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@ -1,13 +1,136 @@
%% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-03-21 21:58:49 +0100
%% Created for Pierre-Francois Loos at 2022-03-23 10:49:10 +0100
%% Saved with string encoding Unicode (UTF-8)
@article{Levine_2006,
author = {Levine, Benjamin G. and Ko, Chaehyuk and Quenneville, Jason and Mart\'inez, Todd J.},
date-added = {2022-03-23 10:49:10 +0100},
date-modified = {2022-03-23 10:49:10 +0100},
doi = {10.1080/00268970500417762},
issn = {0026-8976, 1362-3028},
journal = {Mol. Phys.},
language = {en},
month = mar,
number = {5-7},
pages = {1039-1051},
title = {Conical Intersections and Double Excitations in Time-Dependent Density Functional Theory},
volume = {104},
year = {2006},
bdsk-url-1 = {https://doi.org/10.1080/00268970500417762}}
@article{Maitra_2017,
author = {N. T. Maitra},
date-added = {2022-03-23 10:47:34 +0100},
date-modified = {2022-03-23 10:47:34 +0100},
journal = {J. Phys. Cond. Matt.},
keywords = {10.1088/1361-648X/aa836e},
pages = {423001},
title = {Charge Transfer In Time-Dependent Density Functional Theory},
volume = {29},
year = {2017}}
@article{Maitra_2004,
author = {Maitra, Neepa T. and Zhang, Fan and Cave, Robert J. and Burke, Kieron},
date-added = {2022-03-23 10:47:34 +0100},
date-modified = {2022-03-23 10:47:34 +0100},
doi = {10.1063/1.1651060},
file = {/Users/loos/Zotero/storage/KQFDU7KL/Maitra et al. - 2004 - Double excitations within time-dependent density f.pdf},
issn = {0021-9606, 1089-7690},
journal = {J. Chem. Phys.},
language = {en},
month = apr,
number = {13},
pages = {5932-5937},
title = {Double Excitations within Time-Dependent Density Functional Theory Linear Response},
volume = {120},
year = {2004},
bdsk-url-1 = {https://doi.org/10.1063/1.1651060}}
@inbook{Maitra_2012,
address = {Berlin, Heidelberg},
author = {Maitra, Neepa T.},
booktitle = {Fundamentals of Time-Dependent Density Functional Theory},
date-added = {2022-03-23 10:47:34 +0100},
date-modified = {2022-03-23 10:47:34 +0100},
doi = {10.1007/978-3-642-23518-4_8},
editor = {Marques, Miguel A.L. and Maitra, Neepa T. and Nogueira, Fernando M.S. and Gross, E.K.U. and Rubio, Angel},
file = {/Users/loos/Zotero/storage/MAFNZHIQ/Maitra - 2012 - Memory History , Initial-State Dependence , and D.pdf},
isbn = {978-3-642-23517-7 978-3-642-23518-4},
pages = {167-184},
publisher = {Springer Berlin Heidelberg},
title = {Memory: History , Initial-State Dependence , and Double-Excitations},
volume = {837},
year = {2012},
bdsk-url-1 = {https://doi.org/10.1007/978-3-642-23518-4_8}}
@article{Elliott_2011,
author = {Elliott, Peter and Goldson, Sharma and Canahui, Chris and Maitra, Neepa T.},
date-added = {2022-03-23 10:47:28 +0100},
date-modified = {2022-03-23 10:47:28 +0100},
doi = {10.1016/j.chemphys.2011.03.020},
file = {/Users/loos/Zotero/storage/U6T3LQ8L/Elliott et al. - 2011 - Perspectives on double-excitations in TDDFT.pdf},
issn = {03010104},
journal = {Chem. Phys.},
language = {en},
month = nov,
number = {1},
pages = {110-119},
title = {Perspectives on Double-Excitations in {{TDDFT}}},
volume = {391},
year = {2011},
bdsk-url-1 = {https://doi.org/10.1016/j.chemphys.2011.03.020}}
@article{Tozer_2000,
author = {Tozer, David J. and Handy, Nicholas C.},
date-added = {2022-03-23 10:47:18 +0100},
date-modified = {2022-03-23 10:47:18 +0100},
doi = {10.1039/a910321j},
file = {/Users/loos/Zotero/storage/TFJP3V8Z/Tozer and Handy - 2000 - On the determination of excitation energies using .pdf},
issn = {14639076, 14639084},
journal = {Phys. Chem. Chem. Phys.},
language = {en},
number = {10},
pages = {2117-2121},
title = {On the Determination of Excitation Energies Using Density Functional Theory},
volume = {2},
year = {2000},
bdsk-url-1 = {https://doi.org/10.1039/a910321j}}
@article{Cave_2004,
author = {Cave, Robert J. and Zhang, Fan and Maitra, Neepa T. and Burke, Kieron},
date-added = {2022-03-23 10:47:01 +0100},
date-modified = {2022-03-23 10:47:01 +0100},
doi = {10.1016/j.cplett.2004.03.051},
file = {/Users/loos/Zotero/storage/6L9X6HT4/Cave et al. - 2004 - A dressed TDDFT treatment of the 21Ag states of bu.pdf},
issn = {00092614},
journal = {Chem. Phys. Lett.},
language = {en},
month = may,
number = {1-3},
pages = {39-42},
title = {A Dressed {{TDDFT}} Treatment of the {{21Ag}} States of Butadiene and Hexatriene},
volume = {389},
year = {2004},
bdsk-url-1 = {https://doi.org/10.1016/j.cplett.2004.03.051}}
@article{Loos_2019,
author = {Loos, Pierre-Fran{\c c}ois and Boggio-Pasqua, Martial and Scemama, Anthony and Caffarel, Michel and Jacquemin, Denis},
date-added = {2022-03-23 10:41:29 +0100},
date-modified = {2022-03-23 10:41:29 +0100},
doi = {10.1021/acs.jctc.8b01205},
journal = {J. Chem. Theory Comput.},
pages = {1939--1956},
title = {Reference Energies for Double Excitations},
volume = {15},
year = {2019},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.8b01205}}
@article{Zobel_2021,
author = {Zobel, J. Patrick and Gonz{\'a}lez, Leticia},
date-added = {2022-03-21 21:56:03 +0100},
@ -574,13 +697,10 @@
@article{dreuw_2015,
abstract = {The algebraic diagrammatic construction (ADC) scheme for the polarization propagator provides a series of ab initio methods for the calculation of excited states based on perturbation theory. In recent years, the second-order ADC(2) scheme has attracted attention in the computational chemistry community because of its reliable accuracy and reasonable computational effort in the calculation of predominantly singly excited states. Owing to their size-consistency, ADC methods are suited for the investigation of large molecules. In addition, their Hermitian structure and the availability of the intermediate state representation (ISR) allow for straightforward computation of excited-state properties. Recently, an efficient implementation of ADC(3) has been reported, and its high accuracy for typical valence excited states of organic chromophores has been demonstrated. In this review, the origin of ADC-based excited-state methods in propagator theory is described, and an intuitive route for the derivation of algebraic expressions via the ISR is outlined and comparison to other excited-state methods is made. Existing computer codes and implemented ADC variants are reviewed, but most importantly the accuracy and limits of different ADC schemes are critically examined. WIREs Comput Mol Sci 2015, 5:82\textendash 95. doi: 10.1002/wcms.1206 This article is categorized under: Structure and Mechanism {$>$} Molecular Structures Electronic Structure Theory {$>$} Ab Initio Electronic Structure Methods Theoretical and Physical Chemistry {$>$} Spectroscopy},
annotation = {\_eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/wcms.1206},
author = {Dreuw, Andreas and Wormit, Michael},
date-modified = {2022-03-23 10:30:41 +0100},
doi = {10.1002/wcms.1206},
file = {/Users/monino/Zotero/storage/ULV7SRTF/Dreuw et Wormit - 2015 - The algebraic diagrammatic construction scheme for.pdf;/Users/monino/Zotero/storage/D7CSY4E5/wcms.html},
issn = {1759-0884},
journal = {WIREs Comput. Mol. Sci.},
langid = {english},
number = {1},
pages = {82--95},
title = {The Algebraic Diagrammatic Construction Scheme for the Polarization Propagator for the Calculation of Excited States},
@ -1152,14 +1272,10 @@
@incollection{roos_1996,
abstract = {This chapter contains sections titled: Introduction Multiconfigurational Perturbation Theory Applications in Spectroscopy Summary},
annotation = {\_eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470141526.ch5},
author = {Roos, Bj{\"o}rn O. and Andersson, Kerstin and F{\"u}lscher, Markus P. and Malmqvist, Per-{\^a}ke and {Serrano-Andr{\'e}s}, Luis and Pierloot, Kristin and Merch{\'a}n, Manuela},
author = {Roos, Bjorn O. and Andersson, Kerstin and Fulscher, Markus P. and Malmqvist, Per-{\^a}ke and {Serrano-Andr{\'e}s}, Luis and Pierloot, Kristin and Merch{\'a}n, Manuela},
booktitle = {Advances in {{Chemical Physics}}},
copyright = {Copyright \textcopyright{} 1996 by John Wiley \& Sons, Inc.},
date-modified = {2022-03-23 10:31:28 +0100},
doi = {10.1002/9780470141526.ch5},
file = {/Users/monino/Zotero/storage/KWDFZUBF/9780470141526.html},
isbn = {978-0-470-14152-6},
langid = {english},
pages = {219--331},
publisher = {{John Wiley \& Sons, Ltd}},
shorttitle = {Multiconfigurational {{Perturbation Theory}}},

39
Manuscript/CBD.tex
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@ -247,32 +247,35 @@ A theoretical best estimate is defined for the autoisomerization barrier and for
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Despite the fact that excited states are involved in ubiquitous processes such as photochemistry, \cite{Bernardi_1990,Bernardi_1996,Boggio-Pasqua_2007,Klessinger_1995,Olivucci_2010,Robb_2007,VanderLugt_1969} catalysis or in solar cell technology, \cite{Delgado_2010} none of the currently existing methods has been shown to provide accurate excitation energies in all scenarios due to the complexity of the process, the size of the systems, environment effects and many other possible factors.
Indeed, each computational model has its own theoretical and/or technical issues and the number of possible chemical scenario is so vast that the design of new excited-state methodologies is still a very active field of theoretical quantum chemistry.\cite{Roos_1996,Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Dreuw_2015,Ghosh_2018,Blase_2020,Loos_2020a,Hait_2021,Zobel_2021}
Despite the fact that excited states are involved in ubiquitous processes such as photochemistry, \cite{bernardi_1990,bernardi_1996,boggio-pasqua_2007,klessinger_1995,olivucci_2010,robb_2007,vanderLugt_1969} catalysis or in solar cell technology, \cite{delgado_2010} none of the currently existing methods has been shown to provide accurate excitation energies in all scenarios due to the complexity of the process, the size of the systems, environment effects and many other possible factors.
Indeed, each computational model has its own theoretical and/or technical issues and the number of possible chemical scenario is so vast that the design of new excited-state methodologies is still a very active field of theoretical quantum chemistry.\cite{roos_1996,piecuch_2002,dreuw_2005,krylov_2006,sneskov_2012,gonzales_2012,laurent_2013,adamo_2013,dreuw_2015,ghosh_2018,blase_2020,loos_2020a,hait_2021,zobel_2021}
Speaking of difficult task, the cyclobutadiene (CBD) molecule has been a real challenge for experimental and theoretical chemistry for many decades. \cite{bally_1980} Due to its antiaromaticity \cite{minkin_1994} and large angular strain, \cite{baeyer_1885} CBD presents a high reactivity which made its synthesis a particularly difficult exercise.
The simple H\"uckel molecular orbital theory (wrongly) predicts a triplet ground state at the {\Dfour} square geometry, with two singly-occupied frontier orbitals that are degenerate by symmetry (Hund's rule).
This degeneracy is lifted by the so-called Jahn-Teller effect, \ie, by a descent in symmetry (from {\Dfour} to {\Dtwo} point group) via a geometrical distortion of the molecule.
In such as case, H\"uckel molecular orbital theory (correctly) predicts a closed-shell singlet ground state at the {\Dtwo} rectangular geometry.
\titou{Indeed, synthetic work from Pettis and co-workers \cite{reeves_1969} gives a rectangular geometry to the singlet ground state of CBD and then was confirmed by experimental works. \cite{irngartinger_1983,ermer_1983,kreile_1986}}
The simple H\"uckel molecular orbital theory (wrongly) predicts a triplet ground state at the {\Dfour} square geometry, with two singly-occupied frontier orbitals that are degenerate by symmetry (Hund's rule), while state-of-the-art \textit{ab initio} methods (correctly) predict an open-shell singlet ground state.
This degeneracy is lifted by the so-called Jahn-Teller effect, \ie, by a descent in symmetry (from {\Dfour} to {\Dtwo} point group) via a geometrical distortion of the molecule, leading to a closed-shell singlet ground state.
This was confirmed by several experimental studies by Pettis and co-workers \cite{reeves_1969} and others. \cite{irngartinger_1983,ermer_1983,kreile_1986}
At the {\Dtwo} ground-state structure, the \titou{{\oneAg}} state has a weak multi-configurational character with well-separated frontier orbitals that can be described by single-reference methods.
However, at the {\Dfour} square geometry, the singlet state {\sBoneg} has two singly occupied frontier orbitals that are degenerated so has a two-configurational character and single-reference methods are unreliable to describe it.
The singlet (\Dfour) is a transition state in the automerization reaction between the two rectangular structures (see Fig.~\ref{fig:CBD}).
The autoisomerization barrier (AB) for the CBD molecule is defined as the energy difference between the singlet ground state of the square (\Dfour) structure and the singlet ground state of the rectangular (\Dtwo) geometry.
The energy of this barrier was predicted, experimentally, in the range of 1.6-10 \kcalmol \cite{whitman_1982} and multi-reference calculations gave an energy barrier in the range of 6-7 \kcalmol. \cite{eckert-maksic_2006}
All the specificities of CBD make it a real playground for excited-states methods.
At the {\Dtwo} geometry, the {\oneAg} ground state has a weak multi-configurational character with well-separated frontier orbitals that can be described by single-reference methods.
However, at the {\Dfour} geometry, the {\sBoneg} ground state has two singly occupied frontier orbitals that are degenerate.
Therefore, one must take into account, at least, two electronic configurations to properly model this multi-configurational scenario.
Of course, single-reference methods are naturally unable to describe such situations.
The singlet ground state of the square arrangement is a transition state in the automerization reaction between the two rectangular structures (see Fig.~\ref{fig:CBD}).
The autoisomerization barrier (AB) is defined as the difference between the square and rectangular ground-state energies.
The energy of this barrier is estimated, experimentally, in the range of 1.6-10 \kcalmol \cite{whitman_1982} and previous multi-reference calculations yield an energy barrier in the range of 6-7 \kcalmol. \cite{eckert-maksic_2006}
%All these specificities of CBD make it a real playground for excited-states methods.
The lowest-energy excited states of CBD in both geometries are represented in Fig.~\ref{fig:CBD}.
Are represented {\oneAg} and {\tBoneg} states for the rectangular geometry and {\sBoneg} and {\Atwog} for the square one.
Due to energy scaling, the {\sBoneg}, {\twoAg} and {\Aoneg}, {\Btwog}, states for the {\Dtwo} and {\Dfour} structures, respectively, are not drawn. The {\twoAg} and {\Aoneg} states are doubly excited states and these doubly excited states are known to be challenging to represent for adiabatic time-dependent density functional theory \cite{casida_1995} (TD-DFT) and even for state-of-the-art methods like the approximate third-order coupled-cluster (CC3) \cite{christiansen_1995,koch_1997} or equation-of-motion coupled-cluster with singles, doubles and triples (EOM-CCSDT). \cite{kucharski_1991,kallay_2004,hirata_2000,hirata_2004}
The lowest-energy excited states of CBD in both geometries are represented in Fig.~\ref{fig:CBD}, where
we have reported the {\oneAg} and {\tBoneg} states for the rectangular geometry and {\sBoneg} and {\Atwog} for the square one.
Due to the energy scale, the {\sBoneg}, {\twoAg} and {\Aoneg}, {\Btwog} states for the {\Dtwo} and {\Dfour} structures, respectively, are not shown.
The {\twoAg} and {\Aoneg} states have a strong contribution from doubly-excited configurations and these so-called double excitations \cite{loos_2019} are known to be a challenge for adiabatic time-dependent density-functional theory (TD-DFT) \cite{casida_1995,tozer_2000,maitra_2004,cave_2004,levine_2006,elliott_2011,Maitra_2012,Maitra_2017} and even for state-of-the-art methods like the approximate third-order coupled-cluster (CC3) \cite{christiansen_1995,koch_1997} or equation-of-motion coupled-cluster with singles, doubles and triples (EOM-CCSDT). \cite{kucharski_1991,kallay_2004,hirata_2000,hirata_2004}
In order to tackle the problems of multi-configurational character and double excitations several ways are explored. The most evident way that one can think about to describe multiconfigurational and double excitations are multiconfigurational methods.
In order to tackle the problems of multi-configurational character and double excitations several ways are explored.
The most evident route that one can think about to describe multiconfigurational and double excitations are multiconfigurational methods.
Among these methods, one can find complete active space self-consistent field (CASSCF) \cite{roos_1996}, the second perturbation-corrected variant (CASPT2) \cite{andersson_1990} and the second-order $n$-electron valence state perturbation theory (NEVPT2). \cite{angeli_2001,angeli_2001a,angeli_2002}
The exponential scaling of these methods with the size of the active space is the limitation to the application of these ones to large molecules.
Another way to deal with double excitations is to use high level truncation of the equation-of-motion (EOM) formalism of coupled-cluster (CC) theory.
However, to provide a correct description of doubly excited states one have to take into account contributions from the triple excitations in the CC expansion.
Another way to deal with double excitations is to use high level truncation of the equation-of-motion (EOM) formalism of coupled-cluster (CC) theory. \cite{hirata_2000,sundstrom_2014}
However, to provide a correct description of doubly excited states one have to take into account, at the very least, contributions from the triple excitations in the CC expansion. \cite{watson_2012}
Again, due to the scaling of CC methods with the number of basis functions the applicability of these methods is limited to small molecules.
An alternative to multiconfigurational and CC methods is the use of selected CI (SCI) methods for the computation of transition energies for singly and doubly excited states that are known to reach near full CI energies for small molecules.