668 lines
34 KiB
TeX
668 lines
34 KiB
TeX
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\begin{document}
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% addresses
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\CEISAM}{Universit\'e de Nantes, CNRS, CEISAM UMR 6230, F-44000 Nantes, France}
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\title{Reference energies for cyclobutadiene}
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\author{Enzo \surname{Monino}}
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\affiliation{\LCPQ}
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\author{Martial \surname{Boggio-Pasqua}}
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\affiliation{\LCPQ}
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\author{Anthony \surname{Scemama}}
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\affiliation{\LCPQ}
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\author{Denis \surname{Jacquemin}}
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\affiliation{\CEISAM}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\begin{abstract}
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Write an abstract
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\end{abstract}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Despite the fact that excited states are involved in ubiquitious processes such as photochemistry, catalysis or in solar cell technology, none of the many methods existing is the reference in providing accurate excited states energies. Indeed, each method has its own flaws and there are so many chemical scenario that can occur, so it is still one of the biggest challenge in theoretical chemistry. Speaking of difficult task, cyclobutadiene (CBD) molecule has been a real challenge for experimental and theoretical chemists for many decades \cite{bally_cyclobutadiene_1980}. Due to his antiaromaticity \cite{noauthor_aromaticity_nodate} and his large angular strain \cite{baeyer_ueber_1885} the CBD molecule presents a high reactivity which made the synthesis of this molecule a particularly difficult exercise. Hückel molecular orbital theory gives a triplet state with square ($D_{4h}$) geometry for the ground state of the CBD,with the two singly occupied frontier orbitals that are degenerated by symmetry. This degeneracy is lifted by the Jahn-Teller effect, meaning by distortion of the molecule (lowering symmetry), and gives a singlet state with rectangular ($D_{2h}$) geometry for the ground state. Indeed, synthetic work from Pettis and co-workers \cite{reeves_further_1969} gives a rectangular geometry to the singlet ground state of CBD and then was confirmed by experimental works \cite{irngartinger_bonding_1983,ermer_three_1983,kreile_uv_1986}. At the ground state structrure ($D_{2h}$), the ${}^1A_g$ state has a weak multi-configurational character because of the well separated frontier orbitals and can be described by single-reference methods. But at the square ($D_{4h}$) geometry, the singlet state (${}^1B_{1g}$) 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 ($D_{4h}$) is a transition state in the automerization reaction between the two rectangular structures. The energy barrier for the automerization of the CBD was predicted, experimentally, in the range of 1.6-10 kcal.mol$^{-1}$ \cite{whitman_limits_1982} and multi-reference calculations gave an energy barrier in the range of 6-7 kcal.mol$^{-1}$ \cite{eckert-maksic_automerization_2006}. All the specificities of the CBD molecule make it a real playground for excited-states methods. In the present work we investigate excited states represented in Fig. \ref{fig:CBD} but not only, we also investigate higher states that are not present in Fig. \ref{fig:CBD} due to energy scaling. In those states we have doubly excited states that are known to be challenging to represent for adiabatic time-dependent density functional theory (TD-DFT) and even for state-of-the-art methods like the approximate third-order coupled-cluster (CC3) or equation-of-motion coupled-cluster with singles, doubles and triples (EOM-CCSDT). In order to tackle the problems of multi-configurational character and double excitations we use multi-reference perturbation theory methods like complete active space perturbation theory (CASPT2) and N-electron valence state perturbation theory (NEVPT2), we also use spin-flip formalism established by Krylov in 2001. To briefly introduce the spin-flip idea we can present it like: instead of taking the singlet ground state as reference, the reference configuration is taken as the lowest triplet state. So one can access the singlet ground state and the singlet doubly-excited state via a spin-flip deexcitation and excitation (respectively), the difference of these two excitation energies providing an estimate of the double excitation. Both excited states and automerization barrier of the CBD are studied using a collection of methods.
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\begin{figure}
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\includegraphics[width=0.6\linewidth]{figure2.png}
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\caption{Here comes the caption of the figure.}
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\label{fig:CBD}
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational methods}
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\label{sec:compmet}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Selected Configuration Interaction}
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\label{sec:SCI}
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States energies and excitations energies calculations in the SCI framework are performed using QUANTUM PACKAGE where the CIPSI algorithm is implemented. The CIPSI algorithm allows to avoid the exponential increase of th CI expansion. To treat electronic states in the same way we use a state-averaged formalism meaning that the ground and excited states are represented with the same number and same set of determinants but using different CI coefficients. Then the SCI energy is the sum of two terms, the variational energy obtained by diagonalization of the CI matrix in the reference space and a second-order perturbative correction which estimates the contribution of the determinants not included in the CI space (estimate error in the truncation). It is possible to estimate the FCI limit for the total energies and compute the corresponding transition energies by extrapolating this second-order correction to zero. Extrapolation brings error and we can estime this one by energy difference between excitation energies obtained with the largest SCI wave function and the FCI extrapolated value. These errors provide a rough idea of the quality of the FCI extrapolation and cannot be seen as true bar error, they are reported in the following tables.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Coupled-Cluster}
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\label{sec:CC}
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Different flavours of coupled-cluster (CC) calculations are performed using different codes. Indeed, CC theory provides a hierarchy of methods that provide increasingly accurate energies via the increase of the maximum excitation degree of the cluster operator. However, due to the computational exponential scaling with system size we have to use truncated CC methods. The CC with singles and doubles (CCSD), CC with singles, doubles and triples (CCSDT) calculations are achieved with \textcolor{red}{CFOUR}. The calculations in the context of CC response theory or ``approximate'' series (CC3,CC4) are performed with \textcolor{red}{DALTON}. The CC with singles, doubles, triples and quadruples (CCSDTQ) are done with the \textcolor{red}{CFOUR} code.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Multiconfigurational methods}
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\label{sec:Multi}
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State-averaged complete-active-space self-consistent field (SA-CASSCF) calculations are performed with \textcolor{red}{MOLPRO}. On top of those, NEVPT2 calculations, both partially contracted (PC) and strongly contracted (SC) scheme are considered. The PC-NEVPT2 is theoretically more accurate to the strongly contracted version due to the larger number of pertubers and greater flexibility. CASPT2 and extended multistate (XMS) CASPT2 are also performed.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Spin-Flip}
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\label{sec:sf}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational Details}
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\label{sec:compdet}
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The system under investigation in this work is the cyclobutadiene (CBD) molecule, rectangular ($D_{2h}$) and square ($D_{4h}$) geometries are considered. The ($D_{2h}$) geometry is obtained at the CC3 level without frozen core using the aug-cc-pVTZ and the ($D_{4h}$) geometry is obtained at the RO-CCSD(T) level using aug-cc-pVTZ again without frozen core. In both structures the CBD has a singlet ground state, for the spin-flip calculations we consider the lowest triplet state as reference. Spin-flip techniques are broadly accessible and here, among them, we explore equation-of-motion coupled-cluster singles and doubles (EOM-CCSD), configuration interaction singles (CIS), algebraic-diagrammatic construction (ADC) scheme and TD-DFT. The standard and extended spin-flip ADC(2) (SF-ADC(2)-s and SF-ADC(2)-x respectively) and SF-ADC(3) are performed using Q-CHEM 5.2.1. Spin-flip TD-DFT calculations are also performed using Q-CHEM 5.2.1. The BLYP, B3LYP, PBE0 and BH\&HLYP functionals are considered, they contain 0\%, 20\%, 25$\%$, 50\% of exact exchange and are labeled, respectively, as SF-BLYP, SF-B3LYP, SF-PBE0, SF-BH\&HLYP. We also have done spin-flip TD-DFT calculations using RSH functionals as: CAM-B3LYP, LC-$\omega$PBE08 and $\omega$B97X-V. The main difference here between these RSHs functionals is the amount of exact-exchange at long-range: 75$\%$ for CAM-B3LYP and 100$\%$ for LC-$\omega$PBE08 and $\omega$B97X-V. To complete the use of spin-flip TD-DFT we also considered GH meta-GGA functional M06-2X, RSH meta-GGA functional M11 and DH functionals B2PLYP and B2GPPLYP. EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1. All the calculations are performed using four basis set, the 6-31+G(d) basis and the aug-cc-pVXZ with X$=$D, T, Q. The $\%T_1$ metric that gives the percentage of single excitation calculated at the CC3 level in \textcolor{red}{DALTON} allows to characterize the various states.Throughout all this work, spin-flip and spin-conserved calculations are performed with a UHF reference.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results and discussion}
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\label{sec:res}
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As said in \ref{sec:intro} we study both excited states and automerization barrier. The excited states of interest in this work are the $^1 Ag$, $1 ^3B_{1g}$, $1 ^1B_{1g}$ and $2 ^1A_{g}$ states for the rectangular ($D_{2h}$) structure and the $1 ^1B_{1g}$, $1 ^3A_{2g}$, $2 ^1A_{1g}$ and $1 ^1B_{2g}$ states for the square ($D_{4h}$) structure. For the excited states part we study vertical excitations, as mentioned in \ref{sec:intro} the study of the CBD molecule is a difficult task due to the multi-configurational character of some excited states and there are not reference methods for the description of those. Because of this it is important to define our reference in this work to be able to compare the results of differents methods. To do so we use the Theoretical Best Estimates (TBE)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%================================================
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\subsection{Excited States}
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%%% TABLE I %%%
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\begin{squeezetable}
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\begin{table}
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\caption{
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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 $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
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\label{tab:sf_tddft_D2h}}
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\begin{ruledtabular}
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\begin{tabular}{llrrr}
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& \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\
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\cline{3-5}
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Method & Basis & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{g}$ \\
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\hline
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SF-TD-BLYP & 6-31+G(d) & $1.829$ & $1.926$ & $3.755$ \\
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& aug-cc-pVDZ & $1.828$ & $1.927$ & $3.586$ \\
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& aug-cc-pVTZ & $1.825$ & $1.927$ & $3.546$ \\
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& aug-cc-pVQZ & $1.825$ & $1.927$ & $3.528$ \\[0.1cm]
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SF-TD-B3LYP & 6-31+G(d) & $1.706$ & $2.211$ & $3.993$ \\
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& aug-cc-pVDZ & $1.706$ & $2.204$ & $3.992$ \\
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& aug-cc-pVTZ & $1.703$ & $2.199$ & $3.988$ \\
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& aug-cc-pVQZ & $1.703$ & $2.199$ & $3.989$\\[0.1cm]
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SF-TD-PBE0 & 6-31+G(d) & $1.687$ & $2.314$ & $4.089$ \\
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& aug-cc-pVDZ & $1.684$ & $2.301$ & $4.085$ \\
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& aug-cc-pVTZ & $1.682$ & $2.296$ & $4.081$ \\
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& aug-cc-pVQZ & $1.682$ & $2.296$ & $4.079$\\[0.1cm]
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SF-TD-BH\&HLYP & 6-31+G(d) & $1.552$ & $2.779$ & $4.428$ \\
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& aug-cc-pVDZ & $1.546$ & $2.744$ & $4.422$ \\
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& aug-cc-pVTZ & $1.540$ & $2.732$ & $4.492$ \\
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& aug-cc-pVQZ & $1.540$ & $2.732$ & $4.415$ \\[0.1cm]
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SF-TD-M06-2X & 6-31+G(d) & $1.477$ & $2.835$ & $4.378$ \\
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& aug-cc-pVDZ & $1.467$ & $2.785$ & $4.360$ \\
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& aug-cc-pVTZ & $1.462$ & $2.771$ & $4.357$ \\
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& aug-cc-pVQZ & $1.458$ & $2.771$ & $4.352$ \\[0.1cm]
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SF-TD-CAM-B3LYP & 6-31+G(d) & $1.750$ & $2.337$ & $3.315$ \\
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& aug-cc-pVDZ & $1.745$ & $2.323$ & $4.140$ \\
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& aug-cc-pVTZ & $1.742$ & $2.318$ & $4.138$ \\
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& aug-cc-pVQZ & $1.743$ & $2.319$ & $4.138$ \\[0.1cm]
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SF-TD-$\omega$B97X-V & 6-31+G(d) & $1.810$ & $2.377$ & $4.220$ \\
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& aug-cc-pVDZ & $1.800$ & $2.356$ & $4.217$ \\
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& aug-cc-pVTZ & $1.797$ & $2.351$ & $4.213$ \\[0.1cm]
|
|
SF-TD-M11 & 6-31+G(d) & $1.566$ & $2.687$ & $4.292$ \\
|
|
& aug-cc-pVDZ & $1.546$ & $2.640$ & $4.267$ \\
|
|
& aug-cc-pVTZ & $1.559$ & $2.651$ & $4.300$ \\
|
|
& aug-cc-pVQZ & $1.557$ & $2.650$ & $4.299$ \\[0.1cm]
|
|
SF-TD-LC-$\omega $PBE08 & 6-31+G(d) & $1.917$ & $2.445$ & $4.353$ \\
|
|
& aug-cc-pVDZ & $1.897$ & $2.415$ & $4.346$ \\
|
|
& aug-cc-pVTZ & $1.897$ & $2.415$ & $4.348$ \\
|
|
& aug-cc-pVQZ & $1.897$ & $2.415$ & $4.348$ \\[0.1cm]
|
|
SF-TD-B2PLYP & 6-31+G(d) & $1.538$ & $2.827$ & $4.462$ \\
|
|
& aug-cc-pVDZ & $1.531$ & $2.788$ & $4.455$ \\
|
|
& aug-cc-pVTZ & $1.525$ & $2.776$ & $4.448$ \\[0.1cm]
|
|
SF-TD-B2GPPLYP & 6-31+G(d) & \\
|
|
& aug-cc-pVDZ & \\
|
|
& aug-cc-pVTZ & \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table}
|
|
\end{squeezetable}
|
|
%%% %%% %%% %%%
|
|
|
|
%%% TABLE II %%%
|
|
\begin{squeezetable}
|
|
\begin{table}
|
|
\caption{
|
|
Spin-flip CIS, ADC and CC 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 $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
|
|
\label{tab:sf_adc_D2h}}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{llrrr}
|
|
& \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\
|
|
\cline{3-5}
|
|
Method & Basis & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{g}$ \\
|
|
\hline
|
|
SF-CIS & 6-31+G(d) & $1.514$ & $3.854$ & $5.379$ \\
|
|
& aug-cc-pVDZ & $1.487$ & $3.721$ & $5.348$ \\
|
|
& aug-cc-pVTZ & $1.472$ & $3.701$ & $5.342$ \\
|
|
& aug-cc-pVQZ & $1.471$ & $3.702$ & $5.342$ \\[0.1cm]
|
|
SF-ADC(2)-s & 6-31+G(d) & $1.577$ & $3.303$ & $4.196$ \\
|
|
& aug-cc-pVDZ & $1.513$ & $3.116$ & $4.114$ \\
|
|
& aug-cc-pVTZ & $1.531$ & $3.099$ & $4.131$ \\
|
|
& aug-cc-pVQZ & $1.544$ & $3.101$ & $4.140$ \\[0.1cm]
|
|
SF-ADC(2)-x & 6-31+G(d) & $1.557$ & $3.232$ & $3.728$ \\
|
|
& aug-cc-pVDZ & $1.524$ & $3.039$ & $3.681$ \\
|
|
& aug-cc-pVTZ & $1.539$ & $3.031$ & $3.703$ \\[0.1cm]
|
|
SF-ADC(3) & 6-31+G(d) & $1.435$ & $3.352$ & $4.242$ \\
|
|
& aug-cc-pVDZ & $1.422$ & $3.180$ & $4.208$ \\
|
|
& aug-cc-pVTZ & $1.419$ & $3.162$ & $4.224$ \\[0.1cm]
|
|
SF-EOM-CCSD & 6-31+G(d) & $1.663$ & $3.515$ & $4.275$ \\
|
|
& aug-cc-pVDZ & $1.611$ & $3.315$ & $3.856$ \\
|
|
& aug-cc-pVTZ & $1.609$ & $3.293$ & $4.245$ \\[0.1cm]
|
|
SF-EOM-CC(2,3) & 6-31+G(d) & $1.490$ & $3.333$ & $4.061$ \\
|
|
& aug-cc-pVDZ & $1.464$ & $3.156$ & $4.027$ \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table}
|
|
\end{squeezetable}
|
|
%%% %%% %%% %%%
|
|
|
|
%%% TABLE III %%%
|
|
\begin{squeezetable}
|
|
\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 $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
|
|
\label{tab:D2h}}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{llrrr}
|
|
& \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\
|
|
\cline{3-5}
|
|
Method & Basis & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{g}$ \\
|
|
\hline
|
|
CC3 &6-31+G(d)& $1.42$ & $3.341$ & $4.658$ \\
|
|
& aug-cc-pVDZ & $1.396$ & $3.158$ & $4.711$ \\
|
|
& aug-cc-pVTZ & $1.402$ & $3.119$ & $4.777$ \\
|
|
& aug-cc-pVQZ & $1.409$ & $3.113$ & $4.774$ \\[0.1cm]
|
|
CCSDT &6-31+G(d)& $1.442$ & $3.357$ & $4.311$ \\
|
|
& aug-cc-pVDZ & $1.411$ & $3.175$ & $4.327$ \\
|
|
& aug-cc-pVTZ & $1.411$ & $3.139$ & $4.429$ \\[0.1cm]
|
|
CC4 &6-31+G(d)& & $3.343$ & $4.067$ \\
|
|
& aug-cc-pVDZ & & $3.164$ & $4.041$ \\[0.1cm]
|
|
CCSDTQ &6-31+G(d)& & $3.34$ & $4.073$ \\[0.1cm]
|
|
SA2-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 & $1.67$ & $4.546$ & $4.441$ \\
|
|
& aug-cc-pVQZ & $1.671$ & $4.549$ & $4.44$ \\[0.1cm]
|
|
CASPT2(4,4) &6-31+G(d)& $1.44$ & $3.162$ & $4.115$ \\
|
|
& aug-cc-pVDZ & $1.414$ & $2.971$ & $4.068$ \\
|
|
& aug-cc-pVTZ & $1.412$ & $2.923$ & $4.072$ \\
|
|
& aug-cc-pVQZ & $1.417$ & $2.911$ & $4.081$ \\[0.1cm]
|
|
XMS-CASPT2(4,4) &6-31+G(d)& & & $4.151$ \\
|
|
& aug-cc-pVDZ & & & $4.105$ \\
|
|
& aug-cc-pVTZ & & & $4.114$ \\
|
|
& aug-cc-pVQZ & & & $4.125$ \\[0.1cm]
|
|
SC-NEVPT2(4,4) &6-31+G(d)& $1.407$ & $2.707$ & $4.145$ \\
|
|
& aug-cc-pVDZ & $1.381$ & $2.479$ & $4.109$ \\
|
|
& aug-cc-pVTZ & $1.379$ & $2.422$ & $4.108$ \\
|
|
& aug-cc-pVQZ & $1.384$ & $2.408$ & $4.125$ \\[0.1cm]
|
|
PC-NEVPT2(4,4) &6-31+G(d)& $1.409$ & $2.652$ & $4.12$ \\
|
|
& aug-cc-pVDZ & $1.384$ & $2.424$ & $4.084$ \\
|
|
& aug-cc-pVTZ & $1.382$ & $2.368$ & $4.083$ \\
|
|
& aug-cc-pVQZ & $1.387$ & $2.353$ & $4.091$ \\[0.1cm]
|
|
MRCI(4,4) &6-31+G(d)& $1.564$ & $3.802$ & $4.265$ \\
|
|
& aug-cc-pVDZ & $1.558$ & $3.67$ & $4.254$ \\
|
|
& aug-cc-pVTZ & $1.568$ & $3.678$ & $4.27$ \\
|
|
& aug-cc-pVQZ & $1.574$ & $3.681$ & $4.28$ \\[0.1cm]
|
|
SA2-CASSCF(12,12) &6-31+G(d)& $1.675$ & $3.924$ & $4.22$ \\
|
|
& 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]
|
|
CASPT2(12,12) &6-31+G(d)& $1.508$ & $3.407$ & $4.099$ \\
|
|
& aug-cc-pVDZ & $1.489$ & $3.256$ & $4.044$ \\
|
|
& aug-cc-pVTZ & $1.48$ & $3.183$ & $4.043$ \\
|
|
& aug-cc-pVQZ & $1.482$ & $3.163$ & $4.047$ \\[0.1cm]
|
|
XMS-CASPT2(12,12) &6-31+G(d)& & & $4.111$ \\
|
|
& aug-cc-pVDZ & & & $4.056$ \\
|
|
& aug-cc-pVTZ & & & $4.059$ \\
|
|
& aug-cc-pVQZ & & & $4.065$ \\[0.1cm]
|
|
SC-NEVPT2(12,12) &6-31+G(d)& $1.522$ & $3.409$ & $4.13$ \\
|
|
& aug-cc-pVDZ & $1.511$ & $3.266$ & $4.093$ \\
|
|
& aug-cc-pVTZ & $1.501$ & $3.188$ & $4.086$ \\
|
|
& aug-cc-pVQZ & $1.503$ & $3.167$ & $4.088$ \\[0.1cm]
|
|
PC-NEVPT2(12,12) &6-31+G(d)& $1.487$ & $3.296$ & $4.103$ \\
|
|
& aug-cc-pVDZ & $1.472$ & $3.141$ & $4.064$ \\
|
|
& aug-cc-pVTZ & $1.462$ & $3.063$ & $4.056$ \\
|
|
& aug-cc-pVQZ & $1.464$ & $3.043$ & $4.059$ \\[0.1cm]
|
|
MRCI(12,12) &6-31+G(d)& & & $4.125$ \\[0.1cm]
|
|
CIPSI &6-31+G(d)& $1.486\pm 0.005$ & $3.348\pm 0.024$ & $4.084\pm 0.012$ \\
|
|
& aug-cc-pVDZ & $1.458\pm 0.009$ & $3.187\pm 0.035$ & $4.04\pm 0.04$ \\
|
|
& aug-cc-pVTZ & $1.461\pm 0.030$ & $3.142\pm 0.035$ & $4.03\pm 0.09$ \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table}
|
|
\end{squeezetable}
|
|
%%% %%% %%% %%%
|
|
|
|
%%% TABLE IV %%%
|
|
\begin{squeezetable}
|
|
\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 $D_{4h}$ square-planar equilibrium geometry of the $1\,{}^3A_{2g}$ state.
|
|
\label{tab:D4h}}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{llrrr}
|
|
& \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\
|
|
\cline{3-5}
|
|
Method & Basis & $1\,{}^3A_{2g}$ & $2\,{}^1A_{1g}$ & $1\,{}^1B_{2g}$ \\
|
|
\hline
|
|
CCSD & 6-31+G(d) & $0.148$ & $1.788$ & \\
|
|
& aug-cc-pVDZ & $0.1$ & $1.65$ & \\
|
|
& aug-cc-pVTZ & $0.085$ & $1.6$ & \\
|
|
& aug-cc-pVQZ & $0.084$ & $1.588$ & \\[0.1cm]
|
|
CC3 & 6-31+G(d) & & $1.809$ & $2.836$ \\
|
|
& aug-cc-pVDZ & & $1.695$ & $2.646$ \\
|
|
& aug-cc-pVTZ & & $1.662$ & $2.72$ \\[0.1cm]
|
|
CCSDT & 6-31+G(d) & $0.21$ & $1.751$ & $2.565$ \\
|
|
& aug-cc-pVDZ & $0.165$ & $1.659$ & $2.45$ \\
|
|
& aug-cc-pVTZ & $0.149$ & $1.631$ & $2.537$ \\[0.1cm]
|
|
CC4 & 6-31+G(d) & & $1.604$ & $2.121$ \\
|
|
& aug-cc-pVDZ & & $1.539$ & $1.934$ \\[0.1cm]
|
|
CCSDTQ & 6-31+G(d) & $0.205$ & $1.593$ & $2.134$ \\[0.1cm]
|
|
SA2-CASSCF(4,4) & 6-31+G(d) & $0.447$ & $2.257$ & $3.549$ \\
|
|
& aug-cc-pVDZ & $0.438$ & $2.24$ & $3.443$ \\
|
|
& aug-cc-pVTZ & $0.434$ & $2.234$ & $3.424$ \\
|
|
& aug-cc-pVQZ & $0.435$ & $2.235$ & $3.427$ \\[0.1cm]
|
|
CASPT2(4,4) & 6-31+G(d) & $0.176$ & $1.588$ & $1.899$ \\
|
|
& aug-cc-pVDZ & $0.137$ & $1.54$ & $1.708$ \\
|
|
& aug-cc-pVTZ & $0.128$ & $1.506$ & $1.635$ \\
|
|
& aug-cc-pVQZ & $0.128$ & $1.498$ & $1.612$ \\[0.1cm]
|
|
SC-NEVPT2(4,4) & 6-31+G(d) & $0.083$ & $1.52$ & $1.38$ \\
|
|
& aug-cc-pVDZ & $0.037$ & $1.465$ & $1.14$ \\
|
|
& aug-cc-pVTZ & $0.024$ & $1.428$ & $1.055$ \\
|
|
& aug-cc-pVQZ & $0.024$ & $1.42$ & $1.03$ \\[0.1cm]
|
|
PC-NEVPT2(4,4) & 6-31+G(d) & $0.085$ & $1.496$ & $1.329$ \\
|
|
& aug-cc-pVDZ & $0.039$ & $1.44$ & $1.088$ \\
|
|
& aug-cc-pVTZ & $0.026$ & $1.403$ & $1.003$ \\
|
|
& aug-cc-pVQZ & $0.026$ & $1.395$ & $0.977$ \\[0.1cm]
|
|
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]
|
|
SA2-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 & $0.37$ & $1.943$ & $2.634$ \\
|
|
& aug-cc-pVQZ & $0.371$ & $1.945$ & $2.637$ \\[0.1cm]
|
|
CASPT2(12,12) & 6-31+G(d) & $0.235$ & $1.635$ & $2.17$ \\
|
|
& aug-cc-pVDZ & $0.203$ & $1.588$ & $2.015$ \\
|
|
& aug-cc-pVTZ & $0.183$ & $1.538$ & $1.926$ \\
|
|
& aug-cc-pVQZ & $0.179$ & $1.522$ & $1.898$ \\[0.1cm]
|
|
SC-NEVPT2(12,12) & 6-31+G(d) & $0.218$ & $1.644$ & $2.143$ \\
|
|
& aug-cc-pVDZ & $0.189$ & $1.6$ & $1.991$ \\
|
|
& aug-cc-pVTZ & $0.165$ & $1.546$ & $1.892$ \\
|
|
& aug-cc-pVQZ & $0.16$ & $1.529$ & $1.862$ \\[0.1cm]
|
|
PC-NEVPT2(12,12) & 6-31+G(d) & $0.189$ & $1.579$ & $2.02$ \\
|
|
& aug-cc-pVDZ & $0.156$ & $1.53$ & $1.854$ \\
|
|
& aug-cc-pVTZ & $0.131$ & $1.476$ & $1.756$ \\
|
|
& aug-cc-pVQZ & $0.126$ & $1.46$ & $1.727$ \\[0.1cm]
|
|
CIPSI & 6-31+G(d) & $0.2010\pm 0.0030$ & $1.602\pm 0.007$ & $2.13\pm 0.04$ \\
|
|
& aug-cc-pVDZ & $0.1570\pm 0.0030$ & $1.587\pm 0.005$ & $2.102\pm 0.027$ \\
|
|
& aug-cc-pVTZ & $0.169\pm 0.029$ & $1.63\pm 0.05$ & \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
|
|
\end{table}
|
|
\end{squeezetable}
|
|
%%% %%% %%% %%%
|
|
|
|
%%% TABLE V %%%
|
|
\begin{squeezetable}
|
|
\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 $D_{4h}$ square-planar equilibrium geometry of the $1\,{}^3A_{2g}$ state.
|
|
\label{tab:sf_D4h}}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{llrrr}
|
|
& \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\
|
|
\cline{3-5}
|
|
Method & Basis & $1\,{}^3A_{2g}$ & $2\,{}^1A_{1g}$ & $1\,{}^1B_{2g}$ \\
|
|
\hline
|
|
SF-CIS & 6-31+G(d) & $0.355$ & $2.742$ & $3.101$ \\
|
|
& aug-cc-pVDZ & $0.318$ & $2.593$ & $3.052$ \\
|
|
& aug-cc-pVTZ & $0.305$ & $2.576$ & $3.053$ \\
|
|
& aug-cc-pVQZ & $0.306$ & $2.577$ & $3.056$ \\[0.1cm]
|
|
SF-TD-B3LYP & 6-31+G(d) & $-0.016$ & $0.487$ & $0.542$ \\
|
|
& aug-cc-pVDZ & $-0.019$ & $0.477$ & $0.536$ \\
|
|
& aug-cc-pVTZ & $-0.020$ & $0.472$ & $0.533$ \\
|
|
& aug-cc-pVQZ & $-0.020$ & $0.473$ & $0.533$ \\[0.1cm]
|
|
SF-TD-PBE0 & 6-31+G(d) & $-0.012$ & $0.618$ & $0.689$ \\
|
|
& aug-cc-pVDZ & $-0.016$ & $0.602$ & $0.680$ \\
|
|
& aug-cc-pVTZ & $-0.019$ & $0.597$ & $0.677$ \\
|
|
& aug-cc-pVQZ & $-0.018$ & $0.597$ & $0.677$ \\[0.1cm]
|
|
SF-TD-BH\&HLYP& 6-31+G(d) & $0.064$ & $1.305$ & $1.458$ \\
|
|
& aug-cc-pVDZ & $0.051$ & $1.260$ & $1.437$ \\
|
|
& aug-cc-pVTZ & $0.045$ & $1.249$ & $1.431$ \\
|
|
& aug-cc-pVQZ & $0.046$ & $1.250$ & $1.432$ \\[0.1cm]
|
|
SF-TD-M06-2X & 6-31+G(d) & $0.102$ & $1.476$ & $1.640$ \\
|
|
& aug-cc-pVDZ & $0.086$ & $1.419$ & $1.611$ \\
|
|
& aug-cc-pVTZ & $0.078$ & $1.403$ & $1.602$ \\
|
|
& aug-cc-pVQZ & $0.079$ & $1.408$ & $1.607$ \\[0.1cm]
|
|
SF-TD-CAM-B3LYP & 6-31+G(d) & $0.021$ & $0.603$ & $0.672$ \\
|
|
& aug-cc-pVDZ & $0.012$ & $0.585$ & $0.666$ \\
|
|
& aug-cc-pVTZ & $0.010$ & $0.580$ & $0.664$ \\
|
|
& aug-cc-pVQZ & $0.010$ & $0.580$ & $0.664$ \\[0.1cm]
|
|
SF-TD-$\omega $B97X-V & 6-31+G(d) & $0.040$ & $0.600$ & $0.670$ \\
|
|
& aug-cc-pVDZ & $0.029$ & $0.576$ & $0.664$ \\
|
|
& aug-cc-pVTZ & $0.026$ & $0.572$ & $0.662$ \\
|
|
& aug-cc-pVQZ & $0.026$ & $0.572$ & $0.662$ \\[0.1cm]
|
|
SF-TD-M11 & 6-31+G(d) & $0.102$ & $1.236$ & $1.374$ \\
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& aug-cc-pVDZ & $0.087$ & $1.196$ & $1.362$ \\
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& aug-cc-pVTZ & $0.081$ & $1.188$ & $1.359$ \\
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& aug-cc-pVQZ & $0.080$ & $1.185$ & $1.357$ \\[0.1cm]
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SF-TD-LC-$\omega $PBE08 & 6-31+G(d) & $0.078$ & $0.593$ & $0.663$ \\
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& aug-cc-pVDZ & $0.060$ & $0.563$ & $0.659$ \\
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& aug-cc-pVTZ & $0.058$ & $0.561$ & $0.658$ \\
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& aug-cc-pVQZ & $0.058$ & $0.561$ & $0.659$ \\[0.1cm]
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SF-ADC(2)-s & 6-31+G(d) & $0.345$ & $1.760$ & $2.096$ \\
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& aug-cc-pVDZ & $0.269$ & $1.656$ & $1.894$ \\
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& aug-cc-pVTZ & $0.256$ & $1.612$ & $1.844$ \\[0.1cm]
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SF-ADC(2)-x & 6-31+G(d) & $0.264$ & $1.181$ & $1.972$ \\
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& aug-cc-pVDZ & $0.216$ & $1.107$ & $1.760$ \\
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& aug-cc-pVTZ & $0.212$ & $1.091$ & $1.731$ \\[0.1cm]
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SF-ADC(3) & 6-31+G(d) & $0.123$ & $1.650$ & $2.078$ \\
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& aug-cc-pVDZ & $0.088$ & $1.571$ & $1.878$ \\
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& aug-cc-pVTZ & $0.079$ & $1.575$ & $1.853$ \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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\end{squeezetable}
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%%% %%% %%% %%%
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%================================================
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%================================================
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\subsection{Autoisomerization barrier}
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The autoisomerization barrier for the CBD molecule is defined as the energy difference between the singlet ground state of the square ($D_{4h}$) structure and the singlet ground state of the rectangular ($D_{2h}$) geometry. Results for the calculation of the automerization barrier are shown in Tables \ref{tab:auto_standard} and \ref{tab:auto_spin_flip}. As said in \ref{sec:intro} the range for this barrier is quite large. So again, it is important to define our reference in this work in order to be able to compare our results. Table \ref{tab:auto_standard} gives standard methods results, we can observe a large difference for the autoisomerization barrier between the multi-configurational methods. Indeed, for the CASSCF(12,12) we have a difference of the order of 3 kcal.mol$^{-1}$ with CASPT2(12,12) and NEVPT2(12,12) for all the basis. However, the difference between CASPT2(12,12) and NEVPT2(12,12) is much smaller, of the order of 0.2 kcal.mol$^{-1}$ for all the basis.
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%%% TABLE VI %%%
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\begin{squeezetable}
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\begin{table}
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\caption{Autoisomerization barrier for standard methods in kcal.mol$^{-1}$.}
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\label{tab:auto_standard}
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\begin{ruledtabular}
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\begin{tabular}{llrrrr}
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Method & 6-31+G(d) & aug-cc-pVDZ& aug-cc-pVTZ & aug-cc-pVQZ\\
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\hline
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CASSCF(12,12) & $10.19$ & $10.75$ & $11.59$ & $11.62$ \\
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CASPT2(12,12) & $7.24$ & $7.53$ & $8.51$ & $8.71$ \\
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NEVPT2(12,12) & $7.12$ & $7.33$ & $8.28$ & $8.49$ \\
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CCSD & $8.31$ & $8.8$ & $9.88$ & $10.1$ \\
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CC3 & $6.59$ & $6.89$ & $7.88$ & $8.06$ \\
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CCSDT & $7.26$ & $7.64$ & $8.68$ & \\
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CC4 & $7.4$ & $7.78$ & & \\
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CCSDTQ & $7.51$ & & & \\
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CIPSI & $7.91\pm 0.21$ & $8.58\pm 0.14$ & & \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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\end{squeezetable}
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%%% %%% %%% %%%
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%%% TABLE VII %%%
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\begin{squeezetable}
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\begin{table}
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\caption{Autoisomerization barrier for spin-flip methods in kcal.mol$^{-1}$.}
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\label{tab:auto_spin_flip}
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\begin{ruledtabular}
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\begin{tabular}{llrrrr}
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Method & 6-31+G(d) & AVDZ& AVTZ & AVQZ\\
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|
\hline
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SF-CIS & $2.64$ & $2.82$ & $3.43$ & $3.43$ \\
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SF-TD-BLYP & $23.57$ & $23.62$ & $24.23$ & $24.22$ \\
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SF-TD-B3LYP & $18.84$ & $18.93$ & $19.57$ & $19.57$ \\
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SF-TD-PBE0 & $17.31$ & $17.36$ & $18.01$ & $18.00$ \\
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SF-TD-BH\&HLYP & $11.90$ & $12.07$ & $12.73$ & $12.73$ \\
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SF-TD-M06-2X & $9.34$ & $9.68$ & $10.39$ & $10.40$ \\
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SF-TD-CAM-B3LYP & $18.21$ & $18.30$ & $18.98$ & $18.97$ \\
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SF-TD-$\omega$B97X-V & $18.46$ & $18.48$ & $19.14$ & $19.12$ \\
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SF-TD-M11 & $11.13$ & $10.38$ & $11.28$ & $11.19$ \\
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SF-TD-B2PLYP & $11.37$ & $11.54$ & $12.19$ & \\
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SF-TD-B2GPPLYP & $8.86$ & $9.05$ & $9.70$ & \\
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SF-ADC2-s & $6.69$ & $7.15$ & $8.64$ & $8.85$ \\
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|
SF-ADC2-x & $8.66$ & $9.15$ & $10.40$ & \\
|
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SF-ADC3 & $8.06$ & $8.76$ & $9.58$ & \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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\end{squeezetable}
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%%% %%% %%% %%%
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%================================================
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Conclusion}
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\label{sec:res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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\acknowledgements{
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EM, AS, and PFL acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\bibliography{CBD}
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\end{document}
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