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minor corrections from T2

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Pierre-Francois Loos 2022-03-21 14:22:50 +01:00
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Manuscript/CBD.tex
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@ -191,8 +191,8 @@
\newcommand{\spf}{\text{sf}}
%geometries
\newcommand{\Dtwo}{$D_{2h}$ }
\newcommand{\Dfour}{$D_{4h}$ }
\newcommand{\Dtwo}{$D_{2h}$}
\newcommand{\Dfour}{$D_{4h}$}
\sisetup{range-phrase=--}
\sisetup{range-units=single}
@ -230,7 +230,7 @@
\affiliation{\LCPQ}
\begin{abstract}
The cyclobutadiene molecule represents a playground for ground state and excited states methods. Indeed, due to the high symmetry of the molecule, especially at the square geometry (\Dfour) but also at the rectangular structure (\Dtwo), the ground state and the excited states of cyclobutadiene exhibit multiconfigurational character where single reference methods such as adiabatic time-dependent density functional theory (TD-DFT) or equation-of-motion coupled cluster (EOM-CC) show difficulty to describe them. In this work we provide an extensive study of the autoisomerization barrier, where the rectangular (\Dtwo) and the square geometry (\Dfour) are needed, and of the vertical excitations energies of cyclobutadiene using a large range of methods and basis sets. In order to tackle the problem of multiconfigurational character presents in the autoisomerization barrier and in the vertical excitation energies selected configuration interaction and multireference (CASSCF, CASPT2, and NEVPT2) calculations are performed. Moreover coupled cluster calculations such as CCSD, CC3, CCSDT, CC4 and CCSDTQ are added to the set of methods. To complete the study we provide spin-flip results, which are known to give correct description of multiconfigurational character states, in the TD-DFT framework where numerous exchange-correlation functionals are considered, we also add algebraic diagrammatic construction (ADC) calculations in the spin-flip formalism where we use the ADC(2)-s, ADC(2)-x and ADC(3) schemes. A theoretical best estimate is defined for the autoisomerization barrier and for each vertical energies.
The cyclobutadiene molecule represents a playground for ground state and excited states methods. Indeed, due to the high symmetry of the molecule, especially at the square geometry (\Dfour) but also at the rectangular structure ({\Dtwo}), the ground state and the excited states of cyclobutadiene exhibit multiconfigurational character where single reference methods such as adiabatic time-dependent density functional theory (TD-DFT) or equation-of-motion coupled cluster (EOM-CC) show difficulty to describe them. In this work we provide an extensive study of the autoisomerization barrier, where the rectangular (\Dtwo) and the square geometry (\Dfour) are needed, and of the vertical excitations energies of cyclobutadiene using a large range of methods and basis sets. In order to tackle the problem of multiconfigurational character presents in the autoisomerization barrier and in the vertical excitation energies selected configuration interaction and multireference (CASSCF, CASPT2, and NEVPT2) calculations are performed. Moreover coupled cluster calculations such as CCSD, CC3, CCSDT, CC4 and CCSDTQ are added to the set of methods. To complete the study we provide spin-flip results, which are known to give correct description of multiconfigurational character states, in the TD-DFT framework where numerous exchange-correlation functionals are considered, we also add algebraic diagrammatic construction (ADC) calculations in the spin-flip formalism where we use the ADC(2)-s, ADC(2)-x and ADC(3) schemes. A theoretical best estimate is defined for the autoisomerization barrier and for each vertical energies.
\end{abstract}
\maketitle
@ -259,7 +259,8 @@ In the present work we investigate \Ag, \tBoneg, $1{}^1B_{1g}$, $2{}^1A_{g}$ and
\begin{figure}
\includegraphics[width=0.6\linewidth]{figure2.png}
\caption{Pictorial representation of the excited states of the CBD molecule and its properties under investigation. Black color represent singlet ground state (S) properties and red color triplet (T) properties.}
\caption{Pictorial representation of the ground and excited states of CBD and its properties under investigation.
The singlet ground-state (S) and triplet (T) properties are represented in black and red, respectively.}
\label{fig:CBD}
\end{figure}
@ -334,27 +335,22 @@ Two different sets of geometries obtained with different level of theory are con
%%% TABLE I %%%
\begin{squeezetable}
\begin{table}
\caption{Optimized geometries of the CBD molecule. Bond lengths are in \si{\angstrom} and angles are in degree.}
\caption{Optimized geometries of CBD for various states computed with various levels of theory.
Bond lengths are in \si{\angstrom} and angles are in degree.}
\label{tab:geometries}
\begin{ruledtabular}
\begin{tabular}{llrrr}
Method & \ce{C=C} & \ce{C-C} & \ce{C-H} & \ce{H-C=C}\fnm[1] \\
\hline
\Dtwo\\
\begin{tabular}{lllrrr}
State & Method & \ce{C=C} & \ce{C-C} & \ce{C-H} & \ce{H-C=C}\fnm[1] \\
\hline
\Dtwo $({}^1 A_{g})$&
CASPT2(12,12)/aug-cc-pVTZ & 1.355 & 1.566 & 1.077 & 134.99 \\
CC3/aug-cc-pVTZ & 1.344 & 1.565 & 1.076 & 135.08 \\
CCSD(T)/cc-pVTZ & 1.343 & 1.566 & 1.074 & 135.09 \fnm[2]\\
\hline
\Dfour $({}^1 B_{1g})$ \\
\hline
&CC3/aug-cc-pVTZ & 1.344 & 1.565 & 1.076 & 135.08 \\
&CCSD(T)/cc-pVTZ & 1.343 & 1.566 & 1.074 & 135.09 \fnm[2]\\
\Dfour $({}^1 B_{1g})$&
CASPT2(12,12)/aug-cc-pVTZ & 1.449 & 1.449 & 1.076 & 135.00 \\
\hline
\Dfour $({}^3 A_{2g})$ \\
\hline
\Dfour $({}^3 A_{2g})$&
CASPT2(12,12)/aug-cc-pVTZ & 1.445 & 1.445 & 1.076 & 135.00 \\
RO-CCSD(T)/aug-cc-pVTZ & 1.439 & 1.439 & 1.075 & 135.00
&RO-CCSD(T)/aug-cc-pVTZ & 1.439 & 1.439 & 1.075 & 135.00
\end{tabular}
\end{ruledtabular}
\fnt[1]{Angle between the \ce{C-H} bond and the \ce{C=C} bond.}
@ -374,8 +370,8 @@ Then we compare results for multireference methods, we can see a difference of a
%%% TABLE I %%%
\begin{squeezetable}
\begin{table*}
\caption{Autoisomerization barrier energy in \kcalmol.}
\begin{table}
\caption{Autoisomerization barrier (in \kcalmol) of CBD computed at various levels of theory.}
\label{tab:auto_standard}
\begin{ruledtabular}
@ -412,7 +408,7 @@ CIPSI & $7.91\pm 0.21$ & $8.58\pm 0.14$ & & \\
\fnt[4]{Value obtained using CCSDTQ/6-31+G(d) corrected by the difference between CC4/aug-cc-pVDZ basis and CC4/6-31+G(d).}
\fnt[5]{Value obtained using CCSDTQ/aug-cc-pVDZ corrected by the difference between CC4/aug-cc-pVTZ and CC4/aug-cc-pVDZ.}
\fnt[6]{Value obtained using CCSDTQ/aug-cc-pVTZ corrected by the difference between CC4/aug-cc-pVQZ and CC4/aug-cc-pVTZ.}
\end{table*}
\end{table}
\end{squeezetable}
%%% %%% %%% %%%
@ -420,7 +416,7 @@ CIPSI & $7.91\pm 0.21$ & $8.58\pm 0.14$ & & \\
\begin{figure*}
\includegraphics[scale=0.5]{AB_AVTZ.pdf}
\caption{Autoisomerization barrier (AB) energies for the CBD molecule using the aug-cc-pVTZ basis. Purple histograms are for the SF-TD-DFT functionals, orange histograms are for the SF-ADC schemes, green histograms are for the multireference methods, blue histograms are for the CC methods and the black one is for the TBE.}
\caption{Autoisomerization barrier energies for CBD using the aug-cc-pVTZ basis. Purple histograms are for the SF-TD-DFT functionals, orange histograms are for the SF-ADC schemes, green histograms are for the multireference methods, blue histograms are for the CC methods and the black one is for the TBE.}
\label{fig:AB}
\end{figure*}
@ -448,7 +444,7 @@ For the XMS-CASPT2(4,4) only the $2\,{}^1A_{1g}$ state is described with values
\begin{squeezetable}
\begin{table}
\caption{
Spin-flip TD-DFT vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{g}$ states of CBD at the \Dtwo rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
Spin-flip TD-DFT vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{g}$ states of CBD at the {\Dtwo} rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
\label{tab:sf_tddft_D2h}}
\begin{ruledtabular}
\begin{tabular}{llrrr}
@ -521,13 +517,13 @@ SF-ADC(3) & 6-31+G(d) & $1.435$ & $3.352$ & $4.242$ \\
%%% TABLE IV %%%
\begin{squeezetable}
\begin{table*}
\begin{table}
\caption{
Standard vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{g}$ states of CBD at the \Dtwo rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{g}$ states of CBD at the {\Dtwo} rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
\label{tab:D2h}}
\begin{ruledtabular}
\begin{tabular}{llrrr}
& \mc{4}{c}{Excitation energies (eV)} \\
& & \mc{3}{c}{Excitation energies (eV)} \\
\cline{3-5}
Method & Basis & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{g}$ \\
\hline
@ -597,7 +593,7 @@ CIPSI &6-31+G(d)& $1.486\pm 0.005$ & $3.348\pm 0.024$ & $4.084\pm 0.012$ \\
\fnt[1]{Value obtained using CC4/aug-cc-pVDZ corrected by the difference between CCSDT/aug-cc-pVTZ and CCSDT/aug-cc-pVDZ.}
\fnt[2]{Value obtained using CCSDTQ/6-31+G(d) corrected by the difference between CC4/aug-cc-pVDZ and CC4/6-31+G(d).}
\fnt[3]{Value obtained using CCSDTQ/aug-cc-pVDZ corrected by the difference between CC4/aug-cc-pVTZ and CC4/aug-cc-pVDZ.}
\end{table*}
\end{table}
\end{squeezetable}
%%% %%% %%% %%%
@ -607,7 +603,7 @@ Figure \ref{fig:D2h} shows the vertical energies of the studied excited states d
\begin{figure*}
%width=0.8\linewidth
\includegraphics[scale=0.5]{D2h.pdf}
\caption{Vertical energies of the $1\,{}^3B_{1g} $, $1\,{}^1B_{1g}$ and $2\,{}^1A_{g}$ states for the \Dtwo geometry using the aug-cc-pVTZ basis. Purple lines are for the SF-TD-DFT functionals, orange lines are for the SF-ADC schemes, green lines are for the multireference methods, blue lines are for the CC methods and the black ones are for the TBE.}
\caption{Vertical energies of the $1\,{}^3B_{1g} $, $1\,{}^1B_{1g}$ and $2\,{}^1A_{g}$ states for the {\Dtwo} geometry using the aug-cc-pVTZ basis. Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multireference, CC, and TBE values, respectively.}
\label{fig:D2h}
\end{figure*}
%%% %%% %%% %%%
@ -688,9 +684,9 @@ SF-ADC(3) & 6-31+G(d) & $0.123$ & $1.650$ & $2.078$ \\
%%% TABLE V %%%
\begin{squeezetable}
\begin{table*}
\begin{table}
\caption{
Standard vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states of CBD at the \Dfour square-planar equilibrium geometry of the $1\,{}^3A_{2g}$ state.
Standard vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states of CBD at the {\Dfour} square-planar equilibrium geometry of the $1\,{}^3A_{2g}$ state.
\label{tab:D4h}}
\begin{ruledtabular}
\begin{tabular}{llrrr}
@ -761,7 +757,7 @@ CIPSI & 6-31+G(d) & $0.2010\pm 0.0030$ & $1.602\pm 0.007$ & $2.13\pm 0.04$ \\
\fnt[4]{Value obtained using CCSDTQ/6-31+G(d) corrected by the difference between CC4/aug-cc-pVDZ and CC4/6-31+G(d).}
\fnt[5]{Value obtained using CCSDTQ/aug-cc-pVDZ corrected by the difference between CC4/aug-cc-pVTZ and CC4/aug-cc-pVDZ.}
\end{table*}
\end{table}
\end{squeezetable}
%%% %%% %%% %%%
@ -771,8 +767,8 @@ 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 $1\,{}^3A_{2g} $, $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g} $ states for the \Dfour geometry using the aug-cc-pVTZ basis. Purple lines are for the SF-TD-DFT functionals, orange lines are for the SF-ADC schemes, green lines are for the multireference methods, blue lines are for the CC methods and the black ones are for the TBE.}
\label{fig:D4h}
\caption{Vertical energies of the $1\,{}^3A_{2g} $, $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g} $ 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, multireference, CC, and TBE values, respectively.}
\label{fig:D4h}
\end{figure*}
%%% %%% %%% %%%
@ -850,15 +846,16 @@ Finally we look at the vertical energy errors for the \Dfour structure. First, w
\begin{squeezetable}
\begin{table*}
\caption{Energy differences between the various methods and the TBE considered. Note that AB stands for the autoisomerization barrier and that the energies are in \kcalmol while the vertical energies are given in eV. The number in parenthesis is the percentage T1 calculated at the CC3/aug-cc-pVTZ level.}
\caption{Energy differences between the various methods and the reference TBE values.
Note that AB stands for the autoisomerization barrier.
The numbers reported in parenthesis are the percentage of single excitations involved in the transition ($\%T_1$) calculated at the CC3/aug-cc-pVTZ level.}
\label{tab:TBE}
\begin{ruledtabular}
\begin{tabular}{lrrrrrrr}
%\begin{tabular}{*{1}{*{8}{l}}}
&&\mc{3}{c}{\Dtwo excitation energies (eV)} & \mc{3}{c}{\Dfour excitation energies (eV)} \\
& &\mc{3}{c}{{\Dtwo} excitation energies (eV)} & \mc{3}{c}{{\Dfour} excitation energies (eV)} \\
\cline{3-5} \cline{6-8}
Method & AB & $1\,{}^3B_{1g} $~(99\%) & $1\,{}^1B_{1g} $ (95\%)& $2\,{}^1A_{g} $(1\%) & $1\,{}^3A_{2g} $ & $2\,{}^1A_{1g} $ & $1\,{}^1B_{2g} $ \\
Method & AB (\si{\kcalmol}) & $1\,{}^3B_{1g} $~(99\%) & $1\,{}^1B_{1g} $ (95\%)& $2\,{}^1A_{g} $(1\%) & $1\,{}^3A_{2g} $ & $2\,{}^1A_{1g} $ & $1\,{}^1B_{2g} $ \\
\hline
SF-TD-B3LYP & $10.41$ & $0.241$ & $-0.926$ & $-0.161$ & $-0.164$ & $-1.028$ & $-1.501$ \\
SF-TD-PBE0 & $8.95$ & $0.220$ & $-0.829$ & $-0.068$ & $-0.163$ & $-0.903$ & $-1.357$ \\