2020-08-18 18:27:34 +02:00
\documentclass [aps,prb,reprint,noshowkeys,superscriptaddress] { revtex4-1}
2020-08-24 16:21:54 +02:00
\usepackage { graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,wrapfig,txfonts}
2020-08-18 18:27:34 +02:00
\usepackage [version=4] { mhchem}
\newcommand { \ie } { \textit { i.e.} }
\newcommand { \eg } { \textit { e.g.} }
2020-10-12 19:34:02 +02:00
\newcommand { \alert } [1]{ \textcolor { black} { #1} }
2020-08-18 18:27:34 +02:00
\usepackage [normalem] { ulem}
\newcommand { \titou } [1]{ \textcolor { red} { #1} }
\newcommand { \trashPFL } [1]{ \textcolor { red} { \sout { #1} } }
\newcommand { \PFL } [1]{ \titou { (\underline { \bf PFL} : #1)} }
2020-08-23 10:07:31 +02:00
\newcommand { \toto } [1]{ \textcolor { green} { #1} }
\newcommand { \trashAS } [1]{ \textcolor { green} { \sout { #1} } }
\newcommand { \AS } [1]{ \toto { (\underline { \bf AS} : #1)} }
2020-08-18 18:27:34 +02:00
\newcommand { \mc } { \multicolumn }
\newcommand { \fnm } { \footnotemark }
\newcommand { \fnt } { \footnotetext }
\newcommand { \tabc } [1]{ \multicolumn { 1} { c} { #1} }
\newcommand { \QP } { \textsc { quantum package} }
\usepackage [
colorlinks=true,
citecolor=blue,
breaklinks=true
]{ hyperref}
\urlstyle { same}
\begin { document}
\newcommand { \LCPQ } { Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\' e de Toulouse, CNRS, UPS, France}
2020-10-09 13:47:38 +02:00
\title { The performance of CIPSI on the ground state electronic energy of benzene}
2020-08-18 18:27:34 +02:00
\author { Pierre-Fran\c { c} ois Loos}
\email { loos@irsamc.ups-tlse.fr}
\affiliation { \LCPQ }
2020-08-22 18:05:59 +02:00
\author { Yann Damour}
\affiliation { \LCPQ }
2020-08-18 18:27:34 +02:00
\author { Anthony Scemama}
\email { scemama@irsamc.ups-tlse.fr}
\affiliation { \LCPQ }
2020-08-19 16:54:49 +02:00
% Abstract
\begin { abstract}
2020-10-09 09:55:44 +02:00
Following the recent work of Eriksen \textit { et al.} ~[\href { https://dx.doi.org/10.1021/acs.jpclett.0c02621} { J.~Phys.~Chem.~Lett.~\textbf { 11} , 8922 (2020)} ], we report the performance of the \textit { Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) method on the non-relativistic frozen-core correlation energy of the benzene molecule in the cc-pVDZ basis. Following our usual protocol, we obtain a correlation energy of $ - 863 . 4 $ m$ E _ h $ which agrees with the theoretical estimate of $ - 863 $ m$ E _ h $ proposed by Eriksen \textit { et al.} ~using an extensive array of highly-accurate new electronic structure methods.
2020-08-19 16:54:49 +02:00
\end { abstract}
% Title
2020-08-18 18:27:34 +02:00
\maketitle
2020-08-18 22:49:23 +02:00
% Intro
2020-08-23 09:49:04 +02:00
Although sometimes decried, one cannot deny the usefulness of benchmark sets and their corresponding reference data for the electronic structure community.
2020-08-18 22:49:23 +02:00
These are indeed essential for the validation of existing theoretical models and to bring to light and subsequently understand their strengths and, more importantly, their weaknesses.
In that regard, the previous benchmark datasets provided by the \textit { Simons Collaboration on the Many-Electron Problem} have been extremely valuable. \cite { Leblanc_ 2015,Motta_ 2017,Williams_ 2020}
2020-08-24 16:21:54 +02:00
The same comment applies to the excited-state benchmark set of Thiel and coworkers. \cite { Sauer_ 2009,Schreiber_ 2008,Silva-Junior_ 2010a,Silva-Junior_ 2010b,Silva-Junior_ 2010c}
2020-08-18 22:49:23 +02:00
Following a similar goal, we have recently proposed a large set of highly-accurate vertical transition energies for various types of excited states thanks to the renaissance of selected configuration interaction (SCI) methods \cite { Bender_ 1969,Huron_ 1973,Buenker_ 1974} which can now routinely produce near full configuration interaction (FCI) quality excitation energies for small- and medium-sized organic molecules. \cite { Loos_ 2018a,Loos_ 2019,Loos_ 2020a,Loos_ 2020b,Loos_ 2020c}
2020-08-18 18:27:34 +02:00
% The context
2020-10-20 21:30:07 +02:00
In a recent article, \cite { Eriksen_ 2020} Eriksen \textit { et al.} ~have proposed a blind test for a particular electronic structure problem inviting several groups around the world to contribute to this endeavour.
2020-08-24 22:41:41 +02:00
In addition to coupled cluster theory with singles, doubles, triples, and quadruples (CCSDTQ), \cite { Oliphant_ 1991,Kucharski_ 1992} a large panel of highly-accurate, emerging electronic structure methods were considered:
2020-08-25 18:35:40 +02:00
(i) the many-body expansion FCI (MBE-FCI), \cite { Eriksen_ 2017,Eriksen_ 2018,Eriksen_ 2019a,Eriksen_ 2019b}
2020-10-20 21:30:07 +02:00
(ii) three SCI methods including a second-order perturbative correction [adaptive sampling CI (ASCI), \cite { Tubman_ 2016,Tubman_ 2018,Tubman_ 2020} iterative CI (iCI), \cite { Liu_ 2014,Liu_ 2016,Lei_ 2017,Zhang_ 2020} and semistochastic heat-bath CI (SHCI) \cite { Holmes_ 2016,Holmes_ 2017,Sharma_ 2017} ],
(iii) the full coupled-cluster reduction (FCCR) \cite { Xu_ 2018,Xu_ 2020} which also includes a second-order perturbative correction,
(iv) the density-matrix renornalization group (DMRG) approach, \cite { White_ 1992,White_ 1993,Chan_ 2011} and
2020-10-09 13:47:38 +02:00
(v) two flavors of FCI quantum Monte Carlo (FCIQMC), \cite { Booth_ 2009,Cleland_ 2010} namely AS-FCIQMC \cite { Ghanem_ 2019} and CAD-FCIQMC. \cite { Deustua_ 2017,Deustua_ 2018}
2020-08-24 22:41:41 +02:00
We refer the interested reader to Ref.~\onlinecite { Eriksen_ 2020} and its supporting information for additional details on each method and the complete list of references.
Soon after, Lee \textit { et al.} ~reported phaseless auxiliary-field quantum Monte Carlo \cite { Motta_ 2018} (ph-AFQMC) correlation energies for the very same problem. \cite { Lee_ 2020}
2020-08-18 18:27:34 +02:00
% The system
2020-08-19 16:54:49 +02:00
The target application is the non-relativistic frozen-core correlation energy of the ground state of the benzene molecule in the cc-pVDZ basis.
2020-10-20 21:30:07 +02:00
The geometry of benzene has been optimized at the MP2/6-31G* level \cite { Schreiber_ 2008} and its coordinates can be found in the supporting information of Ref.~\onlinecite { Eriksen_ 2020} alongside its nuclear repulsion and Hartree-Fock energies.
This corresponds to an active space of 30 electrons and 108 orbitals, \ie , the Hilbert space is of the order of $ 10 ^ { 35 } $ Slater determinants.
2020-08-18 22:49:23 +02:00
Needless to say that this size of Hilbert space cannot be tackled by exact diagonalization with current architectures.
2020-08-20 21:38:29 +02:00
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.
2020-08-18 22:49:23 +02:00
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.''}
2020-08-18 18:27:34 +02:00
%%% TABLE 1 %%%
\begin { table}
\caption {
2020-10-20 21:30:07 +02:00
The frozen-core correlation energy $ \Delta E $ (in m$ E _ h $ ) of benzene in the cc-pVDZ basis set using various methods.
2020-08-18 18:27:34 +02:00
\label { tab:energy}
}
\begin { ruledtabular}
2020-08-22 09:46:07 +02:00
\begin { tabular} { llc}
2020-10-20 21:30:07 +02:00
Method & \tabc { $ \Delta E $ } & Ref. \\
2020-08-18 18:27:34 +02:00
\hline
2020-08-26 11:22:06 +02:00
ASCI & $ - 860 . 0 $ & \onlinecite { Eriksen_ 2020} \\
iCI & $ - 861 . 1 $ & \onlinecite { Eriksen_ 2020} \\
2020-08-22 09:46:07 +02:00
CCSDTQ & $ - 862 . 4 $ & \onlinecite { Eriksen_ 2020} \\
2020-08-26 11:22:06 +02:00
DMRG & $ - 862 . 8 $ & \onlinecite { Eriksen_ 2020} \\
FCCR & $ - 863 . 0 $ & \onlinecite { Eriksen_ 2020} \\
2020-08-25 18:35:40 +02:00
MBE-FCI & $ - 863 . 0 $ & \onlinecite { Eriksen_ 2020} \\
2020-08-22 09:46:07 +02:00
CAD-FCIQMC & $ - 863 . 4 $ & \onlinecite { Eriksen_ 2020} \\
2020-08-26 11:22:06 +02:00
AS-FCIQMC & $ - 863 . 7 $ & \onlinecite { Eriksen_ 2020} \\
SHCI & $ - 864 . 2 $ & \onlinecite { Eriksen_ 2020} \\
2020-08-18 18:27:34 +02:00
\hline
2020-08-22 09:46:07 +02:00
ph-AFQMC & $ - 864 . 3 ( 4 ) $ & \onlinecite { Lee_ 2020} \\
2020-08-18 18:27:34 +02:00
\hline
2020-08-26 11:22:06 +02:00
CIPSI & $ - 863 . 4 $ & This work \\
2020-08-18 18:27:34 +02:00
\end { tabular}
\end { ruledtabular}
\end { table}
2020-08-21 22:49:32 +02:00
%%$ FIG. 1 %%%
\begin { figure*}
\includegraphics [width=0.4\linewidth] { fig1a}
\hspace { 0.08\linewidth }
\includegraphics [width=0.4\linewidth] { fig1b}
\caption {
2020-08-24 22:41:41 +02:00
Convergence of the CIPSI correlation energy of benzene using localized orbitals.
Left: $ \Delta E _ \text { var. } $ , $ \Delta E _ \text { var. } + E _ \text { PT 2 } $ , and $ \Delta E _ \text { var. } + E _ \text { rPT 2 } $ (in m$ E _ h $ ) as functions of the number of determinants in the variational space $ N _ \text { det } $ .
Right: $ \Delta E _ \text { var. } $ (in m$ E _ h $ ) as a function of $ E _ \text { PT 2 } $ or $ E _ \text { rPT 2 } $ .
2020-08-25 17:09:45 +02:00
The four-point linear extrapolation curves (dashed lines) are also reported.
2020-08-24 22:41:41 +02:00
The theoretical estimate of $ - 863 $ m$ E _ h $ from Ref.~\onlinecite { Eriksen_ 2020} is marked by a black line for comparison purposes.
2020-10-20 21:30:07 +02:00
The statistical error bars associated with $ E _ \text { PT 2 } $ or $ E _ \text { rPT 2 } $ (not shown) are of the order of the size of the markers.
2020-08-21 22:49:32 +02:00
\label { fig:CIPSI}
}
\end { figure*}
%%% TABLE II %%%
2020-08-22 09:46:07 +02:00
%\begin{squeezetable}
2020-08-21 22:49:32 +02:00
\begin { table*}
\caption { Variational energy $ E _ \text { var. } $ , second-order perturbative correction $ E _ \text { PT 2 } $ and its renormalized version $ E _ \text { rPT 2 } $ (in $ E _ h $ ) as a function of the number of determinants $ N _ \text { det } $ for the ground-state of the benzene molecule computed in the cc-pVDZ basis set.
2020-08-24 22:41:41 +02:00
The statistical error on $ E _ \text { ( r ) PT 2 } $ , corresponding to one standard deviation, are reported in parenthesis.}
2020-08-21 22:49:32 +02:00
\label { tab:NOvsLO}
\begin { ruledtabular}
\begin { tabular} { rcccccc}
& \mc { 3} { c} { Natural orbitals} & \mc { 3} { c} { Localized orbitals} \\
\cline { 2-4} \cline { 5-7}
\tabc { $ N _ \text { det } $ } & \tabc { $ E _ \text { var. } $ } & \tabc { $ E _ \text { var. } + E _ \text { PT 2 } $ } & \tabc { $ E _ \text { var. } + E _ \text { rPT 2 } $ }
& \tabc { $ E _ \text { var. } $ } & \tabc { $ E _ \text { var. } + E _ \text { PT 2 } $ } & \tabc { $ E _ \text { var. } + E _ \text { rPT 2 } $ } \\
\hline
% 5 & $-230.719\,957$ & $-231.754\,877(0)$ & $-231.387\,078(0)$ & $-230.719\,928$ & $-231.718\,694(0)$ & $-231.385\,276(0)$ \\
% 10 & $-230.750\,520$ & $-231.707\,154(0)$ & $-231.411\,127(0)$ & $-230.760\,937$ & $-231.670\,043(0)$ & $-231.402\,535(0)$ \\
% 20 & $-230.767\,479$ & $-231.681\,740(0)$ & $-231.416\,388(0)$ & $-230.807\,967$ & $-231.636\,613(0)$ & $-231.415\,300(0)$ \\
% 40 & $-230.782\,929$ & $-231.664\,092(0)$ & $-231.423\,696(0)$ & $-230.863\,737$ & $-231.588\,750(0)$ & $-231.421\,131(0)$ \\
% 80 & $-230.800\,057$ & $-231.649\,643(0)$ & $-231.429\,207(0)$ & $-230.892\,146$ & $-231.567\,290(0)$ & $-231.427\,639(0)$ \\
% 160 & $-230.818\,727$ & $-231.635\,786(0)$ & $-231.433\,153(0)$ & $-230.922\,173$ & $-231.556\,426(0)$ & $-231.435\,952(0)$ \\
% 320 & $-230.842\,915$ & $-231.619\,618(0)$ & $-231.438\,459(0)$ & $-230.957\,439$ & $-231.541\,301(0)$ & $-231.440\,416(0)$ \\
% 640 & $-230.875\,821$ & $-231.601\,124(0)$ & $-231.445\,624(0)$ & $-231.000\,058$ & $-231.529\,709(0)$ & $-231.449\,099(0)$ \\
1\, 280 & $ - 230 . 978 \, 056 $ & $ - 231 . 559 \, 025 ( 212 ) $ & $ - 231 . 463 \, 633 ( 177 ) $ & $ - 231 . 101 \, 676 $ & $ - 231 . 519 \, 522 ( 149 ) $ & $ - 231 . 472 \, 224 ( 132 ) $ \\
2\, 560 & $ - 231 . 043 \, 712 $ & $ - 231 . 542 \, 344 ( 139 ) $ & $ - 231 . 474 \, 885 ( 120 ) $ & $ - 231 . 161 \, 264 $ & $ - 231 . 515 \, 577 ( 155 ) $ & $ - 231 . 482 \, 477 ( 140 ) $ \\
5\, 120 & $ - 231 . 115 \, 142 $ & $ - 231 . 534 \, 122 ( 213 ) $ & $ - 231 . 488 \, 815 ( 190 ) $ & $ - 231 . 224 \, 632 $ & $ - 231 . 516 \, 375 ( 191 ) $ & $ - 231 . 495 \, 022 ( 177 ) $ \\
10\, 240 & $ - 231 . 188 \, 813 $ & $ - 231 . 531 \, 660 ( 516 ) $ & $ - 231 . 502 \, 992 ( 473 ) $ & $ - 231 . 283 \, 295 $ & $ - 231 . 520 \, 907 ( 271 ) $ & $ - 231 . 507 \, 708 ( 255 ) $ \\
20\, 480 & $ - 231 . 260 \, 065 $ & $ - 231 . 534 \, 172 ( 611 ) $ & $ - 231 . 517 \, 063 ( 573 ) $ & $ - 231 . 330 \, 209 $ & $ - 231 . 526 \, 433 ( 586 ) $ & $ - 231 . 518 \, 045 ( 561 ) $ \\
40\, 960 & $ - 231 . 321 \, 906 $ & $ - 231 . 538 \, 269 ( 501 ) $ & $ - 231 . 528 \, 301 ( 478 ) $ & $ - 231 . 366 \, 008 $ & $ - 231 . 532 \, 288 ( 303 ) $ & $ - 231 . 526 \, 639 ( 293 ) $ \\
81\, 920 & $ - 231 . 366 \, 895 $ & $ - 231 . 541 \, 945 ( 813 ) $ & $ - 231 . 535 \, 785 ( 785 ) $ & $ - 231 . 392 \, 888 $ & $ - 231 . 536 \, 578 ( 614 ) $ & $ - 231 . 532 \, 575 ( 597 ) $ \\
163\, 840 & $ - 231 . 392 \, 866 $ & $ - 231 . 545 \, 499 ( 761 ) $ & $ - 231 . 541 \, 010 ( 739 ) $ & $ - 231 . 414 \, 132 $ & $ - 231 . 541 \, 400 ( 624 ) $ & $ - 231 . 538 \, 378 ( 609 ) $ \\
327\, 680 & $ - 231 . 407 \, 802 $ & $ - 231 . 548 \, 699 ( 662 ) $ & $ - 231 . 544 \, 980 ( 645 ) $ & $ - 231 . 431 \, 952 $ & $ - 231 . 545 \, 873 ( 557 ) $ & $ - 231 . 543 \, 532 ( 545 ) $ \\
655\, 360 & $ - 231 . 418 \, 752 $ & $ - 231 . 551 \, 208 ( 661 ) $ & $ - 231 . 548 \, 004 ( 645 ) $ & $ - 231 . 447 \, 007 $ & $ - 231 . 548 \, 856 ( 498 ) $ & $ - 231 . 547 \, 043 ( 489 ) $ \\
1\, 310\, 720 & $ - 231 . 428 \, 852 $ & $ - 231 . 552 \, 760 ( 616 ) $ & $ - 231 . 550 \, 006 ( 603 ) $ & $ - 231 . 460 \, 970 $ & $ - 231 . 552 \, 137 ( 453 ) $ & $ - 231 . 550 \, 723 ( 446 ) $ \\
2\, 621\, 440 & $ - 231 . 439 \, 324 $ & $ - 231 . 553 \, 845 ( 572 ) $ & $ - 231 . 551 \, 544 ( 560 ) $ & $ - 231 . 473 \, 751 $ & $ - 231 . 555 \, 261 ( 403 ) $ & $ - 231 . 554 \, 159 ( 397 ) $ \\
5\, 242\, 880 & $ - 231 . 450 \, 156 $ & $ - 231 . 557 \, 541 ( 534 ) $ & $ - 231 . 555 \, 558 ( 524 ) $ & $ - 231 . 485 \, 829 $ & $ - 231 . 558 \, 303 ( 362 ) $ & $ - 231 . 557 \, 451 ( 358 ) $ \\
10\, 485\, 760 & $ - 231 . 461 \, 927 $ & $ - 231 . 559 \, 390 ( 481 ) $ & $ - 231 . 557 \, 796 ( 474 ) $ & $ - 231 . 497 \, 515 $ & $ - 231 . 562 \, 568 ( 322 ) $ & $ - 231 . 561 \, 901 ( 319 ) $ \\
2020-08-22 09:46:07 +02:00
20\, 971\, 520 & $ - 231 . 474 \, 019 $ & $ - 231 . 561 \, 315 ( 430 ) $ & $ - 231 . 560 \, 063 ( 424 ) $ & $ - 231 . 508 \, 714 $ & $ - 231 . 564 \, 707 ( 275 ) $ & $ - 231 . 564 \, 223 ( 273 ) $ \\
2020-08-22 15:02:24 +02:00
41\, 943\, 040 & $ - 231 . 487 \, 978 $ & $ - 231 . 564 \, 529 ( 382 ) $ & $ - 231 . 563 \, 593 ( 377 ) $ & $ - 231 . 519 \, 122 $ & $ - 231 . 567 \, 419 ( 240 ) $ & $ - 231 . 567 \, 069 ( 238 ) $ \\
2020-08-23 08:55:40 +02:00
83\, 886\, 080 & $ - 231 . 501 \, 334 $ & $ - 231 . 566 \, 994 ( 317 ) $ & $ - 231 . 566 \, 325 ( 314 ) $ & $ - 231 . 528 \, 568 $ & $ - 231 . 570 \, 084 ( 199 ) $ & $ - 231 . 569 \, 832 ( 198 ) $ \\
2020-08-24 08:57:10 +02:00
167\, 772\, 160 & $ - 231 . 514 \, 009 $ & $ - 231 . 569 \, 939 ( 273 ) $ & $ - 231 . 569 \, 467 ( 271 ) $ & $ - 231 . 536 \, 655 $ & $ - 231 . 571 \, 981 ( 175 ) $ & $ - 231 . 571 \, 804 ( 174 ) $ \\
2020-08-21 22:49:32 +02:00
\end { tabular}
\end { ruledtabular}
\end { table*}
2020-08-22 09:46:07 +02:00
%\end{squeezetable}
2020-08-21 22:49:32 +02:00
%%% %%% %%% %%%
2020-10-12 19:34:02 +02:00
%%% TABLE III %%%
2020-08-18 22:49:23 +02:00
\begin { table}
2020-08-26 11:22:06 +02:00
\caption { Extrapolation distances, $ \Delta E _ { \text { dist } } $ , defined as the difference between the final computed energy, $ \Delta E _ { \text { final } } $ , and the extrapolated energy, $ \Delta E _ { \text { extrap. } } $ associated with ASCI, iCI, SHCI, DMRG, and CIPSI for the best blind-test and post-blind-test estimates of the correlation energy of benzene in the cc-pVDZ basis.
2020-08-19 17:18:42 +02:00
The final variational energies $ \Delta E _ { \text { var. } } $ are also reported.
See Ref.~\onlinecite { Eriksen_ 2020} for more details.
2020-08-24 22:41:41 +02:00
All correlation energies are given in m$ E _ h $ .
2020-08-22 09:46:07 +02:00
\label { tab:extrap_ dist_ table}
2020-08-18 22:49:23 +02:00
}
\begin { ruledtabular}
\begin { tabular} { lcccc}
Method & $ \Delta E _ { \text { var. } } $ & $ \Delta E _ { \text { final } } $ & $ \Delta E _ { \text { extrap. } } $ & $ \Delta E _ { \text { dist } } $ \\
\hline
2020-08-26 11:22:06 +02:00
\mc { 4} { l} { Best blind-test estimates} \\
2020-08-18 22:49:23 +02:00
ASCI & $ - 737 . 1 $ & $ - 835 . 4 $ & $ - 860 . 0 $ & $ - 24 . 6 $ \\
iCI & $ - 730 . 0 $ & $ - 833 . 7 $ & $ - 861 . 1 $ & $ - 27 . 4 $ \\
SHCI & $ - 827 . 2 $ & $ - 852 . 8 $ & $ - 864 . 2 $ & $ - 11 . 4 $ \\
DMRG & $ - 859 . 2 $ & $ - 859 . 2 $ & $ - 862 . 8 $ & $ - 3 . 6 $ \\
2020-08-24 16:21:54 +02:00
\hline
2020-08-26 11:22:06 +02:00
\mc { 4} { l} { Best post-blind-test estimates} \\
ASCI & $ - 772 . 4 $ & $ - 835 . 2 $ & $ - 861 . 3 $ & $ - 26 . 1 $ \\
iCI & $ - 770 . 7 $ & $ - 842 . 8 $ & $ - 864 . 2 $ & $ - 21 . 3 $ \\
2020-08-31 12:38:32 +02:00
SHCI & $ - 835 . 2 $ & $ - 854 . 9 $ & $ - 863 . 6 $ & $ - 8 . 7 $ \\
2020-08-26 11:22:06 +02:00
\hline
2020-08-25 17:09:45 +02:00
CIPSI & $ - 814 . 8 $ & $ - 850 . 2 $ & $ - 863 . 4 $ & $ - 13 . 2 $ \\
2020-08-18 22:49:23 +02:00
\end { tabular}
\end { ruledtabular}
\end { table}
2020-10-12 19:34:02 +02:00
% CIPSI
2020-10-20 21:30:07 +02:00
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 PT2 correction.
2020-10-12 19:34:02 +02:00
In short, the CIPSI algorithm belongs to the family of SCI+PT2 methods.
2020-10-20 21:30:07 +02:00
The idea behind such methods is to slow down 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.
2020-10-12 19:34:02 +02:00
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.
From a 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 our group enabling to perform massively parallel computations. \cite { Garniron_ 2017,Garniron_ 2018,Garniron_ 2019}
In particular, we were able to compute highly-accurate 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 used 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} (which explains the statistical error associated with the PT2 correction in the following).
2020-10-20 21:30:07 +02:00
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.
2020-10-12 19:34:02 +02:00
% Computational details
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} and Lee \textit { et al.} , \cite { Lee_ 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.
2020-10-20 21:30:07 +02:00
Natural orbitals are then computed based on this wave function, and a new SCI calculation is performed with this new natural set of orbitals.
2020-10-12 19:34:02 +02:00
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 { ( r ) PT 2 } $ 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 ) PT 2 } $ provides a qualitative idea of the ``distance'' to the FCI limit.
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 { ( r ) PT 2 } = 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 { Applencourt_ 2018} 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 ($ \sim 5 \times 10 ^ { - 3 } $ a.u.).
The corresponding energies are reported in Table \ref { tab:NOvsLO} as functions of the number of determinants in the variational space $ N _ \text { det } $ .
A second run has been performed with localized orbitals.
Starting from the same natural orbitals, a Boys-Foster localization procedure \cite { Boys_ 1960} was performed in several orbital windows: i) core, ii) valence $ \sigma $ , iii) valence $ \pi $ , iv) valence $ \pi ^ * $ , v) valence $ \sigma ^ * $ , vi) the higher-lying $ \sigma $ orbitals, and vii) the higher-lying $ \pi $ orbitals.
\footnote { Indices of molecular orbitals for Boys-Foster localization procedure:
core [1--6];
$ \sigma $ [7--18];
$ \pi $ [19--21];
$ \pi ^ * $ [22--24];
$ \sigma ^ * $ [25--36];
higher-lying $ \pi $ [39,41--43,46,49,50,53--57,71--74,82--85,87,92,93,98];
higher-lying $ \sigma $ [37,38,40,44,45,47,48,51,52,58--70,75--81,86,88--91,94--97,99--114].}
Like Pipek-Mezey, \cite { Pipek_ 1989} this choice of orbital windows allows to preserve a strict $ \sigma $ -$ \pi $ separation in planar systems like benzene.
As one can see from the energies of Table \ref { tab:NOvsLO} , for a given value of $ N _ \text { det } $ , the variational energy as well as the PT2-corrected energies are much lower with localized orbitals than with natural orbitals.
2020-10-20 21:30:07 +02:00
Indeed, localized orbitals significantly speed up the convergence of SCI calculations by taking benefit of the local character of electron correlation.\cite { Angeli_ 2003,Angeli_ 2009,BenAmor_ 2011,Suaud_ 2017,Chien_ 2018,Eriksen_ 2020}
2020-10-12 19:34:02 +02:00
We, therefore, consider these energies more trustworthy, and we will base our best estimate of the correlation energy of benzene on these calculations.
The convergence of the CIPSI correlation energy using localized orbitals is illustrated in Fig.~\ref { fig:CIPSI} , where one can see the behavior of the correlation energy, $ \Delta E _ \text { var. } $ and $ \Delta E _ \text { var. } + E _ \text { ( r ) PT 2 } $ , as a function of $ N _ \text { det } $ (left panel).
The right panel of Fig.~\ref { fig:CIPSI} is more instructive as it shows $ \Delta E _ \text { var. } $ as a function of $ E _ \text { ( r ) PT 2 } $ , and their corresponding four-point linear extrapolation curves that we have used to get our final estimate of the correlation energy.
2020-10-20 21:30:07 +02:00
(In other words, the four largest variational wave functions are considered to perform the linear extrapolation.)
2020-10-12 19:34:02 +02:00
From this figure, one clearly sees that the rPT2-based correction behaves more linearly than its corresponding PT2 version, and is thus systematically employed in the following.
% Results
Our final number are gathered in Table \ref { tab:extrap_ dist_ table} , where, following the notations of Ref.~\onlinecite { Eriksen_ 2020} , we report, in addition to the final variational energies $ \Delta E _ { \text { var. } } $ , the
2020-10-20 21:30:07 +02:00
extrapolation distances, $ \Delta E _ { \text { dist } } $ , defined as the difference between the final computed energy, $ \Delta E _ { \text { final } } $ , and the extrapolated energy, $ \Delta E _ { \text { extrap. } } $ associated with ASCI, iCI, SHCI, DMRG, and CIPSI.
2020-10-12 19:34:02 +02:00
The three flavours of SCI fall into an interval ranging from $ - 860 . 0 $ m$ E _ h $ (ASCI) to $ - 864 . 2 $ m$ E _ h $ (SHCI), while the other non-SCI methods yield correlation energies ranging from $ - 863 . 7 $ to $ - 862 . 8 $ m$ E _ h $ (see Table \ref { tab:energy} ). Our final CIPSI number (obtained with localized orbitals and rPT2 correction via a four-point linear extrapolation) is $ - 863 . 4 ( 5 ) $ m$ E _ h $ , where the error reported in parenthesis represents the fitting error (not the extrapolation error for which it is much harder to provide a theoretically sound estimate).
2020-10-20 21:30:07 +02:00
\footnote { Using the last 3, 4, 5, and 6 largest wave functions to perform the linear extrapolation yield the following correlation energy estimates: $ - 863 . 1 ( 11 ) $ , $ - 863 . 4 ( 5 ) $ , $ - 862 . 1 ( 8 ) $ , and $ - 863 . 5 ( 11 ) $ mE$ _ h $ , respectively.
2020-10-12 19:34:02 +02:00
These numbers vary by $ 1 . 4 $ mE$ _ h $ .
The four-point extrapolated value of $ - 863 . 4 ( 5 ) $ mE$ _ h $ that we have chosen to report as our best estimate corresponds to the smallest fitting error.
Quadratic fits yield much larger variations and are discarded in practice.
Due to the stochastic nature of $ E _ \text { rPT 2 } $ , the fifth point is slightly off as compared to the others.
Taking into account this fifth point yield a slightly smaller estimate of the correlation energy [$ - 862 . 1 ( 8 ) $ mE$ _ h $ ], while adding a sixth point settles down the correlation energy estimate at $ - 863 . 5 ( 11 ) $ mE$ _ h $
2020-10-20 21:30:07 +02:00
}
2020-10-12 19:34:02 +02:00
For comparison, the best post blind test SHCI estimate is $ - 863 . 3 $ m$ E _ h $ , which agrees almost perfectly with our best CIPSI estimate, while the best post blind test ASCI and iCI correlation energies are $ - 861 . 3 $ and $ - 864 . 15 $ m$ E _ h $ , respectively (see Table \ref { tab:extrap_ dist_ table} ).
2020-08-25 17:09:45 +02:00
% Timings
The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer.
Each Irene's AMD node is a dual-socket AMD Rome (Epyc) CPU@2.60 GHz with 256GiB of RAM, with a total of 64 physical CPU cores per socket.
These nodes are connected via Infiniband HDR100.
The first step of the calculation, \ie , performing a CIPSI calculation up to $ N _ \text { det } \sim 10 ^ 7 $ with Hartree-Fock orbitals in order to produce natural orbitals, takes roughly 24 hours on a single node, and reaching the same number of determinants with natural orbitals or localized orbitals takes roughly the same amount of time.
A second 24-hour run on 10 distributed nodes was performed to push the selection to $ 8 \times 10 ^ 7 $ determinants, and a third distributed run using 40 nodes was used to reach $ 16 \times 10 ^ 7 $ determinants.
In total, the present calculation has required 150k core hours, most of it being spent in the last stage of the computation.
2020-08-19 18:17:06 +02:00
2020-08-18 22:49:23 +02:00
% Acknowledgements
2020-08-26 11:23:10 +02:00
We thank Janus Eriksen and Cyrus Umrigar for useful comments.
2020-08-23 14:26:47 +02:00
This work was performed using HPC resources from GENCI-TGCC (2020-gen1738) and from CALMIP (Toulouse) under allocation 2020-18005.
2020-10-12 19:34:02 +02:00
PFL and AS have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).
2020-08-18 18:27:34 +02:00
2020-08-31 12:38:32 +02:00
% Data availability statement
2020-10-20 21:30:07 +02:00
The data that support the findings of this study are openly available in Zenodo at \href { http://doi.org/10.5281/zenodo.4075286} { http://doi.org/10.5281/zenodo.4075286} .
2020-08-31 12:38:32 +02:00
2020-10-12 19:34:02 +02:00
%merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked
%Control: key (0)
%Control: author (8) initials jnrlst
%Control: editor formatted (1) identically to author
%Control: production of article title (-1) disabled
%Control: page (0) single
%Control: year (1) truncated
%Control: production of eprint (0) enabled
\begin { thebibliography} { 71} %
\makeatletter
\providecommand \@ ifxundefined [1]{ %
\@ ifx{ #1\undefined }
} %
\providecommand \@ ifnum [1]{ %
\ifnum #1\expandafter \@ firstoftwo
\else \expandafter \@ secondoftwo
\fi
} %
\providecommand \@ ifx [1]{ %
\ifx #1\expandafter \@ firstoftwo
\else \expandafter \@ secondoftwo
\fi
} %
\providecommand \natexlab [1]{ #1} %
\providecommand \enquote [1]{ ``#1''} %
\providecommand \bibnamefont [1]{ #1} %
\providecommand \bibfnamefont [1]{ #1} %
\providecommand \citenamefont [1]{ #1} %
\providecommand \href @noop [0]{ \@ secondoftwo} %
\providecommand \href [0]{ \begingroup \@ sanitize@url \@ href} %
\providecommand \@ href[1]{ \@ @startlink{ #1} \@ @href} %
\providecommand \@ @href[1]{ \endgroup #1\@ @endlink} %
\providecommand \@ sanitize@url [0]{ \catcode `\\ 12\catcode `\$ 12\catcode
`\& 12\catcode `\# 12\catcode `\^ 12\catcode `\_ 12\catcode `\% 12\relax } %
\providecommand \@ @startlink[1]{ } %
\providecommand \@ @endlink[0]{ } %
\providecommand \url [0]{ \begingroup \@ sanitize@url \@ url } %
\providecommand \@ url [1]{ \endgroup \@ href { #1} { \urlprefix } } %
\providecommand \urlprefix [0]{ URL } %
\providecommand \Eprint [0]{ \href } %
\providecommand \doibase [0]{ http://dx.doi.org/} %
\providecommand \selectlanguage [0]{ \@ gobble} %
\providecommand \bibinfo [0]{ \@ secondoftwo} %
\providecommand \bibfield [0]{ \@ secondoftwo} %
\providecommand \translation [1]{ [#1]} %
\providecommand \BibitemOpen [0]{ } %
\providecommand \bibitemStop [0]{ } %
\providecommand \bibitemNoStop [0]{ .\EOS \space } %
\providecommand \EOS [0]{ \spacefactor 3000\relax } %
\providecommand \BibitemShut [1]{ \csname bibitem#1\endcsname } %
\let \auto @bib@innerbib\@ empty
%</preamble>
\bibitem [{ \citenamefont { LeBlanc} \ \emph { et~al.} (2015)\citenamefont
{ LeBlanc} , \citenamefont { Antipov} , \citenamefont { Becca} , \citenamefont
{ Bulik} , \citenamefont { Chan} , \citenamefont { Chung} , \citenamefont { Deng} ,
\citenamefont { Ferrero} , \citenamefont { Henderson} , \citenamefont
{ Jim{ \' e} nez-Hoyos} \emph { et~al.} } ]{ Leblanc_ 2015} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~P.~F.} \
\bibnamefont { LeBlanc} } , \bibinfo { author} { \bibfnamefont { A.~E.} \
\bibnamefont { Antipov} } , \bibinfo { author} { \bibfnamefont { F.} ~\bibnamefont
{ Becca} } , \bibinfo { author} { \bibfnamefont { I.~W.} \ \bibnamefont { Bulik} } ,
\bibinfo { author} { \bibfnamefont { G.~K.-L.} \ \bibnamefont { Chan} } , \bibinfo
{ author} { \bibfnamefont { C.-M.} \ \bibnamefont { Chung} } , \bibinfo { author}
{ \bibfnamefont { Y.} ~\bibnamefont { Deng} } , \bibinfo { author} { \bibfnamefont
{ M.} ~\bibnamefont { Ferrero} } , \bibinfo { author} { \bibfnamefont { T.~M.} \
\bibnamefont { Henderson} } , \bibinfo { author} { \bibfnamefont { C.~A.} \
\bibnamefont { Jim{ \' e} nez-Hoyos} } , \emph { et~al.} ,\ } \href { \doibase
10.1103/PhysRevX.5.041041} { \bibfield { journal} { \bibinfo { journal} { Phys.
Rev. X} \ } \textbf { \bibinfo { volume} { 5} } ,\ \bibinfo { pages} { 041041}
(\bibinfo { year} { 2015} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Motta} \ \emph { et~al.} (2017)\citenamefont { Motta} ,
\citenamefont { Ceperley} , \citenamefont { Chan} , \citenamefont { Gomez} ,
\citenamefont { Gull} , \citenamefont { Guo} , \citenamefont { Jim{ \' e} nez-Hoyos} ,
\citenamefont { Lan} , \citenamefont { Li} , \citenamefont { Ma} \emph
{ et~al.} } ]{ Motta_ 2017} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Motta} } , \bibinfo { author} { \bibfnamefont { D.~M.} \ \bibnamefont { Ceperley} } ,
\bibinfo { author} { \bibfnamefont { G.~K.-L.} \ \bibnamefont { Chan} } , \bibinfo
{ author} { \bibfnamefont { J.~A.} \ \bibnamefont { Gomez} } , \bibinfo { author}
{ \bibfnamefont { E.} ~\bibnamefont { Gull} } , \bibinfo { author} { \bibfnamefont
{ S.} ~\bibnamefont { Guo} } , \bibinfo { author} { \bibfnamefont { C.~A.} \
\bibnamefont { Jim{ \' e} nez-Hoyos} } , \bibinfo { author} { \bibfnamefont { T.~N.} \
\bibnamefont { Lan} } , \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont
{ Li} } , \bibinfo { author} { \bibfnamefont { F.} ~\bibnamefont { Ma} } , \emph
{ et~al.} ,\ } \href { \doibase 10.1103/PhysRevX.7.031059} { \bibfield { journal}
{ \bibinfo { journal} { Phys. Rev. X} \ } \textbf { \bibinfo { volume} { 7} } ,\
\bibinfo { pages} { 031059} (\bibinfo { year} { 2017} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Williams} \ \emph { et~al.} (2020)\citenamefont
{ Williams} , \citenamefont { Yao} , \citenamefont { Li} , \citenamefont { Chen} ,
\citenamefont { Shi} , \citenamefont { Motta} , \citenamefont { Niu} ,
\citenamefont { Ray} , \citenamefont { Guo} , \citenamefont { Anderson} \emph
{ et~al.} } ]{ Williams_ 2020} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { K.~T.} \ \bibnamefont
{ Williams} } , \bibinfo { author} { \bibfnamefont { Y.} ~\bibnamefont { Yao} } ,
\bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont { Li} } , \bibinfo { author}
{ \bibfnamefont { L.} ~\bibnamefont { Chen} } , \bibinfo { author} { \bibfnamefont
{ H.} ~\bibnamefont { Shi} } , \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Motta} } , \bibinfo { author} { \bibfnamefont { C.} ~\bibnamefont { Niu} } , \bibinfo
{ author} { \bibfnamefont { U.} ~\bibnamefont { Ray} } , \bibinfo { author}
{ \bibfnamefont { S.} ~\bibnamefont { Guo} } , \bibinfo { author} { \bibfnamefont
{ R.~J.} \ \bibnamefont { Anderson} } , \emph { et~al.} ,\ } \href { \doibase
10.1103/PhysRevX.10.011041} { \bibfield { journal} { \bibinfo { journal} { Phys.
Rev. X} \ } \textbf { \bibinfo { volume} { 10} } ,\ \bibinfo { pages} { 011041}
(\bibinfo { year} { 2020} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Sauer} \ \emph { et~al.} (2009)\citenamefont { Sauer} ,
\citenamefont { Schreiber} , \citenamefont { Silva-Junior} ,\ and\ \citenamefont
{ Thiel} } ]{ Sauer_ 2009} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { S.~P.~A.} \
\bibnamefont { Sauer} } , \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Schreiber} } , \bibinfo { author} { \bibfnamefont { M.~R.} \ \bibnamefont
{ Silva-Junior} } , \ and\ \bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont
{ Thiel} } ,\ } \href { \doibase 10.1021/ct800256j} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume}
{ 5} } ,\ \bibinfo { pages} { 555} (\bibinfo { year} { 2009} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Schreiber} \ \emph { et~al.} (2008)\citenamefont
{ Schreiber} , \citenamefont { Silva-Junior} , \citenamefont { Sauer} ,\ and\
\citenamefont { Thiel} } ]{ Schreiber_ 2008} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Schreiber} } , \bibinfo { author} { \bibfnamefont { M.~R.} \ \bibnamefont
{ Silva-Junior} } , \bibinfo { author} { \bibfnamefont { S.~P.~A.} \ \bibnamefont
{ Sauer} } , \ and\ \bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont
{ Thiel} } ,\ } \href { \doibase 10.1063/1.2889385} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 128} } ,\
\bibinfo { pages} { 134110} (\bibinfo { year} { 2008} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Silva-Junior} \ \emph { et~al.} (2008)\citenamefont
{ Silva-Junior} , \citenamefont { Schreiber} , \citenamefont { Sauer} ,\ and\
\citenamefont { Thiel} } ]{ Silva-Junior_ 2010a} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.~R.} \ \bibnamefont
{ Silva-Junior} } , \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Schreiber} } , \bibinfo { author} { \bibfnamefont { S.~P.~A.} \ \bibnamefont
{ Sauer} } , \ and\ \bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont
{ Thiel} } ,\ } \href { \doibase 10.1063/1.2973541} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 129} } ,\
\bibinfo { pages} { 104103} (\bibinfo { year} { 2008} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Silva-Junior} \ \emph
{ et~al.} (2010{ \natexlab { a} } )\citenamefont { Silva-Junior} , \citenamefont
{ Sauer} , \citenamefont { Schreiber} ,\ and\ \citenamefont
{ Thiel} } ]{ Silva-Junior_ 2010b} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.~R.} \ \bibnamefont
{ Silva-Junior} } , \bibinfo { author} { \bibfnamefont { S.~P.~A.} \ \bibnamefont
{ Sauer} } , \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Schreiber} } , \
and\ \bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont { Thiel} } ,\ } \href
{ \doibase 10.1080/00268970903549047} { \bibfield { journal} { \bibinfo
{ journal} { Mol. Phys.} \ } \textbf { \bibinfo { volume} { 108} } ,\ \bibinfo { pages}
{ 453} (\bibinfo { year} { 2010} { \natexlab { a} } )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Silva-Junior} \ \emph
{ et~al.} (2010{ \natexlab { b} } )\citenamefont { Silva-Junior} , \citenamefont
{ Schreiber} , \citenamefont { Sauer} ,\ and\ \citenamefont
{ Thiel} } ]{ Silva-Junior_ 2010c} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.~R.} \ \bibnamefont
{ Silva-Junior} } , \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Schreiber} } , \bibinfo { author} { \bibfnamefont { S.~P.~A.} \ \bibnamefont
{ Sauer} } , \ and\ \bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont
{ Thiel} } ,\ } \href { \doibase 10.1063/1.3499598} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 133} } ,\
\bibinfo { pages} { 174318} (\bibinfo { year} { 2010} { \natexlab { b} } )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Bender} \ and\ \citenamefont
{ Davidson} (1969)} ]{ Bender_ 1969} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { C.~F.} \ \bibnamefont
{ Bender} } \ and\ \bibinfo { author} { \bibfnamefont { E.~R.} \ \bibnamefont
{ Davidson} } ,\ } \href { \doibase 10.1103/PhysRev.183.23} { \bibfield { journal}
{ \bibinfo { journal} { Phys. Rev.} \ } \textbf { \bibinfo { volume} { 183} } ,\
\bibinfo { pages} { 23} (\bibinfo { year} { 1969} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Huron} \ \emph { et~al.} (1973)\citenamefont { Huron} ,
\citenamefont { Malrieu} ,\ and\ \citenamefont { Rancurel} } ]{ Huron_ 1973} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { B.} ~\bibnamefont
{ Huron} } , \bibinfo { author} { \bibfnamefont { J.~P.} \ \bibnamefont { Malrieu} } ,
\ and\ \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont { Rancurel} } ,\
} \href { \doibase 10.1063/1.1679199} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 58} } ,\ \bibinfo
{ pages} { 5745} (\bibinfo { year} { 1973} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Buenker} \ and\ \citenamefont
{ Peyerimhoff} (1974)} ]{ Buenker_ 1974} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { R.~J.} \ \bibnamefont
{ Buenker} } \ and\ \bibinfo { author} { \bibfnamefont { S.~D.} \ \bibnamefont
{ Peyerimhoff} } ,\ } \href { \doibase 10.1007/BF02394557} { \bibfield { journal}
{ \bibinfo { journal} { Theor. Chim. Acta} \ } \textbf { \bibinfo { volume} { 35} } ,\
\bibinfo { pages} { 33} (\bibinfo { year} { 1974} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Loos} \ \emph { et~al.} (2018)\citenamefont { Loos} ,
\citenamefont { Scemama} , \citenamefont { Blondel} , \citenamefont { Garniron} ,
\citenamefont { Caffarel} ,\ and\ \citenamefont { Jacquemin} } ]{ Loos_ 2018a} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.~F.} \ \bibnamefont
{ Loos} } , \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } ,
\bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Blondel} } , \bibinfo
{ author} { \bibfnamefont { Y.} ~\bibnamefont { Garniron} } , \bibinfo { author}
{ \bibfnamefont { M.} ~\bibnamefont { Caffarel} } , \ and\ \bibinfo { author}
{ \bibfnamefont { D.} ~\bibnamefont { Jacquemin} } ,\ } \href { \doibase
10.1021/acs.jctc.8b00406} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 14} } ,\ \bibinfo { pages}
{ 4360} (\bibinfo { year} { 2018} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Loos} \ \emph { et~al.} (2019)\citenamefont { Loos} ,
\citenamefont { Boggio-Pasqua} , \citenamefont { Scemama} , \citenamefont
{ Caffarel} ,\ and\ \citenamefont { Jacquemin} } ]{ Loos_ 2019} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.-F.} \ \bibnamefont
{ Loos} } , \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Boggio-Pasqua} } ,
\bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } , \bibinfo
{ author} { \bibfnamefont { M.} ~\bibnamefont { Caffarel} } , \ and\ \bibinfo
{ author} { \bibfnamefont { D.} ~\bibnamefont { Jacquemin} } ,\ } \href { \doibase
10.1021/acs.jctc.8b01205} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 15} } ,\ \bibinfo { pages}
{ 1939} (\bibinfo { year} { 2019} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Loos} \ \emph
{ et~al.} (2020{ \natexlab { a} } )\citenamefont { Loos} , \citenamefont { Scemama} ,\
and\ \citenamefont { Jacquemin} } ]{ Loos_ 2020a} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.~F.} \ \bibnamefont
{ Loos} } , \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } , \
and\ \bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont { Jacquemin} } ,\ } \href
{ \doibase 10.1021/acs.jpclett.0c00014} { \bibfield { journal} { \bibinfo
{ journal} { J. Phys. Chem. Lett.} \ } \textbf { \bibinfo { volume} { 11} } ,\
\bibinfo { pages} { 2374} (\bibinfo { year} { 2020} { \natexlab { a} } )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Loos} \ \emph
{ et~al.} (2020{ \natexlab { b} } )\citenamefont { Loos} , \citenamefont { Lipparini} ,
\citenamefont { Boggio-Pasqua} , \citenamefont { Scemama} ,\ and\ \citenamefont
{ Jacquemin} } ]{ Loos_ 2020b} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.~F.} \ \bibnamefont
{ Loos} } , \bibinfo { author} { \bibfnamefont { F.} ~\bibnamefont { Lipparini} } ,
\bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Boggio-Pasqua} } , \bibinfo
{ author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } , \ and\ \bibinfo
{ author} { \bibfnamefont { D.} ~\bibnamefont { Jacquemin} } ,\ } \href { \doibase
10.1021/acs.jctc.9b01216} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 16} } ,\ \bibinfo { pages}
{ 1711} (\bibinfo { year} { 2020} { \natexlab { b} } )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Loos} \ \emph
{ et~al.} (2020{ \natexlab { c} } )\citenamefont { Loos} , \citenamefont { Scemama} ,
\citenamefont { Boggio-Pasqua} ,\ and\ \citenamefont { Jacquemin} } ]{ Loos_ 2020c} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.~F.} \ \bibnamefont
{ Loos} } , \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } ,
\bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Boggio-Pasqua} } , \ and\
\bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont { Jacquemin} } ,\ } \href
{ \doibase 10.1021/acs.jctc.0c00227} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 16} } ,\
\bibinfo { pages} { 3720} (\bibinfo { year} { 2020} { \natexlab { c} } )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Eriksen} \ \emph { et~al.} (2020)\citenamefont
{ Eriksen} , \citenamefont { Anderson} , \citenamefont { Deustua} , \citenamefont
{ Ghanem} , \citenamefont { Hait} , \citenamefont { Hoffmann} , \citenamefont
{ Lee} , \citenamefont { Levine} , \citenamefont { Magoulas} , \citenamefont
{ Shen} , \citenamefont { Tubman} , \citenamefont { Whaley} , \citenamefont { Xu} ,
\citenamefont { Yao} , \citenamefont { Zhang} , \citenamefont { Alavi} ,
\citenamefont { Chan} , \citenamefont { Head-Gordon} , \citenamefont { Liu} ,
\citenamefont { Piecuch} , \citenamefont { Sharma} , \citenamefont { Ten-no} ,
\citenamefont { Umrigar} ,\ and\ \citenamefont { Gauss} } ]{ Eriksen_ 2020} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~J.} \ \bibnamefont
{ Eriksen} } , \bibinfo { author} { \bibfnamefont { T.~A.} \ \bibnamefont
{ Anderson} } , \bibinfo { author} { \bibfnamefont { J.~E.} \ \bibnamefont
{ Deustua} } , \bibinfo { author} { \bibfnamefont { K.} ~\bibnamefont { Ghanem} } ,
\bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont { Hait} } , \bibinfo { author}
{ \bibfnamefont { M.~R.} \ \bibnamefont { Hoffmann} } , \bibinfo { author}
{ \bibfnamefont { S.} ~\bibnamefont { Lee} } , \bibinfo { author} { \bibfnamefont
{ D.~S.} \ \bibnamefont { Levine} } , \bibinfo { author} { \bibfnamefont
{ I.} ~\bibnamefont { Magoulas} } , \bibinfo { author} { \bibfnamefont
{ J.} ~\bibnamefont { Shen} } , \bibinfo { author} { \bibfnamefont { N.~M.} \
\bibnamefont { Tubman} } , \bibinfo { author} { \bibfnamefont { K.~B.} \
\bibnamefont { Whaley} } , \bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont
{ Xu} } , \bibinfo { author} { \bibfnamefont { Y.} ~\bibnamefont { Yao} } , \bibinfo
{ author} { \bibfnamefont { N.} ~\bibnamefont { Zhang} } , \bibinfo { author}
{ \bibfnamefont { A.} ~\bibnamefont { Alavi} } , \bibinfo { author} { \bibfnamefont
{ G.~K.-L.} \ \bibnamefont { Chan} } , \bibinfo { author} { \bibfnamefont
{ M.} ~\bibnamefont { Head-Gordon} } , \bibinfo { author} { \bibfnamefont
{ W.} ~\bibnamefont { Liu} } , \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont
{ Piecuch} } , \bibinfo { author} { \bibfnamefont { S.} ~\bibnamefont { Sharma} } ,
\bibinfo { author} { \bibfnamefont { S.~L.} \ \bibnamefont { Ten-no} } , \bibinfo
{ author} { \bibfnamefont { C.~J.} \ \bibnamefont { Umrigar} } , \ and\ \bibinfo
{ author} { \bibfnamefont { J.} ~\bibnamefont { Gauss} } ,\ } \href { \doibase
10.1021/acs.jpclett.0c02621} { \bibfield { journal} { \bibinfo { journal} { J.
Phys. Chem. Lett.} \ } \textbf { \bibinfo { volume} { 11} } ,\ \bibinfo { pages}
{ 8922} (\bibinfo { year} { 2020} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Oliphant} \ and\ \citenamefont
{ Adamowicz} (1991)} ]{ Oliphant_ 1991} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { N.} ~\bibnamefont
{ Oliphant} } \ and\ \bibinfo { author} { \bibfnamefont { L.} ~\bibnamefont
{ Adamowicz} } ,\ } \href { \doibase 10.1063/1.461534} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 95} } ,\
\bibinfo { pages} { 6645} (\bibinfo { year} { 1991} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Kucharski} \ and\ \citenamefont
{ Bartlett} (1992)} ]{ Kucharski_ 1992} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { S.~A.} \ \bibnamefont
{ Kucharski} } \ and\ \bibinfo { author} { \bibfnamefont { R.~J.} \ \bibnamefont
{ Bartlett} } ,\ } \href { \doibase 10.1063/1.463930} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 97} } ,\
\bibinfo { pages} { 4282} (\bibinfo { year} { 1992} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Eriksen} \ \emph { et~al.} (2017)\citenamefont
{ Eriksen} , \citenamefont { Lipparini} ,\ and\ \citenamefont
{ Gauss} } ]{ Eriksen_ 2017} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~J.} \ \bibnamefont
{ Eriksen} } , \bibinfo { author} { \bibfnamefont { F.} ~\bibnamefont { Lipparini} } ,
\ and\ \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont { Gauss} } ,\ } \href
{ \doibase 10.1021/acs.jpclett.7b02075} { \bibfield { journal} { \bibinfo
{ journal} { J. Phys. Chem. Lett.} \ } \textbf { \bibinfo { volume} { 8} } ,\ \bibinfo
{ pages} { 4633} (\bibinfo { year} { 2017} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Eriksen} \ and\ \citenamefont
{ Gauss} (2018)} ]{ Eriksen_ 2018} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~J.} \ \bibnamefont
{ Eriksen} } \ and\ \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont
{ Gauss} } ,\ } \href { \doibase 10.1021/acs.jctc.8b00680} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume}
{ 14} } ,\ \bibinfo { pages} { 5180} (\bibinfo { year} { 2018} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Eriksen} \ and\ \citenamefont
{ Gauss} (2019{ \natexlab { a} } )} ]{ Eriksen_ 2019a} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~J.} \ \bibnamefont
{ Eriksen} } \ and\ \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont
{ Gauss} } ,\ } \href { \doibase 10.1021/acs.jctc.9b00456} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume}
{ 15} } ,\ \bibinfo { pages} { 4873} (\bibinfo { year}
{ 2019} { \natexlab { a} } )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Eriksen} \ and\ \citenamefont
{ Gauss} (2019{ \natexlab { b} } )} ]{ Eriksen_ 2019b} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~J.} \ \bibnamefont
{ Eriksen} } \ and\ \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont
{ Gauss} } ,\ } \href { \doibase 10.1021/acs.jpclett.9b02968} { \bibfield
{ journal} { \bibinfo { journal} { J. Phys. Chem. Lett.} \ } \textbf { \bibinfo
{ volume} { 27} } ,\ \bibinfo { pages} { 7910} (\bibinfo { year}
{ 2019} { \natexlab { b} } )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Tubman} \ \emph { et~al.} (2016)\citenamefont { Tubman} ,
\citenamefont { Lee} , \citenamefont { Takeshita} , \citenamefont
{ { Head-Gordon} } ,\ and\ \citenamefont { Whaley} } ]{ Tubman_ 2016} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { N.~M.} \ \bibnamefont
{ Tubman} } , \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont { Lee} } ,
\bibinfo { author} { \bibfnamefont { T.~Y.} \ \bibnamefont { Takeshita} } , \bibinfo
{ author} { \bibfnamefont { M.} ~\bibnamefont { { Head-Gordon} } } , \ and\ \bibinfo
{ author} { \bibfnamefont { K.~B.} \ \bibnamefont { Whaley} } ,\ } \href { \doibase
10.1063/1.4955109} { \bibfield { journal} { \bibinfo { journal} { J. Chem.
Phys.} \ } \textbf { \bibinfo { volume} { 145} } ,\ \bibinfo { pages} { 044112}
(\bibinfo { year} { 2016} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Tubman} \ \emph { et~al.} (2018)\citenamefont { Tubman} ,
\citenamefont { Levine} , \citenamefont { Hait} , \citenamefont { Head-Gordon} ,\
and\ \citenamefont { Whaley} } ]{ Tubman_ 2018} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { N.~M.} \ \bibnamefont
{ Tubman} } , \bibinfo { author} { \bibfnamefont { D.~S.} \ \bibnamefont { Levine} } ,
\bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont { Hait} } , \bibinfo { author}
{ \bibfnamefont { M.} ~\bibnamefont { Head-Gordon} } , \ and\ \bibinfo { author}
{ \bibfnamefont { K.~B.} \ \bibnamefont { Whaley} } ,\ } \href @noop { } { \enquote
{ \bibinfo { title} { An efficient deterministic perturbation theory for
selected configuration interaction methods} ,} \ } (\bibinfo { year} { 2018} ),\
\Eprint { http://arxiv.org/abs/1808.02049} { arXiv:1808.02049
[cond-mat.str-el]} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Tubman} \ \emph { et~al.} (2020)\citenamefont { Tubman} ,
\citenamefont { Freeman} , \citenamefont { Levine} , \citenamefont { Hait} ,
\citenamefont { Head-Gordon} ,\ and\ \citenamefont { Whaley} } ]{ Tubman_ 2020} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { N.~M.} \ \bibnamefont
{ Tubman} } , \bibinfo { author} { \bibfnamefont { C.~D.} \ \bibnamefont { Freeman} } ,
\bibinfo { author} { \bibfnamefont { D.~S.} \ \bibnamefont { Levine} } , \bibinfo
{ author} { \bibfnamefont { D.} ~\bibnamefont { Hait} } , \bibinfo { author}
{ \bibfnamefont { M.} ~\bibnamefont { Head-Gordon} } , \ and\ \bibinfo { author}
{ \bibfnamefont { K.~B.} \ \bibnamefont { Whaley} } ,\ } \href { \doibase
10.1021/acs.jctc.8b00536} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 16} } ,\ \bibinfo { pages}
{ 2139} (\bibinfo { year} { 2020} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Liu} \ and\ \citenamefont
{ Hoffmann} (2014)} ]{ Liu_ 2014} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont
{ Liu} } \ and\ \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Hoffmann} } ,\
} \href { \doibase 10.1007/s00214-014-1481-x} { \bibfield { journal} { \bibinfo
{ journal} { Theor. Chem. Acc.} \ } \textbf { \bibinfo { volume} { 133} } ,\ \bibinfo
{ pages} { 1481} (\bibinfo { year} { 2014} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Liu} \ and\ \citenamefont
{ Hoffmann} (2016)} ]{ Liu_ 2016} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont
{ Liu} } \ and\ \bibinfo { author} { \bibfnamefont { M.~R.} \ \bibnamefont
{ Hoffmann} } ,\ } \href { \doibase 10.1021/acs.jctc.5b01099} { \bibfield
{ journal} { \bibinfo { journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo
{ volume} { 12} } ,\ \bibinfo { pages} { 1169} (\bibinfo { year}
{ 2016} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Lei} \ \emph { et~al.} (2017)\citenamefont { Lei} ,
\citenamefont { Liu} ,\ and\ \citenamefont { Hoffmann} } ]{ Lei_ 2017} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { Y.} ~\bibnamefont
{ Lei} } , \bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont { Liu} } , \ and\
\bibinfo { author} { \bibfnamefont { M.~R.} \ \bibnamefont { Hoffmann} } ,\ } \href
{ \doibase 10.1080/00268976.2017.1308029} { \bibfield { journal} { \bibinfo
{ journal} { Mol. Phys.} \ } \textbf { \bibinfo { volume} { 115} } ,\ \bibinfo { pages}
{ 2696} (\bibinfo { year} { 2017} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Zhang} \ \emph { et~al.} (2020)\citenamefont { Zhang} ,
\citenamefont { Liu} ,\ and\ \citenamefont { Hoffmann} } ]{ Zhang_ 2020} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { N.} ~\bibnamefont
{ Zhang} } , \bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont { Liu} } , \ and\
\bibinfo { author} { \bibfnamefont { M.~R.} \ \bibnamefont { Hoffmann} } ,\ } \href
{ \doibase 10.1021/acs.jctc.9b01200} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 16} } ,\
\bibinfo { pages} { 2296} (\bibinfo { year} { 2020} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Holmes} \ \emph { et~al.} (2016)\citenamefont { Holmes} ,
\citenamefont { Changlani} ,\ and\ \citenamefont { Umrigar} } ]{ Holmes_ 2016} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { A.~A.} \ \bibnamefont
{ Holmes} } , \bibinfo { author} { \bibfnamefont { H.~J.} \ \bibnamefont
{ Changlani} } , \ and\ \bibinfo { author} { \bibfnamefont { C.~J.} \ \bibnamefont
{ Umrigar} } ,\ } \href { \doibase 10.1021/acs.jctc.5b01170} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume}
{ 12} } ,\ \bibinfo { pages} { 1561} (\bibinfo { year} { 2016} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Holmes} \ \emph { et~al.} (2017)\citenamefont { Holmes} ,
\citenamefont { Umrigar} ,\ and\ \citenamefont { Sharma} } ]{ Holmes_ 2017} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { A.~A.} \ \bibnamefont
{ Holmes} } , \bibinfo { author} { \bibfnamefont { C.~J.} \ \bibnamefont { Umrigar} } ,
\ and\ \bibinfo { author} { \bibfnamefont { S.} ~\bibnamefont { Sharma} } ,\ } \href
{ \doibase 10.1063/1.4998614} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Phys.} \ } \textbf { \bibinfo { volume} { 147} } ,\ \bibinfo { pages} { 164111}
(\bibinfo { year} { 2017} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Sharma} \ \emph { et~al.} (2017)\citenamefont { Sharma} ,
\citenamefont { Holmes} , \citenamefont { Jeanmairet} , \citenamefont { Alavi} ,\
and\ \citenamefont { Umrigar} } ]{ Sharma_ 2017} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { S.} ~\bibnamefont
{ Sharma} } , \bibinfo { author} { \bibfnamefont { A.~A.} \ \bibnamefont { Holmes} } ,
\bibinfo { author} { \bibfnamefont { G.} ~\bibnamefont { Jeanmairet} } , \bibinfo
{ author} { \bibfnamefont { A.} ~\bibnamefont { Alavi} } , \ and\ \bibinfo { author}
{ \bibfnamefont { C.~J.} \ \bibnamefont { Umrigar} } ,\ } \href { \doibase
10.1021/acs.jctc.6b01028} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 13} } ,\ \bibinfo { pages}
{ 1595} (\bibinfo { year} { 2017} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Xu} \ \emph { et~al.} (2018)\citenamefont { Xu} ,
\citenamefont { Uejima} ,\ and\ \citenamefont { Ten-no} } ]{ Xu_ 2018} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont
{ Xu} } , \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Uejima} } , \ and\
\bibinfo { author} { \bibfnamefont { S.~L.} \ \bibnamefont { Ten-no} } ,\ } \href
{ \doibase 10.1103/PhysRevLett.121.113001} { \bibfield { journal} { \bibinfo
{ journal} { Phys. Rev. Lett.} \ } \textbf { \bibinfo { volume} { 121} } ,\ \bibinfo
{ pages} { 113001} (\bibinfo { year} { 2018} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Xu} \ \emph { et~al.} (2020)\citenamefont { Xu} ,
\citenamefont { Uejima} ,\ and\ \citenamefont { Ten-no} } ]{ Xu_ 2020} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont
{ Xu} } , \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Uejima} } , \ and\
\bibinfo { author} { \bibfnamefont { S.~L.} \ \bibnamefont { Ten-no} } ,\
} \href @noop { } { \enquote { \bibinfo { title} { Towards near-exact solutions of
molecular electronic structure: Full coupled-cluster reduction with a
second-order perturbative correction} ,} \ } (\bibinfo { year} { 2020} ),\ \Eprint
{ http://arxiv.org/abs/2010.01850} { arXiv:2010.01850 [physics.chem-ph]}
\BibitemShut { NoStop} %
\bibitem [{ \citenamefont { White} (1992)} ]{ White_ 1992} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { S.~R.} \ \bibnamefont
{ White} } ,\ } \href { \doibase 10.1103/PhysRevLett.69.2863} { \bibfield
{ journal} { \bibinfo { journal} { Phys. Rev. Lett.} \ } \textbf { \bibinfo
{ volume} { 69} } ,\ \bibinfo { pages} { 2863} (\bibinfo { year}
{ 1992} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { White} (1993)} ]{ White_ 1993} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { S.~R.} \ \bibnamefont
{ White} } ,\ } \href { \doibase 10.1103/PhysRevB.48.10345} { \bibfield { journal}
{ \bibinfo { journal} { Phys. Rev. B} \ } \textbf { \bibinfo { volume} { 48} } ,\
\bibinfo { pages} { 10345} (\bibinfo { year} { 1993} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Chan} \ and\ \citenamefont
{ Sharma} (2011)} ]{ Chan_ 2011} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { G.~K.-L.} \
\bibnamefont { Chan} } \ and\ \bibinfo { author} { \bibfnamefont { S.} ~\bibnamefont
{ Sharma} } ,\ } \href { \doibase 10.1146/annurev-physchem-032210-103338}
{ \bibfield { journal} { \bibinfo { journal} { Annu. Rev. Phys. Chem.} \ } \textbf
{ \bibinfo { volume} { 62} } ,\ \bibinfo { pages} { 465} (\bibinfo { year}
{ 2011} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Booth} \ \emph { et~al.} (2009)\citenamefont { Booth} ,
\citenamefont { Thom} ,\ and\ \citenamefont { Alavi} } ]{ Booth_ 2009} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { G.~H.} \ \bibnamefont
{ Booth} } , \bibinfo { author} { \bibfnamefont { A.~J.~W.} \ \bibnamefont { Thom} } ,
\ and\ \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Alavi} } ,\ } \href
{ \doibase 10.1063/1.3193710} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Phys.} \ } \textbf { \bibinfo { volume} { 131} } ,\ \bibinfo { pages} { 054106}
(\bibinfo { year} { 2009} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Cleland} \ \emph { et~al.} (2010)\citenamefont
{ Cleland} , \citenamefont { Booth} ,\ and\ \citenamefont
{ Alavi} } ]{ Cleland_ 2010} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont
{ Cleland} } , \bibinfo { author} { \bibfnamefont { G.~H.} \ \bibnamefont { Booth} } ,
\ and\ \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Alavi} } ,\ } \href
{ \doibase 10.1063/1.3302277} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Phys.} \ } \textbf { \bibinfo { volume} { 132} } ,\ \bibinfo { pages} { 041103}
(\bibinfo { year} { 2010} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Ghanem} \ \emph { et~al.} (2019)\citenamefont { Ghanem} ,
\citenamefont { Lozovoi} ,\ and\ \citenamefont { Alavi} } ]{ Ghanem_ 2019} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { K.} ~\bibnamefont
{ Ghanem} } , \bibinfo { author} { \bibfnamefont { A.~Y.} \ \bibnamefont { Lozovoi} } ,
\ and\ \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Alavi} } ,\ } \href
{ \doibase 10.1063/1.5134006} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Phys.} \ } \textbf { \bibinfo { volume} { 151} } ,\ \bibinfo { pages} { 224108}
(\bibinfo { year} { 2019} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Deustua} \ \emph { et~al.} (2017)\citenamefont
{ Deustua} , \citenamefont { Shen} ,\ and\ \citenamefont
{ Piecuch} } ]{ Deustua_ 2017} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~E.} \ \bibnamefont
{ Deustua} } , \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont { Shen} } , \
and\ \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont { Piecuch} } ,\ } \href
{ \doibase 10.1103/PhysRevLett.119.223003} { \bibfield { journal} { \bibinfo
{ journal} { Phys. Rev. Lett.} \ } \textbf { \bibinfo { volume} { 119} } ,\ \bibinfo
{ pages} { 223003} (\bibinfo { year} { 2017} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Deustua} \ \emph { et~al.} (2018)\citenamefont
{ Deustua} , \citenamefont { Magoulas} , \citenamefont { Shen} ,\ and\
\citenamefont { Piecuch} } ]{ Deustua_ 2018} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~E.} \ \bibnamefont
{ Deustua} } , \bibinfo { author} { \bibfnamefont { I.} ~\bibnamefont { Magoulas} } ,
\bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont { Shen} } , \ and\ \bibinfo
{ author} { \bibfnamefont { P.} ~\bibnamefont { Piecuch} } ,\ } \href { \doibase
10.1063/1.5055769} { \bibfield { journal} { \bibinfo { journal} { J. Chem.
Phys.} \ } \textbf { \bibinfo { volume} { 149} } ,\ \bibinfo { pages} { 151101}
(\bibinfo { year} { 2018} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Motta} \ and\ \citenamefont
{ Zhang} (2018)} ]{ Motta_ 2018} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Motta} } \ and\ \bibinfo { author} { \bibfnamefont { S.} ~\bibnamefont { Zhang} } ,\
} \href { \doibase 10.1002/wcms.1364} { \bibfield { journal} { \bibinfo
{ journal} { WIREs Comput. Mol. Sci.} \ } \textbf { \bibinfo { volume} { 8} } ,\
\bibinfo { pages} { e1364} (\bibinfo { year} { 2018} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Lee} \ \emph { et~al.} (2020)\citenamefont { Lee} ,
\citenamefont { Malone} ,\ and\ \citenamefont { Reichman} } ]{ Lee_ 2020} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont
{ Lee} } , \bibinfo { author} { \bibfnamefont { F.~D.} \ \bibnamefont { Malone} } , \
and\ \bibinfo { author} { \bibfnamefont { D.~R.} \ \bibnamefont { Reichman} } ,\
} \href { \doibase 10.1063/5.0024835} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 153} } ,\ \bibinfo
{ pages} { 126101} (\bibinfo { year} { 2020} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Evangelisti} \ \emph { et~al.} (1983)\citenamefont
{ Evangelisti} , \citenamefont { Daudey} ,\ and\ \citenamefont
{ Malrieu} } ]{ Evangelisti_ 1983} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { S.} ~\bibnamefont
{ Evangelisti} } , \bibinfo { author} { \bibfnamefont { J.-P.} \ \bibnamefont
{ Daudey} } , \ and\ \bibinfo { author} { \bibfnamefont { J.-P.} \ \bibnamefont
{ Malrieu} } ,\ } \href { \doibase 10.1016/0301-0104(83)85011-3} { \bibfield
{ journal} { \bibinfo { journal} { Chem. Phys.} \ } \textbf { \bibinfo { volume}
{ 75} } ,\ \bibinfo { pages} { 91} (\bibinfo { year} { 1983} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Giner} \ \emph { et~al.} (2013)\citenamefont { Giner} ,
\citenamefont { Scemama} ,\ and\ \citenamefont { Caffarel} } ]{ Giner_ 2013} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont
{ Giner} } , \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } , \
and\ \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Caffarel} } ,\ } \href
{ \doibase 10.1139/cjc-2013-0017} { \bibfield { journal} { \bibinfo { journal}
{ Can. J. Chem.} \ } \textbf { \bibinfo { volume} { 91} } ,\ \bibinfo { pages} { 879}
(\bibinfo { year} { 2013} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Giner} \ \emph { et~al.} (2015)\citenamefont { Giner} ,
\citenamefont { Scemama} ,\ and\ \citenamefont { Caffarel} } ]{ Giner_ 2015} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont
{ Giner} } , \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } , \
and\ \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Caffarel} } ,\ } \href
{ \doibase 10.1063/1.4905528} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Phys.} \ } \textbf { \bibinfo { volume} { 142} } ,\ \bibinfo { pages} { 044115}
(\bibinfo { year} { 2015} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Garniron} \ \emph { et~al.} (2017)\citenamefont
{ Garniron} , \citenamefont { Scemama} , \citenamefont { Loos} ,\ and\
\citenamefont { Caffarel} } ]{ Garniron_ 2017} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { Y.} ~\bibnamefont
{ Garniron} } , \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } ,
\bibinfo { author} { \bibfnamefont { P.-F.} \ \bibnamefont { Loos} } , \ and\
\bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Caffarel} } ,\ } \href
{ \doibase 10.1063/1.4992127} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Phys.} \ } \textbf { \bibinfo { volume} { 147} } ,\ \bibinfo { pages} { 034101}
(\bibinfo { year} { 2017} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Garniron} \ \emph { et~al.} (2018)\citenamefont
{ Garniron} , \citenamefont { Scemama} , \citenamefont { Giner} , \citenamefont
{ Caffarel} ,\ and\ \citenamefont { Loos} } ]{ Garniron_ 2018} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { Y.} ~\bibnamefont
{ Garniron} } , \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } ,
\bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont { Giner} } , \bibinfo
{ author} { \bibfnamefont { M.} ~\bibnamefont { Caffarel} } , \ and\ \bibinfo
{ author} { \bibfnamefont { P.~F.} \ \bibnamefont { Loos} } ,\ } \href { \doibase
10.1063/1.5044503} { \bibfield { journal} { \bibinfo { journal} { J. Chem.
Phys.} \ } \textbf { \bibinfo { volume} { 149} } ,\ \bibinfo { pages} { 064103}
(\bibinfo { year} { 2018} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Garniron} \ \emph { et~al.} (2019)\citenamefont
{ Garniron} , \citenamefont { Applencourt} , \citenamefont { Gasperich} ,
\citenamefont { Benali} , \citenamefont { Fert{ \' e} } , \citenamefont { Paquier} ,
\citenamefont { Pradines} , \citenamefont { Assaraf} , \citenamefont { Reinhardt} ,
\citenamefont { Toulouse} , \citenamefont { Barbaresco} , \citenamefont { Renon} ,
\citenamefont { David} , \citenamefont { Malrieu} , \citenamefont { V{ \' e} ril} ,
\citenamefont { Caffarel} , \citenamefont { Loos} , \citenamefont { Giner} ,\ and\
\citenamefont { Scemama} } ]{ Garniron_ 2019} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { Y.} ~\bibnamefont
{ Garniron} } , \bibinfo { author} { \bibfnamefont { T.} ~\bibnamefont
{ Applencourt} } , \bibinfo { author} { \bibfnamefont { K.} ~\bibnamefont
{ Gasperich} } , \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Benali} } ,
\bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Fert{ \' e} } } , \bibinfo
{ author} { \bibfnamefont { J.} ~\bibnamefont { Paquier} } , \bibinfo { author}
{ \bibfnamefont { B.} ~\bibnamefont { Pradines} } , \bibinfo { author}
{ \bibfnamefont { R.} ~\bibnamefont { Assaraf} } , \bibinfo { author} { \bibfnamefont
{ P.} ~\bibnamefont { Reinhardt} } , \bibinfo { author} { \bibfnamefont
{ J.} ~\bibnamefont { Toulouse} } , \bibinfo { author} { \bibfnamefont
{ P.} ~\bibnamefont { Barbaresco} } , \bibinfo { author} { \bibfnamefont
{ N.} ~\bibnamefont { Renon} } , \bibinfo { author} { \bibfnamefont
{ G.} ~\bibnamefont { David} } , \bibinfo { author} { \bibfnamefont { J.-P.} \
\bibnamefont { Malrieu} } , \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ V{ \' e} ril} } , \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Caffarel} } ,
\bibinfo { author} { \bibfnamefont { P.-F.} \ \bibnamefont { Loos} } , \bibinfo
{ author} { \bibfnamefont { E.} ~\bibnamefont { Giner} } , \ and\ \bibinfo { author}
{ \bibfnamefont { A.} ~\bibnamefont { Scemama} } ,\ } \href { \doibase
10.1021/acs.jctc.9b00176} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 15} } ,\ \bibinfo { pages}
{ 3591} (\bibinfo { year} { 2019} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Caffarel} \ \emph { et~al.} (2014)\citenamefont
{ Caffarel} , \citenamefont { Giner} , \citenamefont { Scemama} ,\ and\
\citenamefont { Ram{ \' \i } rez-Sol{ \' \i } s} } ]{ Caffarel_ 2014} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Caffarel} } , \bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont { Giner} } ,
\bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } , \ and\
\bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont
{ Ram{ \' \i } rez-Sol{ \' \i } s} } ,\ } \href { \doibase 10.1021/ct5004252} { \bibfield
{ journal} { \bibinfo { journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo
{ volume} { 10} } ,\ \bibinfo { pages} { 5286} (\bibinfo { year}
{ 2014} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Caffarel} \ \emph { et~al.} (2016)\citenamefont
{ Caffarel} , \citenamefont { Applencourt} , \citenamefont { Giner} ,\ and\
\citenamefont { Scemama} } ]{ Caffarel_ 2016a} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Caffarel} } , \bibinfo { author} { \bibfnamefont { T.} ~\bibnamefont
{ Applencourt} } , \bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont { Giner} } ,
\ and\ \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } ,\ } \href
{ \doibase 10.1063/1.4947093} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Phys.} \ } \textbf { \bibinfo { volume} { 144} } ,\ \bibinfo { pages} { 151103}
(\bibinfo { year} { 2016} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Caffarel} \ \emph { et~al.} ()\citenamefont { Caffarel} ,
\citenamefont { Applencourt} , \citenamefont { Giner} ,\ and\ \citenamefont
{ Scemama} } ]{ Caffarel_ 2016b} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Caffarel} } , \bibinfo { author} { \bibfnamefont { T.} ~\bibnamefont
{ Applencourt} } , \bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont { Giner} } ,
\ and\ \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } ,\
} \enquote { \bibinfo { title} { Using cipsi nodes in diffusion monte carlo} ,} \
in\ \href { \doibase 10.1021/bk-2016-1234.ch002} { \emph { \bibinfo { booktitle}
{ Recent Progress in Quantum Monte Carlo} } } ,\ Chap.~\bibinfo { chapter} { 2} ,
pp.\ \bibinfo { pages} { 15--46} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Scemama} \ \emph { et~al.} (2015)\citenamefont
{ Scemama} , \citenamefont { Giner} , \citenamefont { Applencourt} ,\ and\
\citenamefont { Caffarel} } ]{ Scemama_ 2015} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont
{ Scemama} } , \bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont { Giner} } ,
\bibinfo { author} { \bibfnamefont { T.} ~\bibnamefont { Applencourt} } , \ and\
\bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Caffarel} } ,\ } \href
{ \doibase 10.13140/RG.2.1.3187.9766} { \enquote { \bibinfo { title} { { QMC using
very large configuration interaction-type expansions} } ,} \ } \bibinfo
{ howpublished} { Pacifichem, Advances in Quantum Monte Carlo} (\bibinfo { year}
{ 2015} )\BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Scemama} \ \emph { et~al.} (2016)\citenamefont
{ Scemama} , \citenamefont { Applencourt} , \citenamefont { Giner} ,\ and\
\citenamefont { Caffarel} } ]{ Scemama_ 2016} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont
{ Scemama} } , \bibinfo { author} { \bibfnamefont { T.} ~\bibnamefont
{ Applencourt} } , \bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont { Giner} } ,
\ and\ \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Caffarel} } ,\
} \href { \doibase 10.1002/jcc.24382} { \bibfield { journal} { \bibinfo
{ journal} { J. Comput. Chem.} \ } \textbf { \bibinfo { volume} { 37} } ,\ \bibinfo
{ pages} { 1866} (\bibinfo { year} { 2016} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Scemama} \ \emph
{ et~al.} (2018{ \natexlab { a} } )\citenamefont { Scemama} , \citenamefont
{ Garniron} , \citenamefont { Caffarel} ,\ and\ \citenamefont
{ Loos} } ]{ Scemama_ 2018} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont
{ Scemama} } , \bibinfo { author} { \bibfnamefont { Y.} ~\bibnamefont { Garniron} } ,
\bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Caffarel} } , \ and\
\bibinfo { author} { \bibfnamefont { P.-F.} \ \bibnamefont { Loos} } ,\ } \href
{ \doibase 10.1021/acs.jctc.7b01250} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 14} } ,\
\bibinfo { pages} { 1395} (\bibinfo { year} { 2018} { \natexlab { a} } )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Scemama} \ \emph
{ et~al.} (2018{ \natexlab { b} } )\citenamefont { Scemama} , \citenamefont { Benali} ,
\citenamefont { Jacquemin} , \citenamefont { Caffarel} ,\ and\ \citenamefont
{ Loos} } ]{ Scemama_ 2018b} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont
{ Scemama} } , \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Benali} } ,
\bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont { Jacquemin} } , \bibinfo
{ author} { \bibfnamefont { M.} ~\bibnamefont { Caffarel} } , \ and\ \bibinfo
{ author} { \bibfnamefont { P.-F.} \ \bibnamefont { Loos} } ,\ } \href { \doibase
10.1063/1.5041327} { \bibfield { journal} { \bibinfo { journal} { J. Chem.
Phys.} \ } \textbf { \bibinfo { volume} { 149} } ,\ \bibinfo { pages} { 034108}
(\bibinfo { year} { 2018} { \natexlab { b} } )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Scemama} \ \emph { et~al.} (2019)\citenamefont
{ Scemama} , \citenamefont { Caffarel} , \citenamefont { Benali} , \citenamefont
{ Jacquemin} ,\ and\ \citenamefont { Loos.} } ]{ Scemama_ 2019} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont
{ Scemama} } , \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Caffarel} } ,
\bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Benali} } , \bibinfo
{ author} { \bibfnamefont { D.} ~\bibnamefont { Jacquemin} } , \ and\ \bibinfo
{ author} { \bibfnamefont { P.~F.} \ \bibnamefont { Loos.} } ,\ } \href { \doibase
10.1016/j.rechem.2019.100002} { \bibfield { journal} { \bibinfo { journal}
{ Res. Chem.} \ } \textbf { \bibinfo { volume} { 1} } ,\ \bibinfo { pages} { 100002}
(\bibinfo { year} { 2019} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Dash} \ \emph { et~al.} (2018)\citenamefont { Dash} ,
\citenamefont { Moroni} , \citenamefont { Scemama} ,\ and\ \citenamefont
{ Filippi} } ]{ Dash_ 2018} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Dash} } , \bibinfo { author} { \bibfnamefont { S.} ~\bibnamefont { Moroni} } ,
\bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } , \ and\
\bibinfo { author} { \bibfnamefont { C.} ~\bibnamefont { Filippi} } ,\ } \href
{ \doibase 10.1021/acs.jctc.8b00393} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 14} } ,\
\bibinfo { pages} { 4176} (\bibinfo { year} { 2018} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Dash} \ \emph { et~al.} (2019)\citenamefont { Dash} ,
\citenamefont { Feldt} , \citenamefont { Moroni} , \citenamefont { Scemama} ,\ and\
\citenamefont { Filippi} } ]{ Dash_ 2019} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Dash} } , \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont { Feldt} } ,
\bibinfo { author} { \bibfnamefont { S.} ~\bibnamefont { Moroni} } , \bibinfo
{ author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } , \ and\ \bibinfo
{ author} { \bibfnamefont { C.} ~\bibnamefont { Filippi} } ,\ } \href { \doibase
10.1021/acs.jctc.9b00476} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 15} } ,\ \bibinfo { pages}
{ 4896} (\bibinfo { year} { 2019} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Applencourt} \ \emph { et~al.} (2018)\citenamefont
{ Applencourt} , \citenamefont { Gasperich} ,\ and\ \citenamefont
{ Scemama} } ]{ Applencourt_ 2018} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { T.} ~\bibnamefont
{ Applencourt} } , \bibinfo { author} { \bibfnamefont { K.} ~\bibnamefont
{ Gasperich} } , \ and\ \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont
{ Scemama} } ,\ } \href @noop { } { \enquote { \bibinfo { title} { Spin adaptation with
determinant-based selected configuration interaction} ,} \ } (\bibinfo { year}
{ 2018} ),\ \Eprint { http://arxiv.org/abs/1812.06902} { arXiv:1812.06902
[physics.chem-ph]} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Foster} \ and\ \citenamefont
{ Boys} (1960)} ]{ Boys_ 1960} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~M.} \ \bibnamefont
{ Foster} } \ and\ \bibinfo { author} { \bibfnamefont { S.~F.} \ \bibnamefont
{ Boys} } ,\ } \href { \doibase 10.1103/RevModPhys.32.300} { \bibfield { journal}
{ \bibinfo { journal} { Rev. Mod. Phys.} \ } \textbf { \bibinfo { volume} { 32} } ,\
\bibinfo { pages} { 300} (\bibinfo { year} { 1960} )} \BibitemShut { NoStop} %
\bibitem [{ Note1()} ]{ Note1} %
\BibitemOpen
\bibinfo { note} { Indices of molecular orbitals for Boys-Foster localization
procedure: core [1--6]; $ \sigma $ [7--18]; $ \pi $ [19--21]; $ \pi ^ * $
[22--24]; $ \sigma ^ * $ [25--36]; higher-lying $ \pi $
[39,41--43,46,49,50,53--57,71--74,82--85,87,92,93,98]; higher-lying $ \sigma $
[37,38,40,44,45,47,48,51,52,58--70,75--81,86,88--91,94--97,99--114].} \BibitemShut
{ Stop} %
\bibitem [{ \citenamefont { Pipek} \ and\ \citenamefont
{ Mezey} (1989)} ]{ Pipek_ 1989} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont
{ Pipek} } \ and\ \bibinfo { author} { \bibfnamefont { P.~G.} \ \bibnamefont
{ Mezey} } ,\ } \href { \doibase 10.1063/1.456588} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 90} } ,\ \bibinfo
{ pages} { 4916} (\bibinfo { year} { 1989} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Angeli} \ \emph { et~al.} (2003)\citenamefont { Angeli} ,
\citenamefont { Calzado} , \citenamefont { Cimiraglia} , \citenamefont
{ Evangelisti} , \citenamefont { Guih\' ery} , \citenamefont { Leininger} ,
\citenamefont { Malrieu} , \citenamefont { Maynau} , \citenamefont { Ruiz} ,\ and\
\citenamefont { Sparta} } ]{ Angeli_ 2003} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { C.} ~\bibnamefont
{ Angeli} } , \bibinfo { author} { \bibfnamefont { C.~J.} \ \bibnamefont { Calzado} } ,
\bibinfo { author} { \bibfnamefont { R.} ~\bibnamefont { Cimiraglia} } , \bibinfo
{ author} { \bibfnamefont { S.} ~\bibnamefont { Evangelisti} } , \bibinfo { author}
{ \bibfnamefont { N.} ~\bibnamefont { Guih\' ery} } , \bibinfo { author}
{ \bibfnamefont { T.} ~\bibnamefont { Leininger} } , \bibinfo { author}
{ \bibfnamefont { J.-P.} \ \bibnamefont { Malrieu} } , \bibinfo { author}
{ \bibfnamefont { D.} ~\bibnamefont { Maynau} } , \bibinfo { author} { \bibfnamefont
{ J.~V.~P.} \ \bibnamefont { Ruiz} } , \ and\ \bibinfo { author} { \bibfnamefont
{ M.} ~\bibnamefont { Sparta} } ,\ } \href { \doibase 10.1080/0026897031000082149}
{ \bibfield { journal} { \bibinfo { journal} { Mol. Phys.} \ } \textbf { \bibinfo
{ volume} { 101} } ,\ \bibinfo { pages} { 1389} (\bibinfo { year}
{ 2003} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Angeli} (2009)} ]{ Angeli_ 2009} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { C.} ~\bibnamefont
{ Angeli} } ,\ } \href { \doibase 10.1002/jcc.21155} { \bibfield { journal}
{ \bibinfo { journal} { J. Comput. Chem.} \ } \textbf { \bibinfo { volume} { 30} } ,\
\bibinfo { pages} { 1319} (\bibinfo { year} { 2009} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Ben~Amor} \ \emph { et~al.} (2011)\citenamefont
{ Ben~Amor} , \citenamefont { Bessac} , \citenamefont { Hoyau} ,\ and\
\citenamefont { Maynau} } ]{ BenAmor_ 2011} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { N.} ~\bibnamefont
{ Ben~Amor} } , \bibinfo { author} { \bibfnamefont { F.} ~\bibnamefont { Bessac} } ,
\bibinfo { author} { \bibfnamefont { S.} ~\bibnamefont { Hoyau} } , \ and\ \bibinfo
{ author} { \bibfnamefont { D.} ~\bibnamefont { Maynau} } ,\ } \href { \doibase
10.1063/1.3600351} { \bibfield { journal} { \bibinfo { journal} { J. Chem.
Phys.} \ } \textbf { \bibinfo { volume} { 135} } ,\ \bibinfo { pages} { 014101}
(\bibinfo { year} { 2011} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Suaud} \ and\ \citenamefont
{ Malrieu} (2017)} ]{ Suaud_ 2017} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { N.} ~\bibnamefont
{ Suaud} } \ and\ \bibinfo { author} { \bibfnamefont { J.-P.} \ \bibnamefont
{ Malrieu} } ,\ } \href { \doibase 10.1080/00268976.2017.1303207} { \bibfield
{ journal} { \bibinfo { journal} { Mol. Phys.} \ } \textbf { \bibinfo { volume}
{ 115} } ,\ \bibinfo { pages} { 2684} (\bibinfo { year} { 2017} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Chien} \ \emph { et~al.} (2018)\citenamefont { Chien} ,
\citenamefont { Holmes} , \citenamefont { Otten} , \citenamefont { Umrigar} ,
\citenamefont { Sharma} ,\ and\ \citenamefont { Zimmerman} } ]{ Chien_ 2018} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { A.~D.} \ \bibnamefont
{ Chien} } , \bibinfo { author} { \bibfnamefont { A.~A.} \ \bibnamefont { Holmes} } ,
\bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Otten} } , \bibinfo
{ author} { \bibfnamefont { C.~J.} \ \bibnamefont { Umrigar} } , \bibinfo { author}
{ \bibfnamefont { S.} ~\bibnamefont { Sharma} } , \ and\ \bibinfo { author}
{ \bibfnamefont { P.~M.} \ \bibnamefont { Zimmerman} } ,\ } \href { \doibase
10.1021/acs.jpca.8b01554} { \bibfield { journal} { \bibinfo { journal} { J.
Phys. Chem. A} \ } \textbf { \bibinfo { volume} { 122} } ,\ \bibinfo { pages} { 2714}
(\bibinfo { year} { 2018} )} \BibitemShut { NoStop} %
\bibitem [{ Note2()} ]{ Note2} %
\BibitemOpen
2020-10-20 21:30:07 +02:00
\bibinfo { note} { \protect \leavevmode { \protect Using the
2020-10-12 19:34:02 +02:00
3, 4, 5, and 6 largest wave functions to perform the linear extrapolation
yield the following correlation energy estimates: $ - 863 . 1 ( 11 ) $ , $ - 863 . 4 ( 5 ) $ ,
$ - 862 . 1 ( 8 ) $ , and $ - 863 . 5 ( 11 ) $ mE$ _ h $ , respectively. These numbers vary by
$ 1 . 4 $ mE$ _ h $ . The four-point extrapolated value of $ - 863 . 4 ( 5 ) $ mE$ _ h $ that we
have chosen to report as our best estimate corresponds to the smallest
fitting error. Quadratic fits yield much larger variations and are discarded
in practice. Due to the stochastic nature of $ E _ \protect \text { rPT 2 } $ , the
fifth point is slightly off as compared to the others. Taking into account
this fifth point yield a slightly smaller estimate of the correlation energy
[$ - 862 . 1 ( 8 ) $ mE$ _ h $ ], while adding a sixth point settles down the correlation
energy estimate at $ - 863 . 5 ( 11 ) $ mE$ _ h $ } } \BibitemShut { NoStop} %
\end { thebibliography} %
2020-08-18 18:27:34 +02:00
2020-08-23 01:31:48 +02:00
\end { document}