From c1af148d020c16482e5b61b2c9ba8fa66d208b23 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Thu, 31 Mar 2022 10:52:56 +0200 Subject: [PATCH] ok up to geom --- Manuscript/CBD.tex | 40 +++++++++++++++++++--------------------- 1 file changed, 19 insertions(+), 21 deletions(-) diff --git a/Manuscript/CBD.tex b/Manuscript/CBD.tex index 131e368..cb95a00 100644 --- a/Manuscript/CBD.tex +++ b/Manuscript/CBD.tex @@ -85,8 +85,8 @@ \begin{abstract} The cyclobutadiene molecule is a well-known playground for theoretical chemists and is particularly suitable to test ground- and excited-state methods. Indeed, due to its high spatial symmetry, especially at the $D_{4h}$ square geometry but also in the $D_{2h}$ rectangular arrangement, the ground and excited states of cyclobutadiene exhibit multi-configurational characters and single-reference methods, such as adiabatic time-dependent density-functional theory (TD-DFT) or equation-of-motion coupled cluster (EOM-CC), are notoriously known to struggle in such situations. -In this work, using a large panel of methods and basis sets, we provide an extensive computational study of the autoisomerization barrier (defined as the difference between the square and rectangular ground-state energies) and the vertical excitation energies of the $D_{2h}$ and $D_{4h}$ arrangements. -In particular, selected configuration interaction (SCI), multi-reference perturbation theory (CASSCF, CASPT2, and NEVPT2), and coupled cluster (CCSD, CC3, CCSDT, CC4, and CCSDTQ) calculations are performed. +In this work, using a large panel of methods and basis sets, we provide an extensive computational study of the autoisomerization barrier (defined as the difference between the square and rectangular ground-state energies) and the vertical excitation energies of the $D_{2h}$ and $D_{4h}$ equilibrium structures. +In particular, selected configuration interaction (SCI), multi-reference perturbation theory (CASSCF, CASPT2, and NEVPT2), and coupled-cluster (CCSD, CC3, CCSDT, CC4, and CCSDTQ) calculations are performed. The spin-flip formalism, which is known to provide a correct description of states with multi-configurational character, is also tested within TD-DFT (where numerous exchange-correlation functionals are considered) and the algebraic diagrammatic construction [ADC(2)-s, ADC(2)-x, and ADC(3)]. A theoretical best estimate is defined for the autoisomerization barrier and for each vertical transition energy. \end{abstract} @@ -102,20 +102,19 @@ Despite the fact that excited states are involved in ubiquitous processes such a Indeed, each computational model has its own theoretical and/or technical limitations and the number of possible chemical scenarios 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 tasks, 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), while state-of-the-art \textit{ab initio} methods (correctly) predict an open-shell singlet ground state. +In the {\Dfour} symmetry, the simple H\"uckel molecular orbital theory (wrongly) predicts a triplet ground state (Hund's rule) with two singly-occupied frontier orbitals that are degenerate by symmetry, 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 (see below). 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} 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. +In the {\Dtwo} symmetry, the {\oneAg} ground state has a weak multi-configurational character with well-separated frontier orbitals that can be described by single-reference methods. +However, in the {\Dfour} symmetry, 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}), while the lowest triplet state {\Atwog} is a minimum on the triplet potential energy surface. +The singlet ground state, {\sBoneg}, 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 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. +The lowest-energy excited states of CBD in both symmetries 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. 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} @@ -180,7 +179,7 @@ This type of extrapolation procedures is now routine in SCI methods as well as o \subsection{Coupled-cluster calculations} \label{sec:CC} 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 diagonalizes the similarity-transformed Hamiltonian in a CI 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_1987a,Koch_1990b,Kucharski_1991,Stanton_1993,Christiansen_1998,Kucharski_2001,Kowalski_2001,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004} +As mentioned above, CC theory can be extended to excited states via the EOM formalism, \cite{Rowe_1968,Stanton_1993} where one diagonalizes the similarity-transformed Hamiltonian in a CI basis of excited determinants yielding the following systematically improvable family of methods for neutral excited states:\cite{Noga_1987a,Koch_1990b,Kucharski_1991,Stanton_1993,Christiansen_1998,Kucharski_2001,Kowalski_2001,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004} EOM-CCSD, EOM-CCSDT, EOM-CCSDTQ, etc. 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. @@ -289,27 +288,26 @@ Table \ref{tab:geometries} reports the key geometrical parameters obtained at th \begin{squeezetable} \begin{table} \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 ($\angle$) are in degree.} \label{tab:geometries} \begin{ruledtabular} \begin{tabular}{lllrrr} - State & Method & \ce{C=C} & \ce{C-C} & \ce{C-H} & \ce{H-C=C}\fnm[1] \\ + State & Method & \ce{C=C} & \ce{C-C} & \ce{C-H} & $\angle\,\ce{H-C=C}$ \\ \hline {\Dtwo} ({\oneAg}) & - CASPT2(12,12)/aug-cc-pVTZ \fnm[2] & 1.355 & 1.566 & 1.077 & 134.99 \\ - &CC3/aug-cc-pVTZ \fnm[2] & 1.344 & 1.565 & 1.076 & 135.08 \\ - &CCSD(T)/cc-pVTZ \fnm[3] & 1.343 & 1.566 & 1.074 & 135.09\\ + CASPT2(12,12)/aug-cc-pVTZ \fnm[1] & 1.355 & 1.566 & 1.077 & 134.99 \\ + &CC3/aug-cc-pVTZ \fnm[1] & 1.344 & 1.565 & 1.076 & 135.08 \\ + &CCSD(T)/cc-pVTZ \fnm[2] & 1.343 & 1.566 & 1.074 & 135.09\\ {\Dfour} ({\sBoneg}) & - CASPT2(12,12)/aug-cc-pVTZ \fnm[2] & 1.449 & 1.449 & 1.076 & 135.00 \\ + CASPT2(12,12)/aug-cc-pVTZ \fnm[1] & 1.449 & 1.449 & 1.076 & 135.00 \\ {\Dfour} ({\Atwog}) & - CASPT2(12,12)/aug-cc-pVTZ \fnm[2] & 1.445 & 1.445 & 1.076 & 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\\ + CASPT2(12,12)/aug-cc-pVTZ \fnm[1] & 1.445 & 1.445 & 1.076 & 135.00 \\ + &RO-CCSD(T)/aug-cc-pVTZ \fnm[1] & 1.439 & 1.439 & 1.075 & 135.00\\ + &RO-CCSD(T)/cc-pVTZ \fnm[2] & 1.439 & 1.439 & 1.073 & 135.00\\ \end{tabular} \end{ruledtabular} - \fnt[1]{Angle between the \ce{C-H} bond and the \ce{C=C} bond.} - \fnt[2]{This work.} - \fnt[3]{From Ref.~\onlinecite{Manohar_2008}.} + \fnt[1]{This work.} + \fnt[2]{From Ref.~\onlinecite{Manohar_2008}.} \end{table} \end{squeezetable} %%% %%% %%% %%%