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final lecture

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Manuscript/FCI2-SI.tex
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@ -1242,7 +1242,7 @@ Pyrrole &$^1A_2 (\Ryd; \pi \ra 3s)$ & 5.51$^a$ \\
&$^1A_2 (\Ryd; \pi \ra 3p)$ & 6.44$^c$ \\ &$^1A_2 (\Ryd; \pi \ra 3p)$ & 6.44$^c$ \\
&$^1B_2 (\Val; (\pi \ra \pis)$ & 6.48$^{e,f}$ \\ &$^1B_2 (\Val; (\pi \ra \pis)$ & 6.48$^{e,f}$ \\
&$^1A_1 (\Val; \pi \ra \pis)$ & 6.53$^d$ \\ &$^1A_1 (\Val; \pi \ra \pis)$ & 6.53$^d$ \\
&$^1B_2 (\Ryd; \pi \ra 3p)$ & 6.50$^d$,6.62$^e$ \\ &$^1B_2 (\Ryd; \pi \ra 3p)$ & 6.50$^d$,6.62$^{e,f}$ \\
&$^3B_2 (\Val; \pi \ra \pis)$ & 4.74$^d$ \\ &$^3B_2 (\Val; \pi \ra \pis)$ & 4.74$^d$ \\
&$^3A_2 (\Ryd; \pi \ra 3s)$ & 5.49$^a$ \\ &$^3A_2 (\Ryd; \pi \ra 3s)$ & 5.49$^a$ \\
&$^3A_1 (\Val; \pi \ra \pis)$ & 5.56$^d$ \\ &$^3A_1 (\Val; \pi \ra \pis)$ & 5.56$^d$ \\
@ -1337,7 +1337,7 @@ Thiophene &$^1A_1 (\Val; \pi \ra \pis)$ & 5.84$^a$ \\
&$^1B_1 (\Ryd; \pi \ra 3p)$ & 6.19$^e$ \\ &$^1B_1 (\Ryd; \pi \ra 3p)$ & 6.19$^e$ \\
&$^1A_2 (\Ryd; \pi \ra 3p)$ & 6.40$^e$,6.52$^f$ \\ &$^1A_2 (\Ryd; \pi \ra 3p)$ & 6.40$^e$,6.52$^f$ \\
&$^1B_1 (\Ryd; \pi \ra 3s)$ & 6.73$^d$, 6.71$^f$ \\ &$^1B_1 (\Ryd; \pi \ra 3s)$ & 6.73$^d$, 6.71$^f$ \\
&$^1B_2 (\Ryd; \pi \ra 3p)$ & 77.42$^b$,7.25$^c$ \\ &$^1B_2 (\Ryd; \pi \ra 3p)$ & 7.42$^b$,7.25$^c$ \\
&$^1A_1 (\Val; \pi \ra \pis)$ & 7.39$^{a,h}$ \\ &$^1A_1 (\Val; \pi \ra \pis)$ & 7.39$^{a,h}$ \\
&$^3B_2 (\Val; \pi \ra \pis)$ & 4.13$^a$ \\ &$^3B_2 (\Val; \pi \ra \pis)$ & 4.13$^a$ \\
&$^3A_1 (\Val; \pi \ra \pis)$ & 4.84$^a$ \\ &$^3A_1 (\Val; \pi \ra \pis)$ & 4.84$^a$ \\

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Manuscript/FCI2.tex
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@ -135,7 +135,7 @@ use equation-of-motion coupled cluster theory up to the highest technically poss
of millions of determinants in the reference space), as well as the multiconfigurational $n$-electron valence state perturbation theory (NEVPT2) method. All these approaches are applied in combination with diffuse-containing of millions of determinants in the reference space), as well as the multiconfigurational $n$-electron valence state perturbation theory (NEVPT2) method. All these approaches are applied in combination with diffuse-containing
atomic basis sets. For all transitions, we report at least CC3/\emph{aug}-cc-pVQZ vertical excitation energies as well as CC3/\emph{aug}-cc-pVTZ oscillator strengths for each dipole-allowed transition. We show that CC3 atomic basis sets. For all transitions, we report at least CC3/\emph{aug}-cc-pVQZ vertical excitation energies as well as CC3/\emph{aug}-cc-pVTZ oscillator strengths for each dipole-allowed transition. We show that CC3
almost systematically delivers transition energies in agreement with higher-level methods with a typical deviation of $\pm 0.04$ eV, except for transitions with a dominant double excitation character where the error is much larger. almost systematically delivers transition energies in agreement with higher-level methods with a typical deviation of $\pm 0.04$ eV, except for transitions with a dominant double excitation character where the error is much larger.
The present contribution gathers a large, diverse and accurate set of more than 200 highly-accurate transition energies for states of various natures (valence, Rydberg, singlet, triplet, $n \ra \pis$, $\pi \ra \pis$\ldots). The present contribution gathers a large, diverse and accurate set of more than 200 highly-accurate transition energies for states of various natures (valence, Rydberg, singlet, triplet, $n \ra \pis$, $\pi \ra \pis$, \ldots).
We use this series of theoretical best estimates to benchmark a series of popular methods for excited state calculations: CIS(D), ADC(2), CC2, STEOM-CCSD, EOM-CCSD, CCSDR(3), CCSDT-3, and CC3. The results We use this series of theoretical best estimates to benchmark a series of popular methods for excited state calculations: CIS(D), ADC(2), CC2, STEOM-CCSD, EOM-CCSD, CCSDR(3), CCSDT-3, and CC3. The results
of these benchmarks are compared to the available literature data. of these benchmarks are compared to the available literature data.
\end{abstract} \end{abstract}
@ -162,13 +162,13 @@ More than 20 years ago, Serrano-Andr\`es, Roos, and collaborators compiled an im
\cite{Ful92,Ser93,Ser93b,Ser93c,Lor95b,Mer96,Mer96b,Roo96,Ser96b} To this end, they relied on experimental GS geometries and the complete-active-space second-order perturbation theory ({\CASPT}) approach with the largest \cite{Ful92,Ser93,Ser93b,Ser93c,Lor95b,Mer96,Mer96b,Roo96,Ser96b} To this end, they relied on experimental GS geometries and the complete-active-space second-order perturbation theory ({\CASPT}) approach with the largest
active spaces and basis sets one could dream of at the time. These {\CASPT} values were later used to assess the performance of TD-DFT combined with various exchange-correlation functionals, \cite{Toz99b,Bur02} and remained for active spaces and basis sets one could dream of at the time. These {\CASPT} values were later used to assess the performance of TD-DFT combined with various exchange-correlation functionals, \cite{Toz99b,Bur02} and remained for
a long time the best theoretical references available on the market. However, beyond comparisons with experiments, which are always challenging when computing vertical transition energies, \cite{San16b} there was no approach a long time the best theoretical references available on the market. However, beyond comparisons with experiments, which are always challenging when computing vertical transition energies, \cite{San16b} there was no approach
available at that time to ascertain the accuracy of these transition energies. Nowadays, it is common knowledge that CASPT2 has the tendency of underestimating vertical excitation energies in organic molecules. available at that time to ascertain the accuracy of these transition energies. Nowadays, it is of common knowledge that CASPT2 has the tendency of underestimating vertical excitation energies in organic molecules.
A decade ago, Thiel and coworkers defined TBE for 104 singlet and 63 triplet valence ES in 28 small and medium conjugated CNOH organic molecules. \cite{Sch08,Sil10b,Sil10c} These TBE were computed on MP2/6-31G(d) structures A decade ago, Thiel and coworkers defined TBE for 104 singlet and 63 triplet valence ES in 28 small and medium conjugated CNOH organic molecules. \cite{Sch08,Sil10b,Sil10c} These TBE were computed on MP2/6-31G(d) structures
with several levels of theories, notably {\CASPT} and various coupled cluster (CC) variants ({\CCD}, {\CCSD}, and {\CCT}). Interestingly, while the default theoretical protocol used by Thiel and coworkers to define their with several levels of theory, notably {\CASPT} and various coupled cluster (CC) variants ({\CCD}, {\CCSD}, and {\CCT}). Interestingly, while the default theoretical protocol used by Thiel and coworkers to define their
first series of TBE was {\CASPT}, \cite{Sch08} the vast majority of their most recent TBE (the so-called ``TBE-2'' in Ref.~\citenum{Sil10c}) were determined at the {\CCT} level of theory with the \emph{aug}-cc-pVTZ (aVTZ) basis set, first series of TBE was {\CASPT}, \cite{Sch08} the vast majority of their most recent TBE (the so-called ``TBE-2'' in Ref.~\citenum{Sil10c}) were determined at the {\CCT} level of theory with the \emph{aug}-cc-pVTZ (aVTZ) basis set,
often using a basis set extrapolation technique. More specifically, CC3/TZVP values were corrected for basis set incompleteness errors by the difference between {\CCD}/{\AVTZ} and {\CCD}/TZVP results. \cite{Sil10b,Sil10c} often using a basis set extrapolation technique. More specifically, CC3/TZVP values were corrected for basis set incompleteness errors by the difference between the {\CCD}/{\AVTZ} and {\CCD}/TZVP results. \cite{Sil10b,Sil10c}
Many works exploited Thiel's TBE for assessing low-order methods, Many works have exploited Thiel's TBE for assessing low-order methods,
\cite{Sil08,Goe09,Jac09c,Roh09,Sau09,Jac10c,Jac10g,Sil10,Mar11,Jac11a,Hui11,Del11,Tra11,Pev12,Dom13,Dem13,Sch13b,Voi14,Har14,Yan14b,Sau15,Pie15,Taj16,Mai16,Ris17,Dut18,Hel19,Haa20} highlighting further \cite{Sil08,Goe09,Jac09c,Roh09,Sau09,Jac10c,Jac10g,Sil10,Mar11,Jac11a,Hui11,Del11,Tra11,Pev12,Dom13,Dem13,Sch13b,Voi14,Har14,Yan14b,Sau15,Pie15,Taj16,Mai16,Ris17,Dut18,Hel19,Haa20} highlighting further
their value for the electronic structure community. In contrast, the number of extensions/improvements of this original set remains quite limited. For example, K\'ann\'ar and Szalay computed, in 2014, {\CCSDT}/TZVP reference energies their value for the electronic structure community. In contrast, the number of extensions/improvements of this original set remains quite limited. For example, K\'ann\'ar and Szalay computed, in 2014, {\CCSDT}/TZVP reference energies
for 17 singlet states of six molecules. \cite{Kan14} Three years later, the same authors reported 46 {\CCSDT}/{\AVTZ} transition energies in small compounds containing two or three non-hydrogen atoms (ethylene, acetylene, formaldehyde, for 17 singlet states of six molecules. \cite{Kan14} Three years later, the same authors reported 46 {\CCSDT}/{\AVTZ} transition energies in small compounds containing two or three non-hydrogen atoms (ethylene, acetylene, formaldehyde,
@ -183,7 +183,7 @@ Although these conclusions agree well with earlier studies, \cite{Wat13,Kan14,Ka
For example, for the ES with a significant yet not dominant double excitation character [such as the infamous $A_g$ ES of butadiene ($\Td = 75\%$)] CC methods including triples deliver rather accurate estimates (MAE of $0.11$ eV with {\CCT} For example, for the ES with a significant yet not dominant double excitation character [such as the infamous $A_g$ ES of butadiene ($\Td = 75\%$)] CC methods including triples deliver rather accurate estimates (MAE of $0.11$ eV with {\CCT}
and $0.06$ eV with {\CCSDT}), surprisingly outperforming second-order multi-reference schemes such as {\CASPT} or the generally robust $n$-electron valence state perturbation theory ({\NEV}). In contrast, for ES with a dominant double and $0.06$ eV with {\CCSDT}), surprisingly outperforming second-order multi-reference schemes such as {\CASPT} or the generally robust $n$-electron valence state perturbation theory ({\NEV}). In contrast, for ES with a dominant double
excitation character, \eg, the low-lying $(n,n) \ra (\pis,\pis)$ excitation in nitrosomethane ($\Td = 2\%$), single-reference methods (not including quadruples) have been found to be unsuitable with MAEs of $0.86$ and $0.42$ eV for {\CCT} excitation character, \eg, the low-lying $(n,n) \ra (\pis,\pis)$ excitation in nitrosomethane ($\Td = 2\%$), single-reference methods (not including quadruples) have been found to be unsuitable with MAEs of $0.86$ and $0.42$ eV for {\CCT}
and {\CCSDT}, respectively. In this case, multireference methods are in practice required to obtain accurate results. \cite{Loo19c} and {\CCSDT}, respectively. In this case, multiconfigurational methods are in practice required to obtain accurate results. \cite{Loo19c}
A clear limit of our 2018 work \cite{Loo18a} was the size of the compounds put together in our set. These were limited to $1$--$3$ non-hydrogen atoms, hence introducing a potential ``chemical'' bias. Therefore, we have decided, in the A clear limit of our 2018 work \cite{Loo18a} was the size of the compounds put together in our set. These were limited to $1$--$3$ non-hydrogen atoms, hence introducing a potential ``chemical'' bias. Therefore, we have decided, in the
present contribution, to consider larger molecules with organic compounds encompassing 4, 5, and 6 non-hydrogen atoms. For such systems, performing {\CCSDTQ} calculations with large one-electron basis sets is elusive. Moreover, present contribution, to consider larger molecules with organic compounds encompassing 4, 5, and 6 non-hydrogen atoms. For such systems, performing {\CCSDTQ} calculations with large one-electron basis sets is elusive. Moreover,
@ -209,13 +209,13 @@ so that we only briefly outline the various theoretical methods that we have emp
\subsection{Geometries} \subsection{Geometries}
Consistently with our previous work, \cite{Loo18a} we systematically use {\CCT}/{\AVTZ} GS geometries obtained without applying the frozen-core approximation. The Cartesian coordinates (in bohr) of each compound can be found in the Consistently with our previous work, \cite{Loo18a} we systematically use {\CCT}/{\AVTZ} GS geometries obtained without applying the frozen-core approximation. The cartesian coordinates (in bohr) of each compound can be found in the
Supporting Information (SI). Several structures have been extracted from previous contributions, \cite{Bud17,Jac18a,Bre18a} whereas the missing structures were optimized using DALTON \cite{dalton} and/or CFOUR, \cite{cfour} applying Supporting Information (SI). Several structures have been extracted from previous contributions, \cite{Bud17,Jac18a,Bre18a} whereas the missing structures were optimized using DALTON \cite{dalton} and/or CFOUR, \cite{cfour} applying
default parameters in both cases. default parameters in both cases.
\subsection{Selected Configuration Interaction} \subsection{Selected Configuration Interaction}
Because sCI methods are less widespread than the other methods mentioned in the Introduction, we shall detail further their main features Because sCI methods are less widespread than the other methods mentioned in the Introduction, we shall detail further their main features.
All the SCI calculations have been performed in the frozen-core approximation with the latest version of QUANTUM PACKAGE \cite{Gar19} using the Configuration Interaction using a Perturbative Selection made Iteratively (CIPSI) algorithm to select the All the SCI calculations have been performed in the frozen-core approximation with the latest version of QUANTUM PACKAGE \cite{Gar19} using the Configuration Interaction using a Perturbative Selection made Iteratively (CIPSI) algorithm to select the
most important determinants in the FCI space. Instead of generating all possible excited determinants like a conventional CI calculation, the iterative CIPSI algorithm performs a sparse exploration of the FCI space via a selection of the most relevant most important determinants in the FCI space. Instead of generating all possible excited determinants like a conventional CI calculation, the iterative CIPSI algorithm performs a sparse exploration of the FCI space via a selection of the most relevant
determinants using a second-order perturbative criterion. At each iteration, the variational (or reference) space is enlarged with new determinants. CIPSI can be seen as a deterministic version of the FCIQMC algorithm developed by Alavi and determinants using a second-order perturbative criterion. At each iteration, the variational (or reference) space is enlarged with new determinants. CIPSI can be seen as a deterministic version of the FCIQMC algorithm developed by Alavi and
@ -259,12 +259,12 @@ true for triplet ES that are known to be characterized by very large $\Td$ value
energies for singlet states. To assign the different ES, we use literature data, as well as the usual criteria, \ie, relative energies, spatial and spin symmetries, compositions from the underlying molecular orbitals, and oscillator strengths. energies for singlet states. To assign the different ES, we use literature data, as well as the usual criteria, \ie, relative energies, spatial and spin symmetries, compositions from the underlying molecular orbitals, and oscillator strengths.
This allows clear-cut assignments for the vast majority of the cases. There are however some state/method combinations for which strong mixing between ES of the same symmetry makes unambiguous assignments almost impossible. This allows clear-cut assignments for the vast majority of the cases. There are however some state/method combinations for which strong mixing between ES of the same symmetry makes unambiguous assignments almost impossible.
\subsection{Four-atom molecules} \subsection{Molecules with four non-hydrogen atoms}
\subsubsection{Cyanoacetylene, cyanogen, and diacetylene} \subsubsection{Cyanoacetylene, cyanogen, and diacetylene}
\begin{sidewaystable}[htp] \begin{sidewaystable}[htp]
\caption{\small Vertical transition energies (in eV) of cyanoacetylene, cyanogen, and diaectylene. All states have a valence $\pi \ra \pis$ character.} \caption{\small Vertical transition energies (in eV) of cyanoacetylene, cyanogen, and diacetylene. All states have a valence $\pi \ra \pis$ character.}
\label{Table-1} \label{Table-1}
\begin{footnotesize} \begin{footnotesize}
\begin{tabular}{l|p{.5cm}p{1.0cm}p{1.2cm}p{1.4cm}|p{.5cm}p{1.0cm}p{1.2cm}p{1.4cm}|p{.5cm}p{1.0cm}p{1.2cm}|p{.5cm}|p{.5cm}|p{.6cm}p{.6cm}} \begin{tabular}{l|p{.5cm}p{1.0cm}p{1.2cm}p{1.4cm}|p{.5cm}p{1.0cm}p{1.2cm}p{1.4cm}|p{.5cm}p{1.0cm}p{1.2cm}|p{.5cm}|p{.5cm}|p{.6cm}p{.6cm}}
@ -315,7 +315,7 @@ state is the onset, whereas an estimate of the vertical energy ($4.2 \pm 0.2$ eV
The ES of these three closely related linear molecules containing two triple bonds have been quite rarely theoretically investigated, \cite{Fis03,Pat06,Luo08,Loo18b,Loo19a} though (rather old) experimental measurements of their The ES of these three closely related linear molecules containing two triple bonds have been quite rarely theoretically investigated, \cite{Fis03,Pat06,Luo08,Loo18b,Loo19a} though (rather old) experimental measurements of their
0-0 energies are available for several ES. \cite{Cal63,Job66a,Job66b,Bel69,Fis72,Har77,Hai79,All84} Our main results are collected in Tables \ref{Table-1} and S1. We consider only low-lying valence $\pi \ra \pis$ transitions, which 0-0 energies are available for several ES. \cite{Cal63,Job66a,Job66b,Bel69,Fis72,Har77,Hai79,All84} Our main results are collected in Tables \ref{Table-1} and S1. We consider only low-lying valence $\pi \ra \pis$ transitions, which
are all characterized by a strongly dominant single excitation nature ($\Td > 90\%$, \emph{vide infra}). For cyanoacetylene, the {\FCI}/{\Pop} estimates come with small error bars, and one notices an excellent agreement between are all characterized by a strongly dominant single excitation nature ($\Td > 90\%$, \emph{vide infra}). For cyanoacetylene, the {\FCI}/{\Pop} estimates come with small error bars, and one notices an excellent agreement between
these values and their {\CCSDTQ} counterparts, a statement holding for the Dunning's double-$\zeta$ basis set results for which the {\FCI} uncertainties are however larger. Using the {\CCSDTQ} values as references, it appears these values and their {\CCSDTQ} counterparts, a statement holding for the Dunning double-$\zeta$ basis set results for which the {\FCI} uncertainties are however larger. Using the {\CCSDTQ} values as references, it appears
that the previously obtained {\CASPT} estimates\cite{Luo08} are, as expected, too low and that the {\CCT} transition energies are slightly more accurate than their CCSDT counterparts, although all CC estimates of Table \ref{Table-1} that the previously obtained {\CASPT} estimates\cite{Luo08} are, as expected, too low and that the {\CCT} transition energies are slightly more accurate than their CCSDT counterparts, although all CC estimates of Table \ref{Table-1}
come, for a given basis set, in a very tight energetic window. There is also a very neat agreement between the CC/{\AVTZ} and {\NEV}/{\AVTZ}. All these facts provide strong evidences that the CC estimates can be fully trusted for come, for a given basis set, in a very tight energetic window. There is also a very neat agreement between the CC/{\AVTZ} and {\NEV}/{\AVTZ}. All these facts provide strong evidences that the CC estimates can be fully trusted for
these three linear systems. The basis set effects are quite significant for the valence ES of cyanoacetylene with successive drops of the transition energies by approximately $0.10$ eV, when going from {\Pop} to {\AVDZ}, and from these three linear systems. The basis set effects are quite significant for the valence ES of cyanoacetylene with successive drops of the transition energies by approximately $0.10$ eV, when going from {\Pop} to {\AVDZ}, and from
@ -329,7 +329,7 @@ as they should. We refer the interested reader to previous works, \cite{Fis03,Lo
\subsubsection{Cyclopropenone, cyclopropenethione, and methylenecyclopropene} \subsubsection{Cyclopropenone, cyclopropenethione, and methylenecyclopropene}
These three related compounds present a three-member $sp^2$ carbon cycle conjugated to an external $\pi$ bond. While the ES of methylenecyclopropene have regularly been investigated with theoretical tools in the past, These three related compounds present a three-membered $sp^2$ carbon cycle conjugated to an external $\pi$ bond. While the ES of methylenecyclopropene have regularly been investigated with theoretical tools in the past,
\cite{Mer96,Roo96,Car10b,Lea12,Gua13,Dad14,Gua14,Sch17,Bud17} the only investigations of vertical transitions we could find for the two other derivatives are a detailed {\CASPT} study of Serrano-Andr\'es and \cite{Mer96,Roo96,Car10b,Lea12,Gua13,Dad14,Gua14,Sch17,Bud17} the only investigations of vertical transitions we could find for the two other derivatives are a detailed {\CASPT} study of Serrano-Andr\'es and
coworkers in 2002, \cite{Ser02} and a more recent work reporting the three lowest-lying singlet states of cyclopropenone at the {\CASPT}/6-31G level.\cite{Liu14b} coworkers in 2002, \cite{Ser02} and a more recent work reporting the three lowest-lying singlet states of cyclopropenone at the {\CASPT}/6-31G level.\cite{Liu14b}
@ -484,7 +484,7 @@ $^e${Electron impact experiment from Refs.~\citenum{Fli78} and \citenum{Doe81} f
note that for the lowest $B_u$ state, there is a vibrational structure with peaks at $5.76$, $5.92$, and $6.05$ eV;} note that for the lowest $B_u$ state, there is a vibrational structure with peaks at $5.76$, $5.92$, and $6.05$ eV;}
$^f${From Ref.~\citenum{Loo19c};} $^f${From Ref.~\citenum{Loo19c};}
$^g${{\CCT} results from Ref.~\citenum{Sch17};} $^g${{\CCT} results from Ref.~\citenum{Sch17};}
$^h${Electron impact experiment from Ref.~\citenum{Ver80} except for the second $^1B_g$ ES for which the value is from another work; \cite{Rob85b} note that $^h${Electron impact experiment from Ref.~\citenum{Ver80} except for the second $^1B_g$ ES for which the value is from another work (see Ref.~\citenum{Rob85b}); note that
for the lowest $^1B_g$ ($^1B_u$) ES, a range of $4.2$--$4.5$ ($7.4$--$7.9$) eV is given in Ref.~\citenum{Ver80}. } for the lowest $^1B_g$ ($^1B_u$) ES, a range of $4.2$--$4.5$ ($7.4$--$7.9$) eV is given in Ref.~\citenum{Ver80}. }
\end{footnotesize} \end{footnotesize}
\end{flushleft} \end{flushleft}
@ -519,7 +519,7 @@ Once more, the experimental data \cite{Ver80,Rob85b} are unhelpful in view of th
\subsubsection{Acetone, cyanoformaldehyde, isobutene, propynal, thioacetone, and thiopropynal} \subsubsection{Acetone, cyanoformaldehyde, isobutene, propynal, thioacetone, and thiopropynal}
Let us now turn towards six other four-atom compounds. There are several earlier studies reporting estimates of the vertical transition energies for both acetone \cite{Gwa95,Mer96b,Roo96,Wib98,Toz99b,Wib02,Sch08,Sil10c,Car10,Pas12,Ise12,Gua13,Sch17} Let us now turn towards six other compounds with four non-hydrogen atoms. There are several earlier studies reporting estimates of the vertical transition energies for both acetone \cite{Gwa95,Mer96b,Roo96,Wib98,Toz99b,Wib02,Sch08,Sil10c,Car10,Pas12,Ise12,Gua13,Sch17}
and isobutene. \cite{Wib02,Car10,Ise12} To the best of our knowledge, for the four other compounds, the previous computational efforts were mainly focussed on the 0-0 energies of the lowest-lying states. \cite{Koh03,Hat05c,Sen11b,Loo18b,Loo19a} and isobutene. \cite{Wib02,Car10,Ise12} To the best of our knowledge, for the four other compounds, the previous computational efforts were mainly focussed on the 0-0 energies of the lowest-lying states. \cite{Koh03,Hat05c,Sen11b,Loo18b,Loo19a}
There are also rather few experimental data available for these six derivatives. \cite{Bir73,Jud83,Bra74,Sta75,Joh79,Jud83,Jud84c,Rob85,Pal87,Kar91b,Xin93} There are also rather few experimental data available for these six derivatives. \cite{Bir73,Jud83,Bra74,Sta75,Joh79,Jud83,Jud84c,Rob85,Pal87,Kar91b,Xin93}
Our main results are reported in Tables \ref{Table-4} and S4. Our main results are reported in Tables \ref{Table-4} and S4.
@ -578,7 +578,7 @@ $^1A_1 (\pi \ra \pis)$ &6.09&6.10&6.07& &5.97&5.98&5.90 &5.91&6.24 &5.64\\
$^1B_2 (n \ra 4p)$ &6.59&6.59&6.59& &6.45&6.44&6.51 & &6.62 &6.40\\ $^1B_2 (n \ra 4p)$ &6.59&6.59&6.59& &6.45&6.44&6.51 & &6.62 &6.40\\
$^1A_1 (n \ra 4p)$ &6.95&6.95&6.96& &6.54&6.53&6.61 &6.60&6.52 &6.52\\ $^1A_1 (n \ra 4p)$ &6.95&6.95&6.96& &6.54&6.53&6.61 &6.60&6.52 &6.52\\
$^3A_2 (n \ra \pis)$ &2.36&2.34& &2.36$\pm$0.00 &2.36&2.35&2.34 & &2.32 &2.14\\ $^3A_2 (n \ra \pis)$ &2.36&2.34& &2.36$\pm$0.00 &2.36&2.35&2.34 & &2.32 &2.14\\
$^3A_1 (\pi \ra \pis)$ &3.45&3.45& & &3.51&3.50&3.46 & &3.46 &\\ $^3A_1 (\pi \ra \pis)$ &3.45&3.45& & &3.51&3.50&3.46 & &3.48 &\\
\hline \hline
\mc{12}{c}{Thiopropynal}\\ \mc{12}{c}{Thiopropynal}\\
& \mc{4}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{1}{c}{Litt.}\\ & \mc{4}{c}{\Pop} & \mc{2}{c}{\AVDZ}& \mc{3}{c}{\AVTZ} & \mc{1}{c}{Litt.}\\
@ -612,7 +612,7 @@ As expected, this error can be partially ascribed to the computational set-up, a
better agreement with ours. Their estimates of the three $n \ra 3p$ transitions, $7.52$, $7.57$, and $7.53$ eV for the $^1A_2$, $^1A_1$, and $^1B_2$ ES, also systematically fall within $0.10$ eV of our current CC values, whereas for these better agreement with ours. Their estimates of the three $n \ra 3p$ transitions, $7.52$, $7.57$, and $7.53$ eV for the $^1A_2$, $^1A_1$, and $^1B_2$ ES, also systematically fall within $0.10$ eV of our current CC values, whereas for these
three ES, the current {\NEV} values are clearly too large. three ES, the current {\NEV} values are clearly too large.
In contrast to acetone, both valence and Rydberg ES of thioacetone are rather insensitive to the excitation order of the CC expansion, as illustrated by the maximal discrepancies of $\pm 0.02$ eV between the {\CCT}/{\Pop} and {\CCSDTQ}/{\Pop} results. In contrast to acetone, both valence and Rydberg ES of thioacetone are rather insensitive to the excitation order of the CC expansion as illustrated by the maximal discrepancies of $\pm 0.02$ eV between the {\CCT}/{\Pop} and {\CCSDTQ}/{\Pop} results.
While the lowest $n \ra \pis$ transition of both spin symmetries are rather basis set insensitive, all the other states need quite large one-electron bases to be correctly described (Table S4). As expected, our theoretical vertical transition energies While the lowest $n \ra \pis$ transition of both spin symmetries are rather basis set insensitive, all the other states need quite large one-electron bases to be correctly described (Table S4). As expected, our theoretical vertical transition energies
show the same ranking but are systematically larger than the available experimental 0-0 energies. show the same ranking but are systematically larger than the available experimental 0-0 energies.
@ -626,7 +626,7 @@ This is further confirmed by the {\FCI} data.
\subsubsection{Intermediate conclusions} \subsubsection{Intermediate conclusions}
\label{sec-ic} \label{sec-ic}
As we have seen for the 15 four-atom molecules considered here, we found extremely consistent transition energies between CC and {\FCI} estimates in the vast majority of the cases. Importantly, we confirm our previous conclusions obtained on For the 15 molecules with four non-hydrogen atoms considered here, we find extremely consistent transition energies between CC and {\FCI} estimates in the vast majority of the cases. Importantly, we confirm our previous conclusions obtained on
smaller compounds: \cite{Loo18a} i) {\CCSDTQ} values systematically fall within (or are extremely close to) the {\FCI} error bar, ii) both {\CCT} and {\CCSDT} are also highly trustable when the considered ES does not exhibit a strong double smaller compounds: \cite{Loo18a} i) {\CCSDTQ} values systematically fall within (or are extremely close to) the {\FCI} error bar, ii) both {\CCT} and {\CCSDT} are also highly trustable when the considered ES does not exhibit a strong double
excitation character. Indeed, considering the 54 ``single'' ES cases for which {\CCSDTQ} estimates could be obtained (only excluding the lowest $^1A_g$ ES of butadiene and glyoxal), we determined negligible MSE $< 0.01$ eV, tiny excitation character. Indeed, considering the 54 ``single'' ES cases for which {\CCSDTQ} estimates could be obtained (only excluding the lowest $^1A_g$ ES of butadiene and glyoxal), we determined negligible MSE $< 0.01$ eV, tiny
MAE ($0.01$ and $0.02$ eV), and small maximal deviations ($0.05$ and $0.04$ eV) for {\CCT} and {\CCSDT}, respectively. This clearly indicates that these two approaches provide chemically-accurate estimates (errors below $1$ MAE ($0.01$ and $0.02$ eV), and small maximal deviations ($0.05$ and $0.04$ eV) for {\CCT} and {\CCSDT}, respectively. This clearly indicates that these two approaches provide chemically-accurate estimates (errors below $1$
@ -635,7 +635,7 @@ When comparing the {\NEV} and {\CCT} ({\CCSDT}) results obtained with {\AVTZ} fo
one obtains a MSE of $+0.09$ ($+0.09$) eV and a MAE of $0.11$ ($0.12$) eV. This seems to indicate that {\NEV}, as applied here, has a slight tendency to overestimate the transition energies. This contrasts with {\CASPT} that is known to one obtains a MSE of $+0.09$ ($+0.09$) eV and a MAE of $0.11$ ($0.12$) eV. This seems to indicate that {\NEV}, as applied here, has a slight tendency to overestimate the transition energies. This contrasts with {\CASPT} that is known to
generally underestimate transition energies, as further illustrated and discussed above and below. generally underestimate transition energies, as further illustrated and discussed above and below.
\subsection{Five-atom molecules} \subsection{Five-membered rings}
We now consider five-membered rings, and, in particular, five common derivatives that have been considered in several previous theoretical studies (\emph{vide infra}): cyclopentadiene, furan, imidazole, pyrrole, and thiophene. We now consider five-membered rings, and, in particular, five common derivatives that have been considered in several previous theoretical studies (\emph{vide infra}): cyclopentadiene, furan, imidazole, pyrrole, and thiophene.
As the most advanced levels of theory employed in the previous section, namely {\CCSDTQ} and {\FCI}, become beyond reach for these compounds (except in very rare occasions), one has to rely on the nature of the ES and the As the most advanced levels of theory employed in the previous section, namely {\CCSDTQ} and {\FCI}, become beyond reach for these compounds (except in very rare occasions), one has to rely on the nature of the ES and the
@ -707,7 +707,7 @@ $^l${Vapour UV spectra from Refs.~\citenum{Pal03b}, \citenum{Hor67}, and \citenu
\end{table} \end{table}
Like furan, pyrrole has been extensively investigated in the literature using a large panel of theoretical methods. \cite{Ser93b,Nak96,Tro97,Pal98,Chr99,Wan00,Roo02,Pal03b,Pas06c,Sch08,She09b,Li10c,Sil10b,Sil10c,Sau11,Nev14,Sch17,Hei19} Like furan, pyrrole has been extensively investigated in the literature using a large panel of theoretical methods. \cite{Ser93b,Nak96,Tro97,Pal98,Chr99,Wan00,Roo02,Pal03b,Pas06c,Sch08,She09b,Li10c,Sil10b,Sil10c,Sau11,Nev14,Sch17,Hei19}
We report six low-lying singlet and four triplet ES in Tables \ref{Table-5} and S5. All these transitions have very large $\Td$ values except for the totally symmetric $\pi \ra \pis$ excitation ($\Td = 86\%$). For all the states, we found We report six low-lying singlet and four triplet ES in Tables \ref{Table-5} and S5. All these transitions have very large $\Td$ values except for the totally symmetric $\pi \ra \pis$ excitation ($\Td = 86\%$). For each state, we found
highly consistent {\CCT} and {\CCSDT} results, often significantly larger than older multi-reference estimates, \cite{Ser93b,Roo02,Li10c} but in nice agreement with the very recent XMS-{\CASPT} results of the Gonzalez highly consistent {\CCT} and {\CCSDT} results, often significantly larger than older multi-reference estimates, \cite{Ser93b,Roo02,Li10c} but in nice agreement with the very recent XMS-{\CASPT} results of the Gonzalez
group, \cite{Hei19} and the present {\NEV} estimates [at the exception of the $^1A_2 (\pi \ra 3p)$ transition]. The match obtained with the twenty years old extrapolated CC values of Christiansen and coworkers \cite{Chr99} is quite remarkable. The only exceptions are the two group, \cite{Hei19} and the present {\NEV} estimates [at the exception of the $^1A_2 (\pi \ra 3p)$ transition]. The match obtained with the twenty years old extrapolated CC values of Christiansen and coworkers \cite{Chr99} is quite remarkable. The only exceptions are the two
$B_2$ transitions that were reported as significantly mixed in this venerable work. For the lowest singlet ES, the {\FCI}/{\Pop} value is $5.24 \pm 0.02$ eV confirming the performance of both {\CCT} and {\CCSDT} for this transition. $B_2$ transitions that were reported as significantly mixed in this venerable work. For the lowest singlet ES, the {\FCI}/{\Pop} value is $5.24 \pm 0.02$ eV confirming the performance of both {\CCT} and {\CCSDT} for this transition.
@ -803,28 +803,28 @@ establish a definitive TBE. This statement is also in line with the results of R
{\CCT} and {\CCSDT} transition energies falling inside these energetic windows in both cases. As one can see in Tables \ref{Table-6} and S6, the basis set effects are rather moderate for the electronic transitions of cyclopentadiene, {\CCT} and {\CCSDT} transition energies falling inside these energetic windows in both cases. As one can see in Tables \ref{Table-6} and S6, the basis set effects are rather moderate for the electronic transitions of cyclopentadiene,
with no variation larger than $0.10$ eV ($0.02$ eV) between {\AVDZ} and {\AVTZ} ({\AVTZ} and {\AVQZ}). When comparing to literature data, our values are unsurprisingly consistent with the {\CCT} values of with no variation larger than $0.10$ eV ($0.02$ eV) between {\AVDZ} and {\AVTZ} ({\AVTZ} and {\AVQZ}). When comparing to literature data, our values are unsurprisingly consistent with the {\CCT} values of
Schwabe and Goerigk, \cite{Sch17} and tend to be significantly larger than earlier {\CASPT} \cite{Ser93b,Sil10c} and MR-MP \cite{Nak96} estimates. As expected, a few gas-phase experiments are available as well for this Schwabe and Goerigk, \cite{Sch17} and tend to be significantly larger than earlier {\CASPT} \cite{Ser93b,Sil10c} and MR-MP \cite{Nak96} estimates. As expected, a few gas-phase experiments are available as well for this
derivative, \cite{Fru79,McD85,McD91b,Sab92} but they hardly represent grounds for valuable comparison. derivative, \cite{Fru79,McD85,McD91b,Sab92} but they hardly represent grounds for comparison.
Due to its lower symmetry, imidazole has been less investigated, the most advanced studies available probably remain the {\CASPT} work of Serrano-Andr\`es and coworkers from 1996, \cite{Ser96b} and the Due to its lower symmetry, imidazole has been less investigated, the most advanced studies available probably remain the {\CASPT} work of Serrano-Andr\`es and coworkers from 1996, \cite{Ser96b} and the
basis-set extrapolated {\CCT} results of Silva-Junior \emph{et al.} for the valence transitions from 2010. \cite{Sil10c} The experimental data in gas-phase are also limited.\cite{Dev06} Our results are displayed in Tables \ref{Table-6} and S6. basis-set extrapolated {\CCT} results of Silva-Junior \emph{et al.} for the valence transitions from 2010. \cite{Sil10c} The experimental data in gas-phase are also limited.\cite{Dev06} Our results are displayed in Tables \ref{Table-6} and S6.
The {\CCT} and {\CCSDT} values are quite consistent despite the fact that the $\Td$ values of the two singlet $A'$ states are slightly smaller than $90\%$. These two states have indeed a (at least partial) Rydberg character, The {\CCT} and {\CCSDT} values are quite consistent despite the fact that the $\Td$ values of the two singlet $A'$ states are slightly smaller than $90\%$. These two states have indeed, at least partially, a Rydberg character
see the footnote in Table \ref{Table-6}. The agreement between the CC estimates and previous {\CASPT}, \cite{Ser96b} and current {\NEV} energies is reasonable, the latter being systematically larger than their {\CCT} (see the footnote in Table \ref{Table-6}). The agreement between the CC estimates and previous {\CASPT}, \cite{Ser96b} and current {\NEV} energies is reasonable, the latter being systematically larger than their {\CCT}
counterparts. For the eight transitions considered here, the basis set effects are moderate and {\AVTZ} yield results within $0.03$ eV of their {\AVQZ} counterparts (Table S6 in the SI). counterparts. For the eight transitions considered here, the basis set effects are moderate and {\AVTZ} yield results within $0.03$ eV of their {\AVQZ} counterparts (Table S6 in the SI).
Finally, the ES of thiophene, which is one of the most important building block in organic electronic devices, were the subject of previous theoretical investigations, \cite{Ser93c,Pal99,Wan01,Kle02,Pas07,Hol14} that unveiled a series of Finally, the ES of thiophene, which is one of the most important building block in organic electronic devices, were the subject of previous theoretical investigations, \cite{Ser93c,Pal99,Wan01,Kle02,Pas07,Hol14} that unveiled a series of
transitions that were not yet characterized in the available measurements. \cite{Dil72,Fli76,Fli76b,Var82,Hab03,Pal99,Hol14} To the best of our knowledge, the present work is the first to report CC calculations obtained with (iterative) transitions that were not yet characterized in the available measurements. \cite{Dil72,Fli76,Fli76b,Var82,Hab03,Pal99,Hol14} To the best of our knowledge, the present work is the first to report CC calculations obtained with (iterative)
triples and therefore constitutes the most accurate estimates to date. Indeed, all the transitions listed in Tables \ref{Table-6} and S6 are characterized by a largely dominant single excitation character, with $\Td$ above triples and therefore constitutes the most accurate estimates to date. Indeed, all the transitions listed in Tables \ref{Table-6} and S6 are characterized by a largely dominant single excitation character, with $\Td$ above
$90\%$ except for the two $^1A_1$ transitions for which $\Td = 88\%$ and $87\%$, respectively. The agreement between {\CCT} and {\CCSDT} remains nevertheless excellent for these low-lying totally symmetric transition. Thiophene is $90\%$ except for the two $^1A_1$ transitions for which $\Td = 88\%$ and $87\%$, respectively. The agreement between {\CCT} and {\CCSDT} remains nevertheless excellent for these low-lying totally symmetric transitions. Thiophene is
also one of these compounds for which the unambiguous characterization of the nature of the ES is difficult, with, \eg, a strong mixing between the second and third singlet ES of $B_2$ symmetry. This makes the assignment of the valence also one of these compounds for which the unambiguous characterization of the nature of the ES is difficult, with, \eg, a strong mixing between the second and third singlet ES of $B_2$ symmetry. This makes the assignment of the valence
($\pi \ra \pis$) or Rydberg ($\pi \ra 3p$) character of this transition particularly tricky at the {\CCT} level. We note that contradictory assignments can be found in the literature. \cite{Ser93c,Wan01,Pas07} As for the ($\pi \ra \pis$) or Rydberg ($\pi \ra 3p$) character of this transition particularly tricky at the {\CCT} level. We note that contradictory assignments can be found in the literature. \cite{Ser93c,Wan01,Pas07} As for the
previously discussed isostructural systems, we note that the only ES that undergoes significant basis set effects beyond {\AVTZ} is the Rydberg $^1B_2 (\pi \ra 3p)$ ($-0.09$ eV when upgrading to {\AVQZ}, see Table S6) and that previously discussed isostructural systems, we note that the only ES that undergoes significant basis set effects beyond {\AVTZ} is the Rydberg $^1B_2 (\pi \ra 3p)$ ($-0.09$ eV when upgrading to {\AVQZ}, see Table S6) and that
the {\NEV} estimates tend to be slightly larger than the {\CCT} values. The data of Table \ref{Table-6} are globally in good agreement with the previously reported values with discrepancies that are significant only for the three highest-lying singlet states. the {\NEV} estimates tend to be slightly larger than the {\CCT} values. The data of Table \ref{Table-6} are globally in good agreement with the previously reported values with discrepancies that are significant only for the three highest-lying singlet states.
\subsection{Six-atom molecules} \subsection{Six-membered rings}
Let us now turn towards seven six-membered rings which play a key role in chemistry: benzene, pyrazine, pyridazine, pyridine, pyrimidine, tetrazine, and triazine. To the best of our knowledge, the present work Let us now turn towards seven six-membered rings which play a key role in chemistry: benzene, pyrazine, pyridazine, pyridine, pyrimidine, tetrazine, and triazine. To the best of our knowledge, the present work
is the first to propose {\CCSDT} reference energies as well as {\CCT}/{\AVQZ} values for all these compounds. Of course, these systems have been investigated before, and beyond Thiel's is the first to propose {\CCSDT} reference energies as well as {\CCT}/{\AVQZ} values for all these compounds. Of course, these systems have been investigated before, and beyond Thiel's
benchmarks, \cite{Sch08,Sil10b,Sil10c} it is worth pointing out the early investigation of Del Bene and coworkers, \cite{Del97b} performed with a CC approach including perturbative corrections for the triples. benchmarks, \cite{Sch08,Sil10b,Sil10c} it is worth pointing out the early investigation of Del Bene and coworkers \cite{Del97b} performed with a CC approach including perturbative corrections for the triples.
Following a theoretically consistent protocol, Nooijen \cite{Noo99} also performed {\STEOM} calculations to study the ES of each of these derivatives. However, these two works only considered singlet ES. Following a theoretically consistent protocol, Nooijen \cite{Noo99} also performed {\STEOM} calculations to study the ES of each of these derivatives. However, these two works only considered singlet ES.
\subsubsection{Benzene, pyrazine, and tetrazine} \subsubsection{Benzene, pyrazine, and tetrazine}
@ -1036,8 +1036,8 @@ Our results for pyridazine and pyridine are gathered in Tables \ref{Table-9} and
focussed on singlet transitions, at the exception of rather old MRCI, \cite{Pal91} and {\CASPT} investigations. \cite{Fis00} Again, the $\Td$ values are larger than $85\%$ ($95\%$) for the singlet (triplet) transitions, focussed on singlet transitions, at the exception of rather old MRCI, \cite{Pal91} and {\CASPT} investigations. \cite{Fis00} Again, the $\Td$ values are larger than $85\%$ ($95\%$) for the singlet (triplet) transitions,
and the only state for which there is a variation larger than $0.03$ eV between the {\CCT}/{\AVDZ} and {\CCSDT}/{\AVDZ} energies is the $^3B_2 (\pi \ra \pis)$ transition. As in the previous six-membered cycles, the basis set and the only state for which there is a variation larger than $0.03$ eV between the {\CCT}/{\AVDZ} and {\CCSDT}/{\AVDZ} energies is the $^3B_2 (\pi \ra \pis)$ transition. As in the previous six-membered cycles, the basis set
effects are rather small and {\AVTZ} provides values close to the CBS limit for the considered transitions. For the singlet valence ES, we find again a rather good match with the results of previous {\STEOM} \cite{Noo99} and effects are rather small and {\AVTZ} provides values close to the CBS limit for the considered transitions. For the singlet valence ES, we find again a rather good match with the results of previous {\STEOM} \cite{Noo99} and
CC. \cite{Del97b,Sil10c} Yet again, these values are significantly higher than the {\CASPT} estimates reported in Refs.~\citenum{Ful92} and \citenum{Sil10c}. For the triplets, the present data represent the most CC \cite{Del97b,Sil10c} calculations. Yet again, these values are significantly higher than the {\CASPT} estimates reported in Refs.~\citenum{Ful92} and \citenum{Sil10c}. For the triplets, the present data represent the most
accurate results published to date. Our {\NEV} values are very close to their {\CCT} analogues for the lowest-lying singlet and triplet, but positively deviate for the higher-lying ES. Interestingly, beyond the popular 20-year old accurate results published to date. Our {\NEV} values are very close to their {\CCT} analogues for the lowest-lying singlet and triplet, but positively deviate for the higher-lying ES. Interestingly, beyond the popular twenty-year old
reference measurements, \cite{Inn88,Pal91} there is a very recent experimental EEL analysis for pyridazine, \cite{Lin19} that locates almost all ES. The transition energies reported in this very recent work are systematically smaller than our reference measurements, \cite{Inn88,Pal91} there is a very recent experimental EEL analysis for pyridazine, \cite{Lin19} that locates almost all ES. The transition energies reported in this very recent work are systematically smaller than our
CC estimates by approximately $-0.20$ eV. Nonetheless, this study provides exactly the same ES ranking as our theoretical protocol. CC estimates by approximately $-0.20$ eV. Nonetheless, this study provides exactly the same ES ranking as our theoretical protocol.
@ -1072,7 +1072,7 @@ $^1A_2 (n \ra \pis)$ &6.07&6.06 &5.98&5.97 &5.93&&6.02 & &5.94&5.98&5.84& 6.0
$^1B_1 (n \ra \pis)$ &6.39& &6.29&6.29 &6.26&&6.40 &6.03&6.18&6.35&6.11& &6.02 \\ $^1B_1 (n \ra \pis)$ &6.39& &6.29&6.29 &6.26&&6.40 &6.03&6.18&6.35&6.11& &6.02 \\
$^1B_2 (n \ra 3s)$ &6.81&6.80 &6.61&6.59 &6.72&&6.77 & &6.85&6.84&6.57& & \\ $^1B_2 (n \ra 3s)$ &6.81&6.80 &6.61&6.59 &6.72&&6.77 & &6.85&6.84&6.57& & \\
$^1A_1 (\pi \ra \pis)$ &7.08&7.09 &6.93&6.94 &6.87&&7.11 &7.10&6.87&6.86&6.57& 6.7 &6.69 \\ $^1A_1 (\pi \ra \pis)$ &7.08&7.09 &6.93&6.94 &6.87&&7.11 &7.10&6.87&6.86&6.57& 6.7 &6.69 \\
$^3B_1 (n \ra \pis)$ &4.20&4.20 &4.12&4.11 &4.10&&4.47 &3.81& &4.11& & &3.85 \\ $^3B_1 (n \ra \pis)$ &4.20&4.20 &4.12&4.11 &4.10&&4.17 &3.81& &4.11& & &3.85 \\
$^3A_1 (\pi \ra \pis)$ &4.55&4.52 &4.56&4.52 &4.55&&4.67 &4.35& &4.39& & &4.42 \\ $^3A_1 (\pi \ra \pis)$ &4.55&4.52 &4.56&4.52 &4.55&&4.67 &4.35& &4.39& & &4.42 \\
$^3A_2 (n \ra \pis)$ &4.77&4.76 &4.67&4.67 &4.66&&4.72 &4.24& &4.71& & &4.18 \\ $^3A_2 (n \ra \pis)$ &4.77&4.76 &4.67&4.67 &4.66&&4.72 &4.24& &4.71& & &4.18 \\
$^3B_2 (\pi \ra \pis)$ &5.08&5.08 &5.00&5.00 &4.96&&5.01 &4.83& &4.81& & &4.93 \\ $^3B_2 (\pi \ra \pis)$ &5.08&5.08 &5.00&5.00 &4.96&&5.01 &4.83& &4.81& & &4.93 \\
@ -1118,13 +1118,13 @@ $^h${CC3-ext.~results from Ref.~\citenum{Sil10c}.}
Table \ref{Table-tbe} reports our two sets of TBE: a set obtained with the {\AVTZ} basis set and one set including an additional correction for the one-electron basis set incompleteness error. The details of our protocol employed to generate these TBE are also provided in Table \ref{Table-tbe}. Table \ref{Table-tbe} reports our two sets of TBE: a set obtained with the {\AVTZ} basis set and one set including an additional correction for the one-electron basis set incompleteness error. The details of our protocol employed to generate these TBE are also provided in Table \ref{Table-tbe}.
For all states with a dominant single-excitation character (that is when $\Td > 80\%$), we rely on CC results using an incremental strategy to generate these TBE. There are only two exceptions to this general rule, two ES in acrolein for which nicely converged FCI values indicated For all states with a dominant single-excitation character (that is when $\Td > 80\%$), we rely on CC results using an incremental strategy to generate these TBE. There are only two exceptions to this general rule, two ES in acrolein for which nicely converged FCI values indicated
non-trifling {\CCSDT} errors. For ES with $\Td$ values between $70\%$ and $80\%$, our previous works indicated that {\CCSDT} tends to overshoot the non-negligeable {\CCSDT} errors. For ES with $\Td$ values between $70\%$ and $80\%$, our previous works indicated that {\CCSDT} tends to overshoot the
transition energies by roughly $0.05$--$0.10$ eV, and that {\NEV} errors tend to be, on average, slightly larger. \cite{Loo19c} Therefore, if {\CCSDTQ} or {\FCI} results are not available, it is extremely difficult to make the final call. For the other transitions, we relied either on the current transition energies by roughly $0.05$--$0.10$ eV, and that {\NEV} errors tend to be, on average, slightly larger. \cite{Loo19c} Therefore, if {\CCSDTQ} or {\FCI} results are not available, it is extremely difficult to make the final call. For the other transitions, we relied either on the current
or previous FCI data or the {\NEV} values as reference. The italicized transition energies in Table \ref{Table-tbe} are believed to be (relatively) less accurate. This is the case when: i) {\NEV} results have to be selected; ii) the CC calculations yield quite large changes in excitation energies or previous FCI data or the {\NEV} values as reference. The italicized transition energies in Table \ref{Table-tbe} are believed to be (relatively) less accurate. This is the case when: i) the {\NEV} result has to be selected; ii) the CC calculations yield quite large changes in excitation energies
while incrementing the excitation order by one unit despite large $\Td$; and iii) there is a very strong ES mixing making hard to follow a specific transition from one method (or one basis) to another. while incrementing the excitation order by one unit despite large $\Td$; and iii) there is a very strong ES mixing making hard to follow a specific transition from one method (or one basis) to another.
To determine the basis set corrections beyond augmented triple-$\zeta$, we use the {\CCT}/{\AVQZ} or {\CCT}/{\AVPZ} results. For several compounds, we also provide in the SI, {\CCT}/d-{\AVQZ} transition energies (\ie, with an additional set of diffuse functions). However, we do not To determine the basis set corrections beyond augmented triple-$\zeta$, we use the {\CCT}/{\AVQZ} or {\CCT}/{\AVPZ} results. For several compounds, we also provide in the SI, {\CCT}/d-{\AVQZ} transition energies (\ie, with an additional set of diffuse functions). However, we do not
consider such values as reference because the addition of a second set of diffuse orbitals only significantly modifies the transition energies when also inducing a stronger ES mixing. We also stick to the frozen-core approximation for two reasons: i) the effect of correlating the core electrons consider such values as reference because the addition of a second set of diffuse orbitals only significantly modifies the transition energies while also inducing a stronger ES mixing. We also stick to the frozen-core approximation for two reasons: i) the effect of correlating the core electrons
is generally negligible (typically $\pm 0.02$ eV) for the compounds under study (see the SI for examples); and ii) it would be, in principle, necessary to add core polarization functions in such a case. is generally negligible (typically $\pm 0.02$ eV) for the compounds under study (see the SI for examples); and ii) it would be, in principle, necessary to add core polarization functions in such a case.
Table \ref{Table-tbe} encompasses 238 ES, each of them obtained, at least, at the {\CCSDT} level. This set can be decomposed as follows: 144 singlet and 94 triplet transitions, or 174 valence (99 $\pi \ra \pis$, 71 $n \ra \pis$ and 4 double excitations) and 64 Rydberg transitions. Table \ref{Table-tbe} encompasses 238 ES, each of them obtained, at least, at the {\CCSDT} level. This set can be decomposed as follows: 144 singlet and 94 triplet transitions, or 174 valence (99 $\pi \ra \pis$, 71 $n \ra \pis$ and 4 double excitations) and 64 Rydberg transitions.
@ -1412,17 +1412,17 @@ Protocol H: {\FCI}/{\Pop} value corrected by the difference between {\CCT}/{\AVT
Having at hand such a large set of accurate transition energies, it seems natural to pursue previous benchmarking efforts. More specifically, we assess here the performance of eight popular wavefunction approaches, namely, Having at hand such a large set of accurate transition energies, it seems natural to pursue previous benchmarking efforts. More specifically, we assess here the performance of eight popular wavefunction approaches, namely,
CIS(D), {\AD}, {\CCD}, {\STEOM}, {\CCSD}, CCSDR(3), CCSDT-3, and {\CCT}. The complete list of results can be found in Table \hl{SXXX} of the SI. To identify the ES for all approaches, we have made, as for the TBE above CIS(D), {\AD}, {\CCD}, {\STEOM}, {\CCSD}, CCSDR(3), CCSDT-3, and {\CCT}. The complete list of results can be found in Table \hl{SXXX} of the SI. To identify the ES for all approaches, we have made, as for the TBE above
choices based on the usual criteria (symmetry, oscillator strength, ordering, and nature of the involved orbitals), and most, yet not all, assignments are unambiguous. In addition, because all tested approaches are single-reference choices based on the usual criteria (symmetry, oscillator strength, ordering, and nature of the involved orbitals). Except for a few cases (see above), assignments are unambiguous. In addition, because all tested approaches are single-reference
methods, we have removed from the reference set the ``unsafe'' transition energies (in italics in Table \ref{Table-tbe}), as well as the four transitions with a dominant double excitation character (with $\Td < 50\%$ as listed in Table methods, we have removed from the reference set the ``unsafe'' transition energies (in italics in Table \ref{Table-tbe}), as well as the four transitions with a dominant double excitation character (with $\Td < 50\%$ as listed in Table
\ref{Table-12}). For the latter transitions, only CCSDT-3 and {\CCT} are able to detect their presence, but with, of course, extremely large errors. A comprehensive list of results are collected in Table \ref{Table-12} which, more \ref{Table-12}). For the latter transitions, only CCSDT-3 and {\CCT} are able to detect their presence, but with, of course, extremely large errors. A comprehensive list of results are collected in Table \ref{Table-12} which, more
specifically, gathers the MSE, MAE, RMSE, SDE, \MaxP, and \MaxN. Figure \ref{Fig-1} shows histograms of the error distributions for these eight methods. Before discussing these, let us stress two obvious biases of this specifically, gathers the MSE, MAE, RMSE, SDE, \MaxP, and \MaxN. Figure \ref{Fig-1} shows histograms of the error distributions for these eight methods. Before discussing these, let us stress two obvious biases of this
benchmark set: i) it encompasses only conjugated organic molecules containing 4 to 6 non-hydrogen atoms; and ii) we mainly used {\CCSDTQ} (4 atoms) or {\CCSDT} (5--6 atoms) reference values. As discussed in molecular set: i) it encompasses only conjugated organic molecules containing 4 to 6 non-hydrogen atoms; and ii) we mainly used {\CCSDTQ} (4 atoms) or {\CCSDT} (5--6 atoms) reference values. As discussed in
Section \ref{sec-ic} and in our previous work, \cite{Loo18a} the MAE obtained with these two methods are of the order of $0.01$ and $0.03$ eV, respectively. This means that any statistical quantity smaller than $\sim 0.02$--$0.03$ eV Section \ref{sec-ic} and in our previous work, \cite{Loo18a} the MAE obtained with these two methods are of the order of $0.01$ and $0.03$ eV, respectively. This means that any statistical quantity smaller than $\sim 0.02$--$0.03$ eV
is very likely to be irrelevant. is very likely to be irrelevant.
\renewcommand*{\arraystretch}{1.0} \renewcommand*{\arraystretch}{1.0}
\begin{table}[htp] \begin{table}[htp]
\caption{Mean signed error (MSE), mean absolute error (MAE), root-mean square error (RMSE), standard deviation of the errors (SDE), as well as the positive [\MaxP] and negative [\MaxN] maximal errors with respect to the TBE. \caption{Mean signed error (MSE), mean absolute error (MAE), root-mean square error (RMSE), standard deviation of the errors (SDE), as well as the positive [\MaxP] and negative [\MaxN] maximal errors with respect to the TBE/{\AVTZ} reported in Table \ref{Table-tbe}.
All these statistical quantities are reported in eV and have been obtained with the {\AVTZ} basis set. All these statistical quantities are reported in eV and have been obtained with the {\AVTZ} basis set.
``Count'' refers to the number of states.} ``Count'' refers to the number of states.}
\label{Table-12} \label{Table-12}
@ -1444,7 +1444,7 @@ CCSDT-3 &127& 0.05 &0.05 &0.07 &0.04 &0.26 &0.00\\
\begin{figure}[htp] \begin{figure}[htp]
\includegraphics[scale=0.98,viewport=2cm 14.5cm 19cm 27.5cm,clip]{Figure-1.pdf} \includegraphics[scale=0.98,viewport=2cm 14.5cm 19cm 27.5cm,clip]{Figure-1.pdf}
\caption{Histograms of the error patterns obtained with various levels of theory, taking the TBE/{\AVTZ} of Table \ref{Table-tbe} as references. \caption{Histograms of the error distribution obtained with various levels of theory, taking the TBE/{\AVTZ} of Table \ref{Table-tbe} as references.
Note the difference of scaling in the vertical axes.} Note the difference of scaling in the vertical axes.}
\label{Fig-1} \label{Fig-1}
\end{figure} \end{figure}
@ -1462,7 +1462,7 @@ The perturbative inclusion of triples as in CCSDR(3) yields a very small MAE ($0
of being too large, an error sign likely inherited from the parent {\CCSD} model. The $0.05$ eV MAE for CCSDR(3) is rather similar to the one obtained for smaller compounds when comparing to {\FCI} ($0.04$ eV), \cite{Loo18a} and is also inline with the of being too large, an error sign likely inherited from the parent {\CCSD} model. The $0.05$ eV MAE for CCSDR(3) is rather similar to the one obtained for smaller compounds when comparing to {\FCI} ($0.04$ eV), \cite{Loo18a} and is also inline with the
2009 benchmark study of Sauer et al. \cite{Sau09} 2009 benchmark study of Sauer et al. \cite{Sau09}
{\CCSD} provides an interesting case study. The calculated MSE ($+0.11$ eV), indicating an overestimation of the transition energies, fits well with several previous recent reports. \cite{Sch08,Car10,Wat13,Kan14,Jac17b,Kan17,Dut18,Jac18a,Loo18a} {\CCSD} provides an interesting case study. The calculated MSE ($+0.11$ eV), indicating an overestimation of the transition energies, fits well with several previous reports. \cite{Sch08,Car10,Wat13,Kan14,Jac17b,Kan17,Dut18,Jac18a,Loo18a}
It is, nonetheless, larger than the one determined for smaller molecules ($+0.05$ eV), \cite{Loo18a} hinting that the performance of {\CCSD} deteriorates for larger compounds. Moreover, the {\CCSD} MAE of $0.13$ eV is much smaller than the one reported by It is, nonetheless, larger than the one determined for smaller molecules ($+0.05$ eV), \cite{Loo18a} hinting that the performance of {\CCSD} deteriorates for larger compounds. Moreover, the {\CCSD} MAE of $0.13$ eV is much smaller than the one reported by
Thiel in his original work ($0.49$ eV) \cite{Sch08} but of the same order of magnitude as in the more recent study of K\'ann\'ar and Szalay performed on Thiel's set ($0.18$ eV for transitions with $\Td > 90\%$). \cite{Kan14} Retrospectively, it is pretty obvious that Thiel in his original work ($0.49$ eV) \cite{Sch08} but of the same order of magnitude as in the more recent study of K\'ann\'ar and Szalay performed on Thiel's set ($0.18$ eV for transitions with $\Td > 90\%$). \cite{Kan14} Retrospectively, it is pretty obvious that
Thiel's much larger MAE is very likely due to the {\CASPT} reference values. \cite{Sch08} Indeed, as we have shown several times in the present study, {\CASPT} transitions energies tend to be significantly too low, therefore exacerbating the usual {\CCSD} overestimation. Thiel's much larger MAE is very likely due to the {\CASPT} reference values. \cite{Sch08} Indeed, as we have shown several times in the present study, {\CASPT} transitions energies tend to be significantly too low, therefore exacerbating the usual {\CCSD} overestimation.
@ -1509,7 +1509,7 @@ CCSDT-3 &0.05 & &0.06 &0.03 &0.08 &0.04\\
\end{tabular} \end{tabular}
\end{table} \end{table}
\section{Conclusions and outlook} \section{Concluding remarks}
We have computed highly-accurate vertical transition energies for a set of 27 organic molecules containing from 4 to 6 (non-hydrogen) atoms. To this end, we employed several state-of-the-art theoretical models with increasingly large diffuse basis sets. We have computed highly-accurate vertical transition energies for a set of 27 organic molecules containing from 4 to 6 (non-hydrogen) atoms. To this end, we employed several state-of-the-art theoretical models with increasingly large diffuse basis sets.
Most of our theoretical best estimates are based on {\CCSDTQ} (4 atoms) or {\CCSDT} (5 and 6 atoms) excitation energies. For the vast majority of the listed excited states, the present contribution is the very first to disclose (sometimes basis-set extrapolated) {\CCSDT}/{\AVTZ} Most of our theoretical best estimates are based on {\CCSDTQ} (4 atoms) or {\CCSDT} (5 and 6 atoms) excitation energies. For the vast majority of the listed excited states, the present contribution is the very first to disclose (sometimes basis-set extrapolated) {\CCSDT}/{\AVTZ}