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ok with geometries for now

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Pierre-Francois Loos 2022-03-30 22:57:16 +02:00
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28
Manuscript/CBD.bib
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@ -1,13 +1,39 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/ %% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-03-30 15:46:27 +0200 %% Created for Pierre-Francois Loos at 2022-03-30 22:18:22 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{BenAmor_2020,
author = {Ben Amor,Nadia and No{\^u}s,Camille and Trinquier,Georges and Malrieu,Jean-Paul},
date-added = {2022-03-30 22:17:29 +0200},
date-modified = {2022-03-30 22:17:45 +0200},
doi = {10.1063/5.0011582},
journal = {J. Chem. Phys},
number = {4},
pages = {044118},
title = {Spin polarization as an electronic cooperative effect},
volume = {153},
year = {2020},
bdsk-url-1 = {https://doi.org/10.1063/5.0011582}}
@article{Davidson_1996,
author = {Davidson, Ernest R.},
date-added = {2022-03-30 22:16:35 +0200},
date-modified = {2022-03-30 22:17:55 +0200},
doi = {10.1021/jp952794n},
journal = {J. Phys. Chem},
number = {15},
pages = {6161-6166},
title = {The Spatial Extent of the V State of Ethylene and Its Relation to Dynamic Correlation in the Cope Rearrangement},
volume = {100},
year = {1996},
bdsk-url-1 = {https://doi.org/10.1021/jp952794n}}
@article{Christiansen_1996b, @article{Christiansen_1996b,
author = {Ove Christiansen and Henrik Koch and Poul J{\o}rgensen}, author = {Ove Christiansen and Henrik Koch and Poul J{\o}rgensen},
date-added = {2022-03-30 15:45:28 +0200}, date-added = {2022-03-30 15:45:28 +0200},

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Manuscript/CBD.tex
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@ -36,160 +36,11 @@
\newcommand{\QP}{\textsc{quantum package}} \newcommand{\QP}{\textsc{quantum package}}
\newcommand{\T}[1]{#1^{\intercal}} \newcommand{\T}[1]{#1^{\intercal}}
% coordinates
\newcommand{\br}{\mathbf{r}}
\newcommand{\dbr}{d\br}
% methods
\newcommand{\evGW}{ev$GW$}
\newcommand{\qsGW}{qs$GW$}
\newcommand{\GOWO}{$G_0W_0$}
\newcommand{\Hxc}{\text{Hxc}}
\newcommand{\xc}{\text{xc}}
\newcommand{\Ha}{\text{H}}
\newcommand{\co}{\text{c}}
\newcommand{\x}{\text{x}}
%
\newcommand{\Norb}{N_\text{orb}}
\newcommand{\Nocc}{O}
\newcommand{\Nvir}{V}
% operators
\newcommand{\hH}{\Hat{H}}
\newcommand{\hS}{\Hat{S}}
% methods
\newcommand{\KS}{\text{KS}}
\newcommand{\HF}{\text{HF}}
\newcommand{\RPA}{\text{RPA}}
\newcommand{\BSE}{\text{BSE}}
\newcommand{\dBSE}{\text{dBSE}}
\newcommand{\GW}{GW}
\newcommand{\stat}{\text{stat}}
\newcommand{\dyn}{\text{dyn}}
\newcommand{\TDA}{\text{TDA}}
% energies
\newcommand{\Enuc}{E^\text{nuc}}
\newcommand{\Ec}{E_\text{c}}
\newcommand{\EHF}{E^\text{HF}}
\newcommand{\EBSE}{E^\text{BSE}}
\newcommand{\EcRPA}{E_\text{c}^\text{RPA}}
\newcommand{\EcBSE}{E_\text{c}^\text{BSE}}
% orbital energies
\newcommand{\e}[1]{\eps_{#1}}
\newcommand{\eHF}[1]{\eps^\text{HF}_{#1}}
\newcommand{\eKS}[1]{\eps^\text{KS}_{#1}}
\newcommand{\eQP}[1]{\eps^\text{QP}_{#1}}
\newcommand{\eGOWO}[1]{\eps^\text{\GOWO}_{#1}}
\newcommand{\eGW}[1]{\eps^{GW}_{#1}}
\newcommand{\eevGW}[1]{\eps^\text{\evGW}_{#1}}
\newcommand{\eGnWn}[2]{\eps^\text{\GnWn{#2}}_{#1}}
\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
\newcommand{\tOm}[2]{\Tilde{\Omega}_{#1}^{#2}}
\newcommand{\homu}{\frac{{\omega}_1}{2}}
% Matrix elements
\newcommand{\A}[2]{A_{#1}^{#2}}
\newcommand{\tA}[2]{\Tilde{A}_{#1}^{#2}}
\newcommand{\B}[2]{B_{#1}^{#2}}
\renewcommand{\S}[1]{S_{#1}}
\newcommand{\ABSE}[2]{A_{#1}^{#2,\text{BSE}}}
\newcommand{\BBSE}[2]{B_{#1}^{#2,\text{BSE}}}
\newcommand{\ARPA}[2]{A_{#1}^{#2,\text{RPA}}}
\newcommand{\BRPA}[2]{B_{#1}^{#2,\text{RPA}}}
\newcommand{\ARPAx}[2]{A_{#1}^{#2,\text{RPAx}}}
\newcommand{\BRPAx}[2]{B_{#1}^{#2,\text{RPAx}}}
\newcommand{\G}[1]{G_{#1}}
\newcommand{\LBSE}[1]{L_{#1}}
\newcommand{\XiBSE}[1]{\Xi_{#1}}
\newcommand{\Po}[1]{P_{#1}}
\newcommand{\W}[2]{W_{#1}^{#2}}
\newcommand{\tW}[2]{\widetilde{W}_{#1}^{#2}}
\newcommand{\Wc}[1]{W^\text{c}_{#1}}
\newcommand{\vc}[1]{v_{#1}}
\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}}
\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
\newcommand{\SigXC}[1]{\Sigma^\text{xc}_{#1}}
\newcommand{\Z}[1]{Z_{#1}}
\newcommand{\MO}[1]{\phi_{#1}}
\newcommand{\ERI}[2]{(#1|#2)}
\newcommand{\rbra}[1]{(#1|}
\newcommand{\rket}[1]{|#1)}
\newcommand{\sERI}[2]{[#1|#2]}
%% bold in Table %% bold in Table
\newcommand{\bb}[1]{\textbf{#1}} \newcommand{\bb}[1]{\textbf{#1}}
\newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}} \newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}}
\newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}} \newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}}
% excitation energies
\newcommand{\OmRPA}[1]{\Omega_{#1}^{\text{RPA}}}
\newcommand{\OmRPAx}[1]{\Omega_{#1}^{\text{RPAx}}}
\newcommand{\OmBSE}[1]{\Omega_{#1}^{\text{BSE}}}
% Matrices
\newcommand{\bO}{\mathbf{0}}
\newcommand{\bI}{\mathbf{1}}
\newcommand{\bvc}{\mathbf{v}}
\newcommand{\bSig}{\mathbf{\Sigma}}
\newcommand{\bSigX}{\mathbf{\Sigma}^\text{x}}
\newcommand{\bSigC}{\mathbf{\Sigma}^\text{c}}
\newcommand{\bSigGW}{\mathbf{\Sigma}^{GW}}
\newcommand{\be}{\mathbf{\epsilon}}
\newcommand{\beGW}{\mathbf{\epsilon}^{GW}}
\newcommand{\beGnWn}[1]{\mathbf{\epsilon}^\text{\GnWn{#1}}}
\newcommand{\bde}{\mathbf{\Delta\epsilon}}
\newcommand{\bdeHF}{\mathbf{\Delta\epsilon}^\text{HF}}
\newcommand{\bdeGW}{\mathbf{\Delta\epsilon}^{GW}}
\newcommand{\bOm}[1]{\mathbf{\Omega}^{#1}}
\newcommand{\bA}[2]{\mathbf{A}_{#1}^{#2}}
\newcommand{\bB}[2]{\mathbf{B}_{#1}^{#2}}
\newcommand{\bX}[2]{\mathbf{X}_{#1}^{#2}}
\newcommand{\bY}[2]{\mathbf{Y}_{#1}^{#2}}
\newcommand{\bZ}[2]{\mathbf{Z}_{#1}^{#2}}
\newcommand{\bK}{\mathbf{K}}
\newcommand{\bP}[1]{\mathbf{P}^{#1}}
% units
\newcommand{\IneV}[1]{#1 eV}
\newcommand{\InAU}[1]{#1 a.u.}
\newcommand{\InAA}[1]{#1 \AA}
%\newcommand{\kcal}{kcal/mol}
% orbitals, gaps, etc
\newcommand{\eps}{\varepsilon}
\newcommand{\IP}{I}
\newcommand{\EA}{A}
\newcommand{\HOMO}{\text{HOMO}}
\newcommand{\LUMO}{\text{LUMO}}
\newcommand{\Eg}{E_\text{g}}
\newcommand{\EgFun}{\Eg^\text{fund}}
\newcommand{\EgOpt}{\Eg^\text{opt}}
\newcommand{\EB}{E_B}
\newcommand{\sig}{\sigma}
\newcommand{\bsig}{{\Bar{\sigma}}}
\newcommand{\sigp}{{\sigma'}}
\newcommand{\bsigp}{{\Bar{\sigma}'}}
\newcommand{\taup}{{\tau'}}
\newcommand{\up}{\uparrow}
\newcommand{\dw}{\downarrow}
\newcommand{\upup}{\uparrow\uparrow}
\newcommand{\updw}{\uparrow\downarrow}
\newcommand{\dwup}{\downarrow\uparrow}
\newcommand{\dwdw}{\downarrow\downarrow}
\newcommand{\spc}{\text{sc}}
\newcommand{\spf}{\text{sf}}
%geometries %geometries
\newcommand{\Dtwo}{$D_{2h}$} \newcommand{\Dtwo}{$D_{2h}$}
\newcommand{\Dfour}{$D_{4h}$} \newcommand{\Dfour}{$D_{4h}$}
@ -264,7 +115,7 @@ Thus, the autoisomerization barrier (AB) is defined as the difference between th
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} 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. %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 the {\sBoneg} and {\Atwog} states for the square one. The lowest-energy excited states of CBD in both symetries are represented in Fig.~\ref{fig:CBD}, where we have reported the {\oneAg} and {\tBoneg} states for the rectangular geometry and the {\sBoneg} and {\Atwog} states 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. 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} 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}
@ -335,7 +186,7 @@ Alternatively to the ``complete'' CC models, one can also employed the CC2, \cit
Here, we have performed CC calculations using various codes. 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} with which only singlet excited states can be computed. Typically, CCSD, CCSDT, and CCSDTQ as well as CC3 and CC4 calculations are achieved with CFOUR, \cite{Matthews_2020} with which only singlet excited states can be computed.
In some case, we have also computed (singlet and triplet) excitation energies and properties (such as the percentage of single excitations involved in a given transition, namely $\%T_1$) at the CC3 and CCSDT levels with DALTON.\cite{Aidas_2014} In some cases, we have also computed (singlet and triplet) excitation energies and properties (such as the percentage of single excitations involved in a given transition, namely $\%T_1$) at the CC3 level with DALTON \cite{Aidas_2014} and CCSDT level with MRCC. \cite{mrcc}
To avoid having to perform multi-reference CC calculations and because one cannot perform high-level CC calculations in the restricted open-shell or unrestricted formalisms, it is worth mentioning that, for the {\Dfour} arrangement, we have considered the lowest \textit{closed-shell} singlet state {\Aoneg} as reference. To avoid having to perform multi-reference CC calculations and because one cannot perform high-level CC calculations in the restricted open-shell or unrestricted formalisms, it is worth mentioning that, for the {\Dfour} arrangement, we have considered the lowest \textit{closed-shell} singlet state {\Aoneg} as reference.
Hence, the open-shell ground state {\sBoneg} and the {\Btwog} state appear as a deexcitation and an excitation, respectively. Hence, the open-shell ground state {\sBoneg} and the {\Btwog} state appear as a deexcitation and an excitation, respectively.
@ -349,7 +200,7 @@ With respect to {\Aoneg}, {\sBoneg} has a dominant double excitation character,
State-averaged CASSCF (SA-CASSCF) calculations are performed with MOLPRO. \cite{Werner_2020} State-averaged CASSCF (SA-CASSCF) calculations are performed with MOLPRO. \cite{Werner_2020}
For each excited state, a set of state-averaged orbitals is computed by taking into account the excited state of interest as well as the ground state (even if it has a different symmetry). For each excited state, a set of state-averaged orbitals is computed by taking into account the excited state of interest as well as the ground state (even if it has a different symmetry).
Two active spaces have been considered: (i) a minimal (4e,4o) active space including valence $\pi$ orbitals, and (ii) an extended (12e,12o) active space where we have additionally included the $\sigma_\text{CC}$ and $\sigma_\text{CC}^*$ orbitals. Two active spaces have been considered: (i) a minimal (4e,4o) active space including valence $\pi$ orbitals, and (ii) an extended (12e,12o) active space where we have additionally included the $\sigma_\text{CC}$ and $\sigma_\text{CC}^*$ orbitals.
For ionic states, like the {\sBoneg} state of CBD, it is particularly important to take into account the $\sigma$-$\pi$ coupling. For ionic states, like the {\sBoneg} state of CBD, it is particularly important to take into account the $\sigma$-$\pi$ coupling. \cite{Davidson_1996,Angeli_2009,BenAmor_2020}
On top of this CASSCF treatment, CASPT2 calculations are performed within the RS2 contraction scheme, while the NEVPT2 energies are computed within both the partially contracted (PC) and strongly contracted (SC) schemes. \cite{Angeli_2001,Angeli_2001a,Angeli_2002} On top of this CASSCF treatment, CASPT2 calculations are performed within the RS2 contraction scheme, while the NEVPT2 energies are computed within both the partially contracted (PC) and strongly contracted (SC) schemes. \cite{Angeli_2001,Angeli_2001a,Angeli_2002}
Note that PC-NEVPT2 is theoretically more accurate than SC-NEVPT2 due to the larger number of external configurations and greater flexibility. Note that PC-NEVPT2 is theoretically more accurate than SC-NEVPT2 due to the larger number of external configurations and greater flexibility.
@ -379,8 +230,6 @@ 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. 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} 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. %EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -427,18 +276,19 @@ Note that, due to error bar inherently linked to the CIPSI calculations (see Sub
%================================================ %================================================
\subsection{Geometries} \subsection{Geometries}
\label{sec:geometries} \label{sec:geometries}
Two different sets of geometries obtained with different levels of theory are considered for the ground state property and for the excited states of the CBD molecule. Two different sets of geometries obtained with different levels of theory are considered for the autoisomerization barrier and the excited states of the CBD molecule.
First, because the autoisomerization barrier is computed as an energy difference between distinct geometries, it is paramount to obtain these at the same level of theory. First, because the autoisomerization barrier is obtained as a difference of energies computed at distinct geometries, it is paramount to obtain these at the same level of theory.
However, due to the fact that the ground state of the square arrangement is a transition state of singlet open-shell nature, it is technically difficult to optimize the geometry with high-order CC methods. However, due to the fact that the ground state of the square arrangement is a transition state of singlet open-shell nature, it is technically difficult to optimize the geometry with high-order CC methods.
Therefore, we have chosen to rely on CASPT2(12,12)/aug-cc-pVTZ for both the {\Dtwo} and {\Dfour} structures. Therefore, we rely on CASPT2(12,12)/aug-cc-pVTZ for both the {\Dtwo} and {\Dfour} ground-state structures.
Second, because the vertical transition energies are computed for a particular geometry, we can afford to use different methods for the rectangular and square structures. Second, because the vertical transition energies are computed for a particular equilibrium geometry, we can afford to use different methods for the rectangular and square structures.
Hence, we rely on CC3/aug-cc-pVTZ for the rectangular ({\Dtwo}) geometry and restricted open-shell version of CCSD(T)/aug-cc-pVTZ for the square ({\Dfour}) geometry. Hence, we rely on CC3/aug-cc-pVTZ to compute the equilibrium geometry of the {\oneAg} state in the rectangular ({\Dtwo}) arrangement and the restricted open-shell (RO) version of CCSD(T)/aug-cc-pVTZ to obtain the equilibrium geometry of the {\Atwog} state in the square ({\Dfour}) arrangement.
Table \ref{tab:geometries} reports the key geometrical parameters obtained with these different methods. These two geometries are the lowest-energy equilibrium structure of their respective spin manifold (see Fig.~\ref{fig:CBD}).
Table \ref{tab:geometries} reports the key geometrical parameters obtained at these levels of theory as well as previous geometries computed by Manohar and Krylov at the CCSD(T)/cc-pVTZ level.
%%% TABLE I %%% %%% TABLE I %%%
\begin{squeezetable} \begin{squeezetable}
\begin{table} \begin{table}
\caption{Optimized geometries of CBD for various states computed with various levels of theory. \caption{Optimized geometries associated with several states of CBD computed with various levels of theory.
Bond lengths are in \si{\angstrom} and angles are in degree.} Bond lengths are in \si{\angstrom} and angles are in degree.}
\label{tab:geometries} \label{tab:geometries}
\begin{ruledtabular} \begin{ruledtabular}
@ -453,7 +303,8 @@ Table \ref{tab:geometries} reports the key geometrical parameters obtained with
CASPT2(12,12)/aug-cc-pVTZ \fnm[2] & 1.449 & 1.449 & 1.076 & 135.00 \\ CASPT2(12,12)/aug-cc-pVTZ \fnm[2] & 1.449 & 1.449 & 1.076 & 135.00 \\
{\Dfour} ({\Atwog}) & {\Dfour} ({\Atwog}) &
CASPT2(12,12)/aug-cc-pVTZ \fnm[2] & 1.445 & 1.445 & 1.076 & 135.00 \\ CASPT2(12,12)/aug-cc-pVTZ \fnm[2] & 1.445 & 1.445 & 1.076 & 135.00 \\
&\alert{RO-CCSD(T)/aug-cc-pVTZ} \fnm[2] & 1.439 & 1.439 & 1.075 & 135.00 &RO-CCSD(T)/aug-cc-pVTZ \fnm[2] & 1.439 & 1.439 & 1.075 & 135.00\\
&RO-CCSD(T)/cc-pVTZ \fnm[3] & 1.439 & 1.439 & 1.073 & 135.00\\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Angle between the \ce{C-H} bond and the \ce{C=C} bond.} \fnt[1]{Angle between the \ce{C-H} bond and the \ce{C=C} bond.}