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Manuscript/CBD.bib
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@ -1,13 +1,28 @@
%% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2022-04-07 21:53:18 +0200
%% Created for Pierre-Francois Loos at 2022-04-11 13:38:06 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Kancherla_2019,
abstract = {In recent years, visible light-induced excited-state transition-metal (TM) (Mn, Co, Cu, and Pd) catalysis has attracted significant attention for the development of various chemical transformations. In contrast to metal/photoredox dual catalysis that uses conventional photosensitizers and TMs cooperatively, photoexcited-state TM catalysis uses a single TM complex as both the photocatalyst (PC) and the cross-coupling catalyst, resulting in more sustainable and efficient reactions. Unlike the outer-sphere mechanism active in conventional photocatalysis, these TM catalysts operate through a photoinduced inner-sphere mechanism in which the substrate--TM interaction is crucial for the bond-breaking or bond-forming steps, making this system an important advance in efficient carbon--carbon (C--C) bond formation reactions. Given the importance of these TM complexes as next-generation PCs with distinct mechanisms, in this review we highlight recent developments in photoexcited TM catalysis for C--C bond formation.},
author = {Rajesh Kancherla and Krishnamoorthy Muralirajan and Arunachalam Sagadevan and Magnus Rueping},
date-added = {2022-04-11 13:37:20 +0200},
date-modified = {2022-04-11 13:38:06 +0200},
doi = {https://doi.org/10.1016/j.trechm.2019.03.012},
journal = {Trends Chem.},
number = {5},
pages = {510-523},
title = {Visible Light-Induced Excited-State Transition-Metal Catalysis},
volume = {1},
year = {2019},
bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S258959741930084X},
bdsk-url-2 = {https://doi.org/10.1016/j.trechm.2019.03.012}}
@article{Dutta_2013,
author = {Dutta,Achintya Kumar and Pal,Sourav and Ghosh,Debashree},
date-added = {2022-04-07 21:53:18 +0200},

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Manuscript/CBD.tex
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@ -99,7 +99,7 @@ A theoretical best estimate is defined for the automerization barrier and for ea
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Despite the fact that excited states are involved in ubiquitous processes such as photochemistry, \cite{Bernardi_1990,Bernardi_1996,Boggio-Pasqua_2007,Klessinger_1995,Olivucci_2010,Robb_2007,VanderLugt_1969} catalysis, and solar cells, \cite{Delgado_2010} none of the currently existing methods has shown to provide accurate excitation energies in all scenarios due to the complexity of the process, the size of the systems, the impact of the environment, and many other factors.
Despite the fact that excited states are involved in ubiquitous processes such as photochemistry, \cite{Bernardi_1990,Bernardi_1996,Boggio-Pasqua_2007,Klessinger_1995,Olivucci_2010,Robb_2007,VanderLugt_1969} catalysis, \cite{Kancherla_2019} and solar cells, \cite{Delgado_2010} none of the currently existing methods has shown to provide accurate excitation energies in all scenarios due to the complexity of the process, the size of the systems, the impact of the environment, and many other factors.
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 remains a very active field of theoretical quantum chemistry.\cite{Roos_1996,Piecuch_2002b,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 both experimental and theoretical chemistry for many decades. \cite{Bally_1980}
@ -109,11 +109,11 @@ This degeneracy is lifted by the so-called pseudo Jahn-Teller effect, \ie, by a
This was confirmed by several experimental studies by Pettis and co-workers \cite{Reeves_1969} and others. \cite{Irngartinger_1983,Ermer_1983,Kreile_1986}
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.
However, in the {\Dfour} symmetry, the {\sBoneg} ground state is a diradical that has two degenerate singly occupied frontier orbitals.
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.
Interestingly singlet {\sBoneg} 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 in the {\Dfour} arrangement.
The automerization barrier (AB) is defined as the difference between the square and rectangular ground-state energies.
Interestingly, the {\sBoneg} 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 in the {\Dfour} arrangement.
The automerization barrier (AB) is thus defined as the difference between the square and rectangular ground-state energies.
The energy of this barrier is estimated, experimentally, in the range of \SIrange{1.6}{10}{\kcalmol}, \cite{Whitman_1982} while previous state-of-the-art \textit{ab initio} calculations yield values in the \SIrange{7}{9}{\kcalmol} range. \cite{Eckert-Maksic_2006,Li_2009,Shen_2012,Zhang_2019}
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.
@ -132,7 +132,7 @@ Although multi-reference CC methods have been designed, \cite{Jeziorski_1981,Mah
In this context, an interesting alternative to multi-configurational and CC methods is provided by selected configuration interaction (SCI) methods, \cite{Bender_1969,Whitten_1969,Huron_1973,Giner_2013,Evangelista_2014,Giner_2015,Caffarel_2016b,Holmes_2016,Tubman_2016,Liu_2016,Ohtsuka_2017,Zimmerman_2017,Coe_2018,Garniron_2018} which are able to provide near full CI (FCI) ground- and excited-state energies of small molecules. \cite{Caffarel_2014,Caffarel_2016a,Scemama_2016,Holmes_2017,Li_2018,Scemama_2018,Scemama_2018b,Li_2020,Loos_2018a,Chien_2018,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020e,Garniron_2019,Eriksen_2020,Yao_2020,Williams_2020,Veril_2021,Loos_2021,Damour_2021}
For example, the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) method limits the exponential increase of the size of the CI expansion by retaining the most energetically relevant determinants only, using a second-order energetic criterion to select perturbatively determinants in the FCI space. \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2017,Garniron_2018,Garniron_2019}
%Nonetheless, SCI methods remain very expensive and can be applied to a limited number of situations.
Nonetheless, SCI methods remain very expensive and can be applied to a limited number of situations.
Finally, another option to deal with these chemical scenarios is to rely on the cheaper spin-flip formalism, established by Krylov in 2001, \cite{Krylov_2001a,Krylov_2001b,Krylov_2002,Casanova_2020} where one accesses the ground and doubly-excited states via a single (spin-flip) de-excitation and excitation from the lowest triplet state, respectively.
Obviously, spin-flip methods have their own flaws, especially spin contamination (\ie, the artificial mixing of electronic states with different spin multiplicities) due not only to the spin incompleteness in the spin-flip expansion but also to the potential spin contamination of the reference configuration. \cite{Casanova_2020}
@ -187,7 +187,7 @@ Typically, CCSD, CCSDT, and CCSDTQ as well as CC3 and CC4 calculations are achie
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 at the CCSDT level with MRCC. \cite{mrcc}
To avoid having to perform multi-reference CC calculations or 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 de-excitation and an excitation, respectively.
Hence, the open-shell ground state, {\sBoneg}, and the {\Btwog} state appear as a de-excitation and an excitation, respectively.
With respect to {\Aoneg}, {\sBoneg} has a dominant double excitation character, while {\Btwog} has a dominant single excitation character, hence their contrasting convergence behaviors with respect to the order of the CC expansion (see below).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -197,7 +197,7 @@ With respect to {\Aoneg}, {\sBoneg} has a dominant double excitation character,
State-averaged CASSCF (SA-CASSCF) calculations are performed for vertical transition energies, whereas state-specific CASSCF is used for computing the automerization barrier. \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).
Two active spaces have been considered: (i) a minimal (4e,4o) active space including the 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.
\titou{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}
For ionic excited 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}
Note that PC-NEVPT2 is theoretically more accurate than SC-NEVPT2 due to the larger number of external configurations and greater flexibility.
@ -234,7 +234,7 @@ Although there also exist spin-flip extension of EOM-CC methods, \cite{Krylov_20
\subsection{Theoretical best estimates}
\label{sec:TBE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
When technically possible, each level of theory is tested with four gaussian basis sets, namely, 6-31+G(d) and aug-cc-pVXZ with X $=$ D, T, and Q. \cite{Dunning_1989}
When technically possible, each level of theory is tested with four Gaussian basis sets, namely, 6-31+G(d) and aug-cc-pVXZ with X $=$ D, T, and Q. \cite{Dunning_1989}
This helps us to assess the convergence of each property with respect to the size of the basis set.
More importantly, for each studied quantity (i.e., the automerization barrier and the vertical excitation energies), we provide a theoretical best estimate (TBE) established in the aug-cc-pVTZ basis.
These TBEs are defined using extrapolated CCSDTQ/aug-cc-pVTZ values except in a single occasion where the NEVPT2(12,12) value is used.
@ -382,18 +382,18 @@ CCSDTQ & $7.51$ & $[ 7.89]$\fnm[4]& $[\bf 8.93]$\fnm[5]& $[ 9.11]$\fnm[6]\\
%%% FIGURE II %%%
\begin{figure*}
\includegraphics[width=0.8\linewidth]{AB_AVTZ}
\caption{\titou{Automerization barrier (in \si{\kcalmol}) of CBD at various levels of theory using the aug-cc-pVTZ basis.
See {\SupInf} for the total energies.}}
\caption{Error (with respect to the TBE) in the automerization barrier (in \si{\kcalmol}) of CBD at various levels of theory using the aug-cc-pVTZ basis.
See {\SupInf} for the total energies.}
\label{fig:AB}
\end{figure*}
%%% %%% %%% %%%
The results concerning the automerization barrier are reported in Table \ref{tab:auto_standard} for various basis sets and shown in Fig.~\ref{fig:AB} for the aug-cc-pVTZ basis.
Our TBE with this basis set is 8.93 \kcalmol.
Our TBE with this basis set is \SI{8.93}{\kcalmol}, which is in excellent agreement with previous studies \cite{Eckert-Maksic_2006,Li_2009,Shen_2012,Zhang_2019} (see {\SupInf}).
First, one can see large variations of the energy barrier at the SF-TD-DFT level, with differences as large as \SI{10}{\kcalmol} between the different functionals for a given basis set.
Nonetheless, it is clear that the performance of a given functional is directly linked to the amount of exact exchange at short range.
Indeed, hybrid functionals with a ca.~50\%\ fraction of short-range exact exchange (\eg, BH\&HLYP, M06-2X, and M11) perform significantly better than the functionals having a small fraction of short-range exact exchange (\eg, B3LYP, PBE0, CAM-B3LYP, $\omega$B97X-V, and LC-$\omega$PBE08).
Indeed, hybrid functionals with approximately 50\% of short-range exact exchange (\eg, BH\&HLYP, M06-2X, and M11) perform significantly better than the functionals having a small fraction of short-range exact exchange (\eg, B3LYP, PBE0, CAM-B3LYP, $\omega$B97X-V, and LC-$\omega$PBE08).
However, they are still off by \SIrange{1}{4}{\kcalmol} from the TBE reference value, the most accurate result being obtained with M06-2X.
For the RSH functionals, the automerization barrier is much less sensitive to the amount of longe-range exact exchange.
Another important feature of SF-TD-DFT is the fast convergence of the energy barrier with the size of the basis set. \cite{Loos_2019d}
@ -402,7 +402,7 @@ With the augmented double-$\zeta$ basis, the SF-TD-DFT results are basically con
For the SF-ADC family of methods, the energy differences are much smaller with a maximum deviation of \SI{2}{\kcalmol} between different versions.
In particular, we observe that SF-ADC(2)-s and SF-ADC(3), which respectively scale as $\order*{N^5}$ and $\order*{N^6}$ (where $N$ is the number of basis functions), under- and overestimate the automerization barrier, making SF-ADC(2.5) a good compromise with an error of only \SI{0.18}{\kcalmol} compared to the TBE/aug-cc-pVTZ basis reference value.
Nonetheless, at a $\order*{N^5}$ computational scaling, SF-ADC(2)-s is particularly accurate, even compared to high-order CC methods (see below).
We note that SF-ADC(2)-x is probably not worth its extra cost [as compared to SF-ADC(2)-s] as it overestimates the energy barrier even more than SF-ADC(3).
We note that SF-ADC(2)-x [which scales as $\order*{N^6}$] is probably not worth its extra cost [as compared to SF-ADC(2)-s] as it overestimates the energy barrier even more than SF-ADC(3).
This behavior was previously reported by Dreuw's group. \cite{Wormit_2014,Harbach_2014,Dreuw_2015}
Overall, even with the best exchange-correlation functional, SF-TD-DFT is clearly outperformed by the more expensive SF-ADC models.
@ -597,7 +597,6 @@ CIPSI &6-31+G(d)& $1.486\pm 0.005$ & $3.348\pm 0.024$ & $4.084\pm 0.012$ \\
\includegraphics[width=0.8\linewidth]{D2h}
\caption{Vertical excitation energies of the {\tBoneg}, {\sBoneg}, and {\twoAg} states at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state using the aug-cc-pVTZ basis.
See {\SupInf} for the raw data.}
%Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multi-reference, CC, and TBE values, respectively.}
\label{fig:D2h}
\end{figure*}
%%% %%% %%% %%%
@ -626,12 +625,12 @@ Again, the extended version, SF-ADC(2)-x, does not seem to be relevant in the pr
Also, as reported previously, \cite{Loos_2020d} SF-ADC(2)-s and SF-ADC(3) have mirror error patterns making SF-ADC(2.5) particularly accurate except for the doubly-excited state {\twoAg} where the error with respect to the TBE (\SI{0.140}{\eV}) is larger than the SF-ADC(2)-s error (\SI{0.093}{\eV}).
Let us now move to the discussion of the results obtained with standard wave function methods that are reported in Table \ref{tab:D2h}.
Regarding the multi-configurational calculations, the most striking result is the poor description of the {\sBoneg} \titou{ionic} state, especially with the (4e,4o) active space where CASSCF predicts this state higher in energy than the {\twoAg} state.
Regarding the multi-configurational calculations, the most striking result is the poor description of the {\sBoneg} ionic state, especially with the (4e,4o) active space where CASSCF predicts this state higher in energy than the {\twoAg} state.
Of course, the PT2 correction is able to correct the state ordering problem but cannot provide quantitative excitation energies due to the poor zeroth-order treatment.
Another ripple effect of the unreliability of the reference wave function is the large difference between CASPT2 and NEVPT2 that differ by half an \si{\eV}.
This feature is characteristic of the inadequacy of the active space to model such a state.
For the two other states, {\tBoneg} and {\twoAg}, the errors at the CASPT2(4,4) and NEVPT2(4,4) levels are much smaller (below \SI{0.1}{\eV}).
Using a larger active space resolves most of these issues: CASSCF predicts the correct state ordering (though the \titou{ionic} state is still badly described in term of energetics), CASPT2 and NEVPT2 excitation energies are much closer, and their accuracy is often improved (especially for the triplet and doubly-excited states) although it is difficult to reach chemical accuracy (\ie, an error below \SI{0.043}{\eV}) on a systematic basis.
Using a larger active space resolves most of these issues: CASSCF predicts the correct state ordering (though the ionic state is still badly described in term of energetics), CASPT2 and NEVPT2 excitation energies are much closer, and their accuracy is often improved (especially for the triplet and doubly-excited states) although it is difficult to reach chemical accuracy (\ie, an error below \SI{0.043}{\eV}) on a systematic basis.
Finally, for the CC models (Table \ref{tab:D2h}), the two states with a large $\%T_1$ value, {\tBoneg} and {\sBoneg}, are already extremely accurate at the CC3 level, and systematically improved by CCSDT and CC4.
This trend is in line with the observations made on the QUEST database. \cite{Veril_2021}
@ -797,7 +796,6 @@ CIPSI & 6-31+G(d) & $0.201\pm 0.003$ & $1.602\pm 0.007$ & $2.13\pm 0.04$ \\
\includegraphics[width=0.8\linewidth]{D4h}
\caption{Vertical excitation energies (in \si{\eV}) of the {\Atwog}, {\Aoneg}, and {\Btwog} states at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state using the aug-cc-pVTZ basis.
See {\SupInf} for the raw data.}
%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*}
%%% %%% %%% %%%
@ -829,9 +827,9 @@ Let us turn to the multi-reference results (Table \ref{tab:D4h}).
For both active spaces, expectedly, CASSCF does not provide a quantitive energetic description, although it is worth mentioning that the right state ordering is preserved.
This is, of course, magnified with the (4e,4o) active space for which the second-order perturbative treatment is unable to provide a satisfying description due to the limited active space.
In particular SC-NEVPT2(4,4)/aug-cc-pVTZ and PC-NEVPT2(4,4)/aug-cc-pVTZ underestimate the singlet-triplet gap by \SI{0.072}{} and \SI{0.097}{\eV} and, more importantly, flip the ordering of {\Aoneg} and {\Btwog}.
Although {\Aoneg} is not badly described, the excitation energy of the \titou{ionic} state {\Btwog} is off by almost \SI{1}{\eV}.
Thanks to the IPEA shift in CASPT2(4,4), the singlet-triplet gap is accurate and the state ordering remains correct but the \titou{ionic} state is still far from being well described.
The (12e,12o) active space significantly alleviates these effects, and, as usual now, the agreement between CASPT2 and NEVPT2 is very much improved for each state, though the accuracy of multi-configurational approaches remains questionable for the \titou{ionic} state with, \emph{e.g.,} an error up to \SI{-0.093}{\eV} at the PC-NEVPT2(12,12)/aug-cc-pVTZ level.
Although {\Aoneg} is not badly described, the excitation energy of the ionic state {\Btwog} is off by almost \SI{1}{\eV}.
Thanks to the IPEA shift in CASPT2(4,4), the singlet-triplet gap is accurate and the state ordering remains correct but the ionic state is still far from being well described.
The (12e,12o) active space significantly alleviates these effects, and, as usual now, the agreement between CASPT2 and NEVPT2 is very much improved for each state, though the accuracy of multi-configurational approaches remains questionable for the ionic state with, \emph{e.g.,} an error up to \SI{-0.093}{\eV} at the PC-NEVPT2(12,12)/aug-cc-pVTZ level.
Finally, let us analyze the excitation energies computed with various CC models that are gathered in Table \ref{tab:D4h}.
As mentioned in Sec.~\ref{sec:CC}, we remind the reader that these calculations are performed by considering the {\Aoneg} state as reference, and that, therefore,
@ -865,7 +863,7 @@ However, it was satisfying to see that the spin-flip version of ADC can lower th
\item Concerning the multi-configurational methods, we have found that while NEVPT2 and CASPT2 can provide different excitation energies for the small (4e,4o) active space, the results become highly similar when the larger (12e,12o) active space is considered.
From a more general perspective, a significant difference between NEVPT2 and CASPT2 is usually not a good omen and can be seen as a clear warning sign that the active space is too small or poorly chosen.
The \titou{ionic} states remain a struggle for both CASPT2 and NEVPT2, even with the (12e,12o) active space.
The ionic states remain a struggle for both CASPT2 and NEVPT2, even with the (12e,12o) active space.
\item In the context of CC methods, although the inclusion of triple excitations (via CC3 or CCSDT) yields very satisfactory results in most cases, the inclusion of quadruples excitation (via CC4 or CCSDTQ) is mandatory to reach high accuracy (especially in the case of doubly-excited states).
Finally, we point out that, considering the error bar related to the CIPSI extrapolation procedure, CCSDTQ and CIPSI yield equivalent excitation energies, hence confirming the outstanding accuracy of CCSDTQ in the context of molecular excited states.