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Manuscript/CBD.bib
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@ -1,13 +1,26 @@
%% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-03-24 22:36:43 +0100
%% Created for Pierre-Francois Loos at 2022-03-28 22:40:05 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Matthews_2020,
author = {Matthews,Devin A. and Cheng,Lan and Harding,Michael E. and Lipparini,Filippo and Stopkowicz,Stella and Jagau,Thomas-C. and Szalay,P{\'e}ter G. and Gauss,J{\"u}rgen and Stanton,John F.},
date-added = {2022-03-28 21:47:45 +0200},
date-modified = {2022-03-28 21:47:45 +0200},
doi = {10.1063/5.0004837},
journal = {J. Chem. Phys.},
number = {21},
pages = {214108},
title = {Coupled-cluster techniques for computational chemistry: The CFOUR program package},
volume = {152},
year = {2020},
bdsk-url-1 = {https://doi.org/10.1063/5.0004837}}
@article{Sarkar_2022,
author = {R. Sarkar and P. F. Loos and M. Boggio-Pasqua and D. Jacquemin.},
date-added = {2022-03-24 22:00:41 +0100},
@ -4274,23 +4287,6 @@
year = {2014},
bdsk-url-1 = {https://doi.org/10.1039/C3CP54374A}}
@article{Matthews_2020,
abstract = {The analytic gradient theory for both iterative and noniterative coupled-cluster approximations that include connected quadruple excitations is presented. These methods include, in particular, CCSDT(Q), which is an analog of the well-known CCSD(T) method which starts from the full CCSDT method rather than CCSD. The resulting methods are implemented in the CFOUR program suite, and pilot applications are presented for the equilibrium geometries and harmonic vibrational frequencies of the simplest Criegee intermediate, CH2OO, as well as to the isomerization pathway between dimethylcarbene and propene. While all methods are seen to approximate the full CCSDTQ results well for ``well-behaved'' systems, the more difficult case of the Criegee intermediate shows that CCSDT(Q), as well as certain iterative approximations, display problematic behavior.},
author = {Matthews, Devin A.},
date-modified = {2022-03-23 11:33:54 +0100},
doi = {10.1021/acs.jctc.0c00522},
file = {/Users/monino/Zotero/storage/LCIZ3YB9/Matthews - 2020 - Analytic Gradients of Approximate Coupled Cluster .pdf;/Users/monino/Zotero/storage/ZZZQCDI4/acs.jctc.html},
issn = {1549-9618},
journal = {J. Chem. Theory Comput.},
month = oct,
number = {10},
pages = {6195--6206},
publisher = {{American Chemical Society}},
title = {Analytic {{Gradients}} of {{Approximate Coupled Cluster Methods}} with {{Quadruple Excitations}}},
volume = {16},
year = {2020},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.0c00522}}
@book{Minkin_1994,
author = {Minkin, Vladimir I and Glukhovtsev, Mikhail N. and Simkin, Boris Ya.},
date-modified = {2022-03-23 11:52:27 +0100},

57
Manuscript/CBD.tex
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@ -259,13 +259,12 @@ At the {\Dtwo} geometry, the {\oneAg} ground state has a weak multi-configuratio
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 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}), while the lowest triplet state {\Atwog} is a minimum on the triplet potential energy surface.
Thus, 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} while 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}, where
we have reported the {\oneAg} and {\tBoneg} states for the rectangular geometry and {\sBoneg} and {\Atwog} for the square one.
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 higher-energy states ({\sBoneg} and {\twoAg} for {\Dtwo} and {\Aoneg} and {\Btwog} for {\Dfour}) are not shown.
Interestingly, 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 an absolute nightmare for adiabatic time-dependent density-functional theory (TD-DFT) \cite{Runge_1984,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}
@ -289,9 +288,10 @@ One can address part of this issue by increasing the excitation order or by comp
both solutions being associated with an additional computational cost.
In the present work, we investigate the accuracy of a large panel of computational methods on the autoisomerization barrier and the low-lying excited states of CBD at the {\Dtwo} and {\Dfour} ground-state geometries.
Computational details are reported in Section \ref{sec:compmet} for SCI (Subsection \ref{sec:SCI}), EOM-CC (Subsection \ref{sec:CC}), multi-configurational (Subsection \ref{sec:Multi}) and spin-flip (Subsection \ref{sec:sf}) methods.
Computational details are reported in Section \ref{sec:compmet}.
% for SCI (Subsection \ref{sec:SCI}), EOM-CC (Subsection \ref{sec:CC}), multi-configurational (Subsection \ref{sec:Multi}) and spin-flip (Subsection \ref{sec:sf}) methods.
Section \ref{sec:res} is devoted to the discussion of our results.
First, we consider the autoisomerization barrier (Subsection \ref{sec:auto}) and then the excited states (Subsection \ref{sec:states}) of the CBD molecule for both geometries.
%First, we consider the autoisomerization barrier (Subsection \ref{sec:auto}) and then the excited states (Subsection \ref{sec:states}) of the CBD molecule for both geometries.
%%% FIGURE 1 %%%
\begin{figure}
@ -327,14 +327,21 @@ This type of extrapolation procedures is now routine in SCI methods as well as o
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Coupled-cluster calculations}
\label{sec:CC}
\alert{Coupled-cluster theory provides a hierarchy of methods that yields increasingly accurate energies by ramping up the maximum excitation degree of the cluster operator: CC with singles and doubles (CCSD), CC with singles, doubles, and triples (CCSDT), CC with singles, doubles, triples, and quadruples (CCSDTQ), etc.
%Without any truncation of the cluster operator, one has the full CC (FCC) that is equivalent to the full configuration interaction (FCI) giving the exact energy and wave function of the system for a fixed atomic basis set.
%However, due to the computational exponential scaling with system size we have to use truncated CC methods.
Here, we performed different types of CC calculations using different codes.
CCSD, CCSDT, and CCSDTQ calculations are achieved with \textcolor{red}{CFOUR}.
The calculations in the context of CC response theory or ``approximate'' series (CC3 and CC4) are performed with \textcolor{red}{DALTON}.\cite{Aidas_2014}
The CC with singles, doubles, triples and quadruples (CCSDTQ) are done with the \titou{CFOUR} code.
The CC2, \cite{Christiansen_1995,Hattig_2000} CC3 \cite{Christiansen_1995,Koch_1995} and CC4 \cite{Kallay_2005,Matthews_2020} methods can be seen as cheaper approximations of CCSD, \cite{Purvis_1982} CCSDT \cite{Noga_1987a} and CCSDTQ \cite{Kucharski_1991a} by skipping the most expensive terms and avoiding the storage of higher-excitations amplitudes.}
Coupled-cluster theory provides a hierarchy of methods that yields increasingly accurate ground state energies by ramping up the maximum excitation degree of the cluster operator: \cite{Cizek_1966,Paldus_1972,Crawford_2000,Piecuch_2002b,Bartlett_2007,Shavitt_2009} CC with singles and doubles (CCSD), \cite{Cizek_1966,Purvis_1982} CC with singles, doubles, and triples (CCSDT), \cite{Noga_1987a,Scuseria_1988} CC with singles, doubles, triples, and quadruples (CCSDTQ), \cite{Oliphant_1991,Kucharski_1991a,Kucharski_1992} etc.
As mentioned above, CC theory can be extended to excited states via the EOM formalism, \cite{Rowe_1968,Stanton_1993} where one projects out the similarity-transformed Hamiltonian in a basis of excited determinants yielding the following systematically improvable family of methods for neutral excited states: EOM-CCSD, EOM-CCSDT, EOM-CCSDTQ, etc.\cite{Noga_1987,Koch_1990,Kucharski_1991,Stanton_1993,Christiansen_1998b,Kucharski_2001,Kowalski_2001,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004}
In the following, we will omit the prefix EOM for the sake of conciseness.
Alternatively to the ``complete'' CC models, one can also employed the CC2, \cite{Christiansen_1995,Hattig_2000} CC3 \cite{Christiansen_1995,Koch_1995} and CC4 \cite{Kallay_2005,Matthews_2020,Loos_2021} methods which can be seen as cheaper approximations of CCSD, CCSDT, and CCSDTQ by skipping the most expensive terms and avoiding the storage of higher-excitations amplitudes.
Here, we have performed CC calculations using various codes.
Typically, CCSD, CCSDT, and CCSDTQ as well as CC3 and CC4 calculations are achieved with CFOUR. \cite{Matthews_2020}
In some case, we have also computed CC3 energies and properties with DALTON.\cite{Aidas_2014}
\alert{To avoid having to perform multi-reference CC calculations and because one cannot perform high-level CC calculations in the restricted open-shell formalism, it is worth mentioning that, for the {\Dfour} arrangement, we have considered the lowest \textit{closed-shell} singlet state of {\Aoneg} symmetry as reference.
The open-shell ground state {\sBoneg} and the {\Btwog} states are obtained as deexcitations.}
%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 higher-energy states ({\sBoneg} and {\twoAg} for {\Dtwo} and {\Aoneg} and {\Btwog} for {\Dfour}) are not shown.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -357,9 +364,13 @@ All these calculations are also carried out with MOLPRO. \cite{Werner_2020}
\subsection{Spin-flip calculations}
\label{sec:sf}
Within the spin-flip formalism, one considers the lowest triplet state as reference instead of the singlet ground state.
Ground-state energies are then computed as the sum of the triplet reference state and the corresponding deexcitation energy.
Likewise, excitation energies with respect to the singlet ground state are computed as differences of excitation energies with respect to the reference triplet state.
Nowadays, spin-flip techniques are broadly accessible thanks to intensive developments in the electronic structure community (see Ref.~\onlinecite{Casanova_2020} and references therein).
Here, we explore the spin-flip version \cite{Lefrancois_2015} of the algebraic-diagrammatic construction \cite{Schirmer_1982} (ADC) using the standard and extended second-order ADC schemes, SF-ADC(2)-s \cite{Trofimov_1997,Dreuw_2015} and SF-ADC(2)-x, \cite{Dreuw_2015} as well as its third-order version, SF-ADC(3). \cite{Dreuw_2015,Trofimov_2002,Harbach_2014}
These calculations are performed using Q-CHEM 5.2.1. \cite{qchem}
\titou{ADC(2.5)?}
We have also carried out spin-flip calculations within the TD-DFT framework (SF-TD-DFT), \cite{Shao_2015} and these are also performed with Q-CHEM 5.2.1. \cite{qchem}
The B3LYP, \cite{Becke_1988b,Lee_1988a,Becke_1993b} PBE0 \cite{Adamo_1999a,Ernzerhof_1999} and BH\&HLYP hybrid functionals are considered, which contain 20\%, 25\%, 50\% of exact exchange, respectively.
@ -369,6 +380,8 @@ The main difference between these range-separated functionals is their amount of
Finally, the hybrid meta-GGA functional M06-2X \cite{Zhao_2008} and the range-separated hybrid meta-GGA functional M11 \cite{Peverati_2011} are also employed.
Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximation. \cite{Hirata_1999}
%EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -630,7 +643,7 @@ CC4 &6-31+G(d)& & $3.343$ & $4.067$ \\
CCSDTQ &6-31+G(d)& & $3.340$ & $4.073$ \\
& aug-cc-pVDZ & & $\left[3.161\right]$\fnm[2]& $\left[4.047\right]$\fnm[2] \\
& aug-cc-pVTZ & & $\left[3.125\right]$\fnm[3]& $\left[4.149\right]$\fnm[3]\\[0.1cm]
SA-CASSCF(4,4) &6-31+G(d)& $1.662$ & $4.657$ & $4.439$ \\
CASSCF(4,4) &6-31+G(d)& $1.662$ & $4.657$ & $4.439$ \\
& aug-cc-pVDZ & $1.672$ & $4.563$ & $4.448$ \\
& aug-cc-pVTZ & $1.670$ & $4.546$ & $4.441$ \\
& aug-cc-pVQZ & $1.671$ & $4.549$ & $4.440$ \\[0.1cm]
@ -654,7 +667,7 @@ MRCI(4,4) &6-31+G(d)& $1.564$ & $3.802$ & $4.265$ \\
& aug-cc-pVDZ & $1.558$ & $3.670$ & $4.254$ \\
& aug-cc-pVTZ & $1.568$ & $3.678$ & $4.270$ \\
& aug-cc-pVQZ & $1.574$ & $3.681$ & $4.280$ \\[0.1cm]
SA-CASSCF(12,12) &6-31+G(d)& $1.675$ & $3.924$ & $4.220$ \\
CASSCF(12,12) &6-31+G(d)& $1.675$ & $3.924$ & $4.220$ \\
& aug-cc-pVDZ & $1.685$ & $3.856$ & $4.221$ \\
& aug-cc-pVTZ & $1.686$ & $3.844$ & $4.217$ \\
& aug-cc-pVQZ & $1.687$ & $3.846$ & $4.216$ \\[0.1cm]
@ -800,7 +813,7 @@ CC4 & 6-31+G(d) & & $1.604$ & $2.121$ \\
CCSDTQ & 6-31+G(d) & $0.205$ & $1.593$ & $2.134$ \\
& aug-cc-pVDZ & $\left[0.160\right]$\fnm[2] & $\left[1.528 \right]$\fnm[4]&$\left[1.947\right]$\fnm[4] \\
& aug-cc-pVTZ & $\left[0.144\right]$\fnm[3] & $\left[1.500 \right]$\fnm[5]&$\left[2.034\right]$\fnm[5] \\ [0.1cm]
SA-CASSCF(4,4) & 6-31+G(d) & $0.447$ & $2.257$ & $3.549$ \\
CASSCF(4,4) & 6-31+G(d) & $0.447$ & $2.257$ & $3.549$ \\
& aug-cc-pVDZ & $0.438$ & $2.240$ & $3.443$ \\
& aug-cc-pVTZ & $0.434$ & $2.234$ & $3.424$ \\
& aug-cc-pVQZ & $0.435$ & $2.235$ & $3.427$ \\[0.1cm]
@ -820,7 +833,7 @@ MRCI(4,4) & 6-31+G(d) & $0.297$ & $1.861$ & $2.571$ \\
& aug-cc-pVDZ & $0.273$ & $1.823$ & $2.419$ \\
& aug-cc-pVTZ & $0.271$ & $1.824$ & $2.415$ \\
& aug-cc-pVQZ & $0.273$ & $1.825$ & $2.413$ \\[0.1cm]
SA-CASSCF(12,12) & 6-31+G(d) & $0.386$ & $1.974$ & $2.736$ \\
CASSCF(12,12) & 6-31+G(d) & $0.386$ & $1.974$ & $2.736$ \\
& aug-cc-pVDZ & $0.374$ & $1.947$ & $2.649$ \\
& aug-cc-pVTZ & $0.370$ & $1.943$ & $2.634$ \\
& aug-cc-pVQZ & $0.371$ & $1.945$ & $2.637$ \\[0.1cm]
@ -857,7 +870,7 @@ Figure \ref{fig:D4h} shows the vertical energies of the studied excited states d
\begin{figure*}
%width=0.8\linewidth
\includegraphics[scale=0.5]{D4h.pdf}
\caption{Vertical energies of the {\Atwog}, {\Aoneg} and {\Btwog} states for the \Dfour geometry using the aug-cc-pVTZ basis. Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multi-reference, CC, and TBE values, respectively.}
\caption{Vertical energies of the {\Atwog}, {\Aoneg} and {\Btwog} states for the {\Dfour} geometry using the aug-cc-pVTZ basis. Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multi-reference, CC, and TBE values, respectively.}
\label{fig:D4h}
\end{figure*}
%%% %%% %%% %%%
@ -909,13 +922,13 @@ Finally we look at the vertical energy errors for the \Dfour structure. First, w
%CCSDT & $-0.25$ & $-0.051$ & $0.014$ & $0.280$ & $0.005$ & $0.131$ & $0.503$ \\
%CC4 & $-0.11$ & & $0.003$ & $-0.006$ & & $0.011$ & $-0.013$ \\
%CCSDTQ & $0.00$ & & $0.000$ & $0.000$ & $0.000$ & $0.000$ & $0.000$ \\[0.1cm]
%SA-CASSCF(4,4) & & $0.208$ & $1.421$ & $0.292$ & $0.290$ & $0.734$ & $1.390$ \\
%CASSCF(4,4) & & $0.208$ & $1.421$ & $0.292$ & $0.290$ & $0.734$ & $1.390$ \\
%CASPT2(4,4) & & $-0.050$ & $-0.202$ & $-0.077$ & $-0.016$ & $0.006$ & $-0.399$ \\
%XMS-CASPT2(4,4) & & & & $-0.035$ & & & \\
%SC-NEVPT2(4,4) & & $-0.083$ & $-0.703$ & $-0.041$ & $-0.120$ & $-0.072$ & $-0.979$ \\
%PC-NEVPT2(4,4) & & $-0.080$ & $-0.757$ & $-0.066$ & $-0.118$ & $-0.097$ & $-1.031$ \\
%MRCI(4,4) & & $0.106$ & $0.553$ & $0.121$ & $0.127$ & $0.324$ & $0.381$ \\[0.1cm]
%SA-CASSCF(12,12) & $2.66$ & $0.224$ & $0.719$ & $0.068$ & $0.226$ & $0.443$ & $0.600$ \\
%CASSCF(12,12) & $2.66$ & $0.224$ & $0.719$ & $0.068$ & $0.226$ & $0.443$ & $0.600$ \\
%CASPT2(12,12) & & $0.018$ & $0.058$ & $-0.106$ & $0.039$ & $0.038$ & $-0.108$ \\
%XMS-CASPT2(12,12) & & & & $-0.090$ & & & \\
%SC-NEVPT2(12,12) & $-0.65$ & $0.039$ & $0.063$ & $-0.063$ & $0.021$ & $0.046$ & $-0.142$ \\
@ -963,13 +976,13 @@ CC3 & $-1.05$ & $-0.060$ & $-0.006$ & $0.628$ & & $0.162$ & $0.686$ \\
CCSDT & $-0.25$ & $-0.051$ & $0.014$ & $0.280$ & $0.005$ & $0.131$ & $0.503$ \\
CC4 & $-0.11$ & & $0.003$ & $-0.006$ & & $0.011$ & $-0.013$ \\
CCSDTQ & $0.00$ & & $0.000$ & $0.000$ & $0.000$ & $0.000$ & $0.000$ \\[0.1cm]
SA-CASSCF(4,4) & & $0.208$ & $1.421$ & $0.292$ & $0.290$ & $0.734$ & $1.390$ \\
CASSCF(4,4) & & $0.208$ & $1.421$ & $0.292$ & $0.290$ & $0.734$ & $1.390$ \\
CASPT2(4,4) & & $-0.050$ & $-0.202$ & $-0.077$ & $-0.016$ & $0.006$ & $-0.399$ \\
%XMS-CASPT2(4,4) & & & & $-0.035$ & & & \\
SC-NEVPT2(4,4) & & $-0.083$ & $-0.703$ & $-0.041$ & $-0.120$ & $-0.072$ & $-0.979$ \\
PC-NEVPT2(4,4) & & $-0.080$ & $-0.757$ & $-0.066$ & $-0.118$ & $-0.097$ & $-1.031$ \\
MRCI(4,4) & & $0.106$ & $0.553$ & $0.121$ & $0.127$ & $0.324$ & $0.381$ \\[0.1cm]
SA-CASSCF(12,12) & $2.66$ & $0.224$ & $0.719$ & $0.068$ & $0.226$ & $0.443$ & $0.600$ \\
CASSCF(12,12) & $2.66$ & $0.224$ & $0.719$ & $0.068$ & $0.226$ & $0.443$ & $0.600$ \\
CASPT2(12,12) & & $0.018$ & $0.058$ & $-0.106$ & $0.039$ & $0.038$ & $-0.108$ \\
%XMS-CASPT2(12,12) & & & & $-0.090$ & & & \\
SC-NEVPT2(12,12) & $-0.65$ & $0.039$ & $0.063$ & $-0.063$ & $0.021$ & $0.046$ & $-0.142$ \\