {69},
Year = {1992}}
Year = {1992},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.69.2863}}
@article{Ghanem_2019,
Author = {Ghanem, K. and Lozovoi, A. Y. and Alavi, A.},
@ -128,7 +131,8 @@
Pages = {224108},
Title = {Unbiasing the Initiator Approximation in Full Configuration Interaction Quantum Monte Carlo},
Volume = {151},
Year = {2019}}
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5134006}}
@article{Deustua_2018,
Author = {Deustua, J. E. and Magoulas, I. and Shen, J. and Piecuch, P.},
@ -139,16 +143,16 @@
Pages = {151101},
Title = {Communication: Approaching Ex- act Quantum Chemistry by Cluster Analysis of Full Configuration Interaction Quan- tum Monte Carlo Wave Functions},
Volume = {149},
Year = {2018}}
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5055769}}
@misc{Tubman_2018,
title={An efficient deterministic perturbation theory for selected configuration interaction methods},
author={Norm M. Tubman and Daniel S. Levine and Diptarka Hait and Martin Head-Gordon and K. Birgitta Whaley},
year={2018},
eprint={1808.02049},
archivePrefix={arXiv},
primaryClass={cond-mat.str-el}
}
Archiveprefix = {arXiv},
Author = {Norm M. Tubman and Daniel S. Levine and Diptarka Hait and Martin Head-Gordon and K. Birgitta Whaley},
Eprint = {1808.02049},
Primaryclass = {cond-mat.str-el},
Title = {An efficient deterministic perturbation theory for selected configuration interaction methods},
Year = {2018}}
@article{Tubman_2020,
Author = {Tubman, N. M. and Freeman, C. D. and Levine, D. S. and Hait, D. and Head-Gordon, M. and Whaley, K. B.},
@ -159,16 +163,17 @@
Pages = {2139},
Title = {Modern Approaches to Exact Diagonalization and Selected Configuration Interaction with the Adaptive Sampling CI Method},
Volume = {16},
Year = {2020}}
Year = {2020},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b00536}}
@article{Tubman_2016,
Author = {Tubman, Norm M. and Lee, Joonho and Takeshita, Tyler Y. and {Head-Gordon}, Martin and Whaley, K. Birgitta},
Date-Added = {2020-08-18 21:12:46 +0200},
Date-Modified = {2020-08-18 21:12:46 +0200},
Date-Modified = {2020-08-20 10:17:19 +0200},
Doi = {10.1063/1.4955109},
File = {/Users/loos/Zotero/storage/VDKR3CTF/Tubman16.pdf},
Issn = {0021-9606, 1089-7690},
Journal = {The Journal of Chemical Physics},
Journal = {J. Chem. Phys.},
Language = {en},
Month = jul,
Number = {4},
@ -196,7 +201,8 @@
Pages = {011041},
Title = {Direct comparison of many-body methods for realistic electronic Hamiltonians},
Volume = {10},
Year = {2020}}
Year = {2020},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevX.10.011041}}
@article{Qin_2020,
Author = {Qin, Mingpu and Chung, Chia-Min and Shi, Hao and Vitali, Ettore and Hubig, Claudius and Schollw{\"o}ck, Ulrich and White, Steven R and Zhang, Shiwei and others},
@ -240,7 +246,8 @@
Pages = {031059},
Title = {Towards the solution of the many-electron problem in real materials: Equation of state of the hydrogen chain with state-of-the-art many-body methods},
Volume = {7},
Year = {2017}}
Year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevX.7.031059}}
@article{Leblanc_2015,
Author = {LeBlanc, J. P. F. and Antipov, Andrey E and Becca, Federico and Bulik, Ireneusz W and Chan, Garnet Kin-Lic and Chung, Chia-Min and Deng, Youjin and Ferrero, Michel and Henderson, Thomas M and Jim{\'e}nez-Hoyos, Carlos A and others},
@ -252,7 +259,8 @@
Pages = {041041},
Title = {Solutions of the two-dimensional hubbard model: benchmarks and results from a wide range of numerical algorithms},
Volume = {5},
Year = {2015}}
Year = {2015},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevX.5.041041}}
@article{Scemama_2016,
Author = {Scemama, Anthony and Applencourt, Thomas and Giner, Emmanuel and Caffarel, Michel},
@ -309,7 +317,8 @@
Pages = {879},
Title = {Using perturbatively selected configuration interaction in quantum Monte Carlo calculations},
Volume = {91},
Year = {2013}}
Year = {2013},
Bdsk-Url-1 = {https://doi.org/10.1139/cjc-2013-0017}}
@article{Ceperley_1991,
Author = {D. M. Ceperley},
@ -529,7 +538,8 @@
Pages = {2374--2383},
Title = {The Quest for Highly-Accurate Excitation Energies: a Computational Perspective},
Volume = {11},
Year = {2020}}
Year = {2020},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.0c00014}}
@article{Loos_2020b,
Author = {P. F. Loos and F. Lipparini and M. Boggio-Pasqua and A. Scemama and D. Jacquemin},
@ -804,12 +814,12 @@
Year = {2015},
Bdsk-Url-1 = {https://doi.org/10.13140/RG.2.1.3187.9766}}
@article{Applencourt_2018,
Author = {Applencourt, Thomas and Gasperich, Kevin and Scemama, Anthony},
Eprint = {1812.06902},
Journal = {arXiv},
Month = {Dec},
Title = {{Spin adaptation with determinant-based selected configuration interaction}},
Url = {https://arxiv.org/abs/1812.06902v1},
Year = {2018},
Bdsk-Url-1 = {https://arxiv.org/abs/1812.06902v1}}
@misc{Applencourt_2018,
title={Spin adaptation with determinant-based selected configuration interaction},
author={Thomas Applencourt and Kevin Gasperich and Anthony Scemama},
year={2018},
eprint={1812.06902},
archivePrefix={arXiv},
primaryClass={physics.chem-ph}
}

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benzene.nb
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benzene.tex
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@ -67,7 +67,7 @@ The target application is the non-relativistic frozen-core correlation energy of
The geometry of benzene has been computed at the MP2/6-31G* level and it can be found in the supporting information of Ref.~\onlinecite{Eriksen_2020}.
This corresponds to an active space of 30 electrons and 108 orbitals, \ie, the Hilbert space of benzene is of the order of $10^{35}$ Slater determinants.
Needless to say that this size of Hilbert space cannot be tackled by exact diagonalization with current architectures.
The correlation energies reported in Ref.~\onlinecite{Eriksen_2020} are gathered in Table \ref{tab:energy} alongside the best ph-AFQMC estimate from Ref.~\onlinecite{Lee_2020}.
The correlation energies reported in Ref.~\onlinecite{Eriksen_2020} are gathered in Table \ref{tab:energy} alongside the best ph-AFQMC estimate from Ref.~\onlinecite{Lee_2020} based on a CAS(6,6) trial wave function.
The outcome of this work is nicely summarized in the abstract of Ref.~\onlinecite{Eriksen_2020}:
\textit{``In our assessment, the evaluated high-level methods are all found to qualitatively agree on a final correlation energy, with most methods yielding an estimate of the FCI value around $-863$ m$E_h$. However, we find the root-mean-square deviation of the energies from the studied methods to be considerable ($1.3$ m$E_h$), which in light of the acclaimed performance of each of the methods for smaller molecular systems clearly displays the challenges faced in extending reliable, near-exact correlation methods to larger systems.''}
@ -98,7 +98,7 @@ The outcome of this work is nicely summarized in the abstract of Ref.~\onlinecit
\end{table}
% CIPSI
In this Note, we report the frozen-core correlation energy obtained with a fourth flavor of SCI known as \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI), \cite{Huron_1973} which also includes a second-order perturbative (PT2) correction.
For the sake of completeness and our very own curiosity, we report in this Note the frozen-core correlation energy obtained with a fourth flavor of SCI known as \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI), \cite{Huron_1973} which also includes a second-order perturbative (PT2) correction.
In short, the CIPSI algorithm belongs to the family of SCI+PT2 methods.
The idea behind such methods is to avoid the exponential increase of the size of the CI expansion by retaining the most energetically relevant determinants only, thanks to the use of a second-order energetic criterion to select perturbatively determinants in the FCI space.
However, performing SCI calculations rapidly becomes extremely tedious when one increases the system size as one hits the exponential wall inherently linked to these methods.
@ -106,31 +106,32 @@ However, performing SCI calculations rapidly becomes extremely tedious when one
From an historical point of view, CIPSI is probably one of the oldest SCI algorithm.
It was developed in 1973 by Huron, Rancurel, and Malrieu \cite{Huron_1973} (see also Ref.~\onlinecite{Evangelisti_1983}).
Recently, the determinant-driven CIPSI algorithm has been efficiently implemented \cite{Giner_2013,Giner_2015} in the open-source programming environment {\QP} by one of us (AS) enabling to perform massively parallel computations. \cite{Garniron_2017,Garniron_2018,Garniron_2019}
In particular, we were able to compute highly-accurate calculations of ground- and excited-state energies of small- and medium-sized molecules (including benzene). \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c}
In particular, we were able to compute highly-accurate calculations of ground- and excited-state energies for small- and medium-sized molecules (including benzene). \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c}
CIPSI is also frequently use to provide accurate trial wave function for QMC calculations. \cite{Caffarel_2014,Caffarel_2016a,Caffarel_2016b,Giner_2013,Giner_2015,Scemama_2015,Scemama_2016,Scemama_2018,Scemama_2018b,Scemama_2019,Dash_2018,Dash_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}
Moreover, a renormalized version of the PT2 correction (dubbed rPT2 in the following) has been recently implemented for a more efficient extrapolation to the FCI limit (see below). \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 CIPSI algorithm.
% Computational details
The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer.
Each Irene's AMD node is a dual-socket \titou{Intel(R) Xeon(R) Platinum 8168 CPU@2.70 GHz with 192GiB of RAM}, with a total of 128 physical CPU cores.
% Discussion
Being late to the party, we obviously cannot report blindly our CIPSI results.
However, following the philosophy of Eriksen \textit{et al.}, \cite{Eriksen_2020} we will report our results with the most neutral tone, leaving the freedom to the reader to make up his/her mind.
We then follow our usual ``protocol'' \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} by performing a preliminary SCI calculation using Hartree-Fock orbitals in order to generate a SCI wave function with at least $10^7$ determinants.
Natural orbitals (NOs) are then computed based on this wave function, and a new, larger SCI calculation is performed with this new set of orbitals.
This 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 $E_\text{PT2}$ 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 $E_\text{PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit.
As mentioned above, SCI+PT2 methods rely heavily on extrapolation when one deals with medium-sized systems.
As mentioned above, SCI+PT2 methods rely heavily on extrapolation, especially when one deals with medium-sized systems.
We then linearly extrapolate the total SCI energy to $E_\text{PT2} = 0$ (which effectively corresponds to the FCI limit) using the two largest SCI wave functions.
Although it is not possible to provide a theoretically sound error bar, we estimate the extrapolation error by the difference in excitation energy between the largest SCI wave function and its corresponding extrapolated value.
We believe that it provides a very safe estimate of the extrapolation error.
Note that all the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as described in Ref.~\onlinecite{Applencourt_2018}.
% Results and discussion
The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer.
Each Irene's AMD node is a dual-socket \titou{Intel(R) Xeon(R) Platinum 8168 CPU@2.70 GHz with 192GiB of RAM}, with a total of 128 physical CPU cores.
The three flavours of SCI fall into an interval ranging from $-863.7$ to $-862.8$ m$E_h$.
The CIPSI number is ?
\titou{Note that, even though benzene is big, we have already reported excitation energies of benzene with the 6-31+G(d) basis in Ref.~\onlinecite{Loos_2019}.}
%%% TABLE II %%%
\begin{table}
@ -163,7 +164,7 @@ The CIPSI number is ?
Left: $\Delta E_\text{var.}$, $\Delta E_\text{var.} + E_\text{PT2}$, and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of the number of determinants in the variational space.
Right: $\Delta E_\text{var.} + E_\text{PT2}$ and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of $E_\text{PT2}$ or $E_\text{rPT2}$.
The two-point linear extrapolation curves (dashed lines) are also reported.
The theoretical best estimate of $-863$ m$E_h$ from Ref.~\onlinecite{Eriksen_2020} is reported for comparison purposes.
The theoretical best estimate of $-863$ m$E_h$ from Ref.~\onlinecite{Eriksen_2020} is marked by a black line for comparison purposes.
\label{fig:CIPSI}
}
\end{figure*}