diff --git a/Manuscript/FCI2.tex b/Manuscript/FCI2.tex index be0ce84..27a1f77 100644 --- a/Manuscript/FCI2.tex +++ b/Manuscript/FCI2.tex @@ -25,6 +25,7 @@ \newcommand{\Ndet}{N_\text{det}} \newcommand{\ex}[6]{$^{#1}#2_{#3}^{#4}(#5 \ra #6)$} +\newcommand{\Td}{\%T_1} % methods \newcommand{\TDDFT}{TD-DFT} @@ -152,11 +153,11 @@ Following the same philosophy, two years ago, we reported a set of 106 transitio CI (SCI) calculations on {\CCT}/{\AVTZ} GS structures. \cite{Loo18a} We exploited these TBE to benchmark many ES theories. \cite{Loo18a} Amongst our conclusions, we found that {\CCSDTQ} yields near-{\FCI} quality excitation energies, whereas we could not detect any significant differences between {\CCT} and {\CCSDT} transition energies, both being very accurate with mean absolute errors (MAE) as small as $0.03$ eV compared to {\FCI}. -Although these conclusions agree well with earlier studies, \cite{Wat13,Kan14,Kan17} they obviously only hold for single excitations, \ie, transitions with $\%T_1$ in the range $90$--$100\%$. Therefore, we also recently proposed a set of 20 TBE for transitions exhibiting a significant double-excitation character (\ie, with $\%T_1$ typically below $80\%$). \cite{Loo19c} -Unsurprisingly, our results clearly evidenced that the error in CC methods is intimately related to the $\%T_1$ value. -For example, ES with a significant yet \titou{not dominant} double excitation character, such as the infamous $A_g$ ES of butadiene ($\%T_1 = 75\%$), +Although these conclusions agree well with earlier studies, \cite{Wat13,Kan14,Kan17} they obviously only hold for single excitations, \ie, transitions with $\Td$ in the range $80$--$100\%$. Therefore, we also recently proposed a set of 20 TBE for transitions exhibiting a significant double-excitation character (\ie, with $\Td$ typically below $80\%$). \cite{Loo19c} +Unsurprisingly, our results clearly evidenced that the error in CC methods is intimately related to the $\Td$ value. +For example, ES with a significant yet \titou{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 excitation character, \eg, the low-lying $(n,n) \ra (\pis,\pis)$ excitation in nitrosomethane ($\%T_1 = 2\%$), +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 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 required to obtain accurate results. \cite{Loo19c} @@ -166,7 +167,7 @@ compounds encompassing 4, 5, and 6 non-hydrogen atoms. For such systems, perfor extrapolating to the {\FCI} limit with an error of $\sim 0.01$ eV is rarely achievable in practice. Consequently, the ``brute-force'' determination of {\FCI}/CBS estimates, as in our earlier work, \cite{Loo18a} is definitely out of reach here. Anticipating this problem, we have recently investigated bootstrap CBS extrapolation techniques. \cite{Loo18a,Loo19c} In particular, we have demonstrated that, following an ONIOM-like scheme, \cite{Chu15} one can very accurately estimate such limit by correcting high-level values obtained in a small basis by the difference between {\CCT} results obtained in a larger basis and in the same small basis.\cite{Loo18a} -We globally follow such strategy here. In addition, we also perform {\NEV} calculations in an effort to check the consistency of our estimates. It is especially critical for ES with intermediate $\%T_1$ values. +We globally follow such strategy here. In addition, we also perform {\NEV} calculations in an effort to check the consistency of our estimates. It is especially critical for ES with intermediate $\Td$ values. Using this protocol, we define a set of more than 200 {\AVTZ} reference transition energies, most being within $\pm 0.03$ eV of the {\FCI} limit. These reference energies are obtained on {\CCT}/{\AVTZ} geometries and further basis set corrections (up to quadruple-$\zeta$ at least) are also provided for {\CCT}. Together with the results obtained in our two earlier works, \cite{Loo18a,Loo19c} the present TBE will hopefully contribute to climb a rung higher on the ES accuracy ladder. @@ -594,14 +595,13 @@ $^i${0-0 energies from Ref.~\citenum{Jud84c}.} \end{flushleft} \end{table} -\titou{HERE} For acetone, one should clearly distinguish the valence ES, for which both methodological and basis set effects are small, and the Rydberg transitions that both very sensitive to the basis set, and upshifted by ca.~$0.04$ eV with {\CCSDTQ} as compared to {\CCT} and {\CCSDT}. For this compound, the 1996 {\CASPT} transition energies of Merch\'an and coworkers listed on the right panel of Table \ref{Table-4} are clearly too low, especially for the three valence ES. \cite{Mer96b} As expected, this error can be partially ascribed to the \titou{details of the calculations}, as the Urban group obtained {\CASPT} excitation energies of $4.40$, $4.09$ and $6.22$ eV for the $^1A_2$, $^3A_2$, and $^3A_1$ ES, \cite{Pas12} in much 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. -In contrast to acetone, both valence and Rydberg ES of thioacetone are rather insensitive to 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 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 show the same ranking but are systematically larger than the available experimental 0-0 energies. @@ -628,14 +628,14 @@ This seems to indicate that {\NEV}, as applied here, has a slight tendency to ov \subsection{Five-atom molecules} -Let us now turn towards five-member rings. We treat here five classical 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 used in the previous Section, namely {\CCSDTQ} and {\FCI}, become beyond reach for these compounds, one has to rely on the nature of the ES and the consistency between the results of +We now consider five-member 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 used in the previous Section, namely {\CCSDTQ} and {\FCI}, become beyond reach for these compounds, one has to rely on the nature of the ES and the consistency between the results of different approaches to deduce TBE. For furan, previous theoretical works have been performed with almost all possible wavefunction approaches, \cite{Ser93b,Nak96,Tro97b,Chr98b,Chr98c,Wan00,Gro03,Pas06b,Sch08,She09b,Li10c,Sau11,Sil10b,Sil10c,Hol15,Sch17} but the present work is, to the best of our knowledge, the first to disclose {\CCSDT} values as well as {\CCT} energies obtained with a quadruple-$\zeta$ basis set. Our results for ten low-lying states are listed in Tables \ref{Table-5} and S5. -All computed singlet (triplet) transitions show $\%T_1$ in the 92--94 \%\ (97--99) \%\ range, and consistently the maximal discrepancies between the {\CCT} and {\CCSDT} transition energies are small ($0.04$ eV). In addition there is a -good consistency between the present data and both the {\NEV} results of Ref.~\citenum{Pas06b} and the MR-CC values of Ref.~\citenum{Li10c} for almost all transitions, but the $^1B_2 (\pi \ra 3p)$ excitation that we predict to +All computed singlet (triplet) transitions show $\Td$ in the $92$--$94\%$ ($97$--$99\%$) range, and consistently the maximal discrepancies between the {\CCT} and {\CCSDT} transition energies are small ($0.04$ eV). In addition there is a +good consistency between the present data and both the {\NEV} results of Ref.~\citenum{Pas06b} and the MR-CC values of Ref.~\citenum{Li10c} for almost all transitions, but the $^1B_2$ ($\pi \ra 3p$) excitation that we predict to be significantly higher than in most previous works, even after accounting for the quite large basis set effects ($-0.10$ eV between {\AVTZ} and {\AVQZ} estimates, see Table S5). We trust that our estimate is the most accurate to date for that ES. Interestingly, the recent {\AT} values of Ref.~\citenum{Hol15} are smaller by ca.~$-0.2$ eV as compared to {\CCSDT} values for all transitions (see Table \ref{Table-6}), consistent with the error sign we found in smaller compounds with ADC(3). \cite{Loo18a} Eventually, we note that the experimental data, \cite{Vee76b,Fli76,Rob85b} provide the same state ordering as our calculations. @@ -687,7 +687,7 @@ $^c${MR-CC results from Ref.~\citenum{Li10c};} $^d${{\AT} results from Ref.~\citenum{Hol15};} $^e${{\CCT} results from Ref.~\citenum{Sch17};} $^f${Various experiments summarized in Ref.~\citenum{Wan00};} -$^g${Electron impact from Ref.~\citenum{Vee76b}: for the $^1A_1$ state two values (6.44 and 6.61 eV) are reported, whereas for the two lowest triplet states, Two values (3.99 eV and 5.22 eV) can be found in Ref.~\citenum{Fli76};} +$^g${Electron impact from Ref.~\citenum{Vee76b}: for the $^1A_1$ state two values ($6.44$ and $6.61$ eV) are reported, whereas for the two lowest triplet states, Two values ($3.99$ eV and $5.22$ eV) can be found in Ref.~\citenum{Fli76};} $^h${{\NEV} results from Ref.~\citenum{Pas06c};} $^i${Best estimate from Ref.~\citenum{Chr99}, based on CC calculations;} $^j${XMS-{\CASPT} results from Ref.~\citenum{Hei19};} @@ -698,7 +698,7 @@ $^l${Vapour UV spectra from Refs.~\citenum{Pal03b}, \citenum{Hor67}, and \citenu \end{table} Like furan, pyrrole has been extensively investigated previously using a large palette of approaches. \cite{Ser93b,Nak96,Tro97,Pal98,Chr99,Wan00,Roo02,Pal03b,Pas06c,Sch08,She09b,Li10c,Sau11,Sil10b,Sil10c,Nev14,Sch17,Hei19} -We report six low-lying singlet and four triplet ES in Tables \ref{Table-5} and S5. All considered transitions have very large $\%T_1$ but for the totally symmetric $\pi \ra \pis$ excitation ($\%T_1 = 86\%$). For all states, we found +We report six low-lying singlet and four triplet ES in Tables \ref{Table-5} and S5. All considered transitions have very large $\Td$ but for the totally symmetric $\pi \ra \pis$ excitation ($\Td = 86\%$). For all states, 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' group, \cite{Hei19} 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 also remarkable but for the two $B_2$ transitions that were reported as significantly mixed in that venerable work. For the lowest singlet ES, the {\FCI}/{\Pop} value is $5.24 \pm 0.02$ eV confirming the performances of both {\CCT} and {\CCSDT} for that transition. @@ -784,28 +784,28 @@ $^q${0-0 energies from Ref.~\citenum{Hol14}.} Although a quite significant array of previous wavefunction studies has been performed for cyclopentadiene not only at the {\CASPT}, \cite{Ser93b,Sch08,Sil10c} and CC \cite{Sch08,Sil10b,Sch17} levels but also with SAC-CI \cite{Wan00b} and various multi-reference approaches, \cite{Nak96,She09b} this compound has been less intensively studied than furan and pyrrole (\emph{vide infra}), probably due to the presence of the -methylene group that renders the computations significantly more expensive. All transitions listed in Tables \ref{Table-6} and S6 are characterized by $\%T_1$ exceeding 93\%\ but for the $^1A_1 (\pi \ra \pis)$ -excitation that has a nature similar to the lowest $A_g$ state of butadiene ($\%T_1 = 79\%$). Consistently, the {\CCT} and {\CCSDT} results are nearly identical for all ES but for that transition. By comparing the results -obtained for this $A_1 (\pi \ra \pis)$ transition to its butadiene counterpart, one can infer that the {\CCSDT} estimate is probably too large by ca. 0.04--0.08 eV, and that the {\NEV} value is likely not more accurate -than the {\CCSDT} one. This statement is also in line with the results of Ref.~\citenum{Loo19c}. For the two $B_2 (\pi \ra \pis)$ transitions, we could obtain a {\FCI}/{\Pop} estimates of 5.78$\pm$0.02 eV and -(singlet) 3.33$\pm$0.05 eV (triplet); the {\CCT} and {\CCSDT} transition energies falling inside these energetic windows in both cases. As can be seen in Tables \ref{Table-6} and S6, the basis set effects are rather moderate -for all transitions, with no variations 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 +methylene group that renders the computations significantly more expensive. All transitions listed in Tables \ref{Table-6} and S6 are characterized by $\Td$ exceeding 93\%\ but for the $^1A_1 (\pi \ra \pis)$ +excitation that has a nature similar to the lowest $A_g$ state of butadiene ($\Td = 79\%$). Consistently, the {\CCT} and {\CCSDT} results are nearly identical for all ES but for that transition. By comparing the results +obtained for this $A_1 (\pi \ra \pis)$ transition to its butadiene counterpart, one can infer that the {\CCSDT} estimate is probably too large by ca.~$0.04$--$0.08$ eV, and that the {\NEV} value is likely not more accurate +than the {\CCSDT} one. This statement is also in line with the results of Ref.~\citenum{Loo19c}. For the two $B_2 (\pi \ra \pis)$ transitions, we could obtain a {\FCI}/{\Pop} estimates of $5.78 \pm 0.02$ eV and +(singlet) $3.33 \pm 0.05$ eV (triplet); the {\CCT} and {\CCSDT} transition energies falling inside these energetic windows in both cases. As can be seen in Tables \ref{Table-6} and S6, the basis set effects are rather moderate +for all transitions, with no variations 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 derivative, \cite{Fru79,McD85,McD91b,Sab92} but hardly allow to make the final call. Due to its lower symmetry, imidazole has been less investigated, the most advanced studies available probably remaining the 1996 {\CASPT} work of Serrano-Andres \emph{et al}, \cite{Ser96b} and the basis-set extrapolated {\CCT} investigation of Silva-Junior \emph{et al} for the valence transitions. \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 $\%T_1$ of the two singlet $A'$ states -are slightly smaller 90\%. For all eight considered transitions, the basis set effects are moderate and {\AVTZ} yield results within 0.03 eV of {\AVQZ} (Table S6 in the SI). \hl{NEV ?} +Our results are displayed in Tables \ref{Table-6} and S6. The {\CCT} and {\CCSDT} values are quite consistent despite the fact that the $\Td$ of the two singlet $A'$ states +are slightly smaller 90\%. For all eight considered transitions, the basis set effects are moderate and {\AVTZ} yield results within $0.03$ eV of {\AVQZ} (Table S6 in the SI). \hl{NEV ?} Finally, the ES thiophene, which is one of the most important building block in organic electronic devices, were the subject of a few 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 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 $\%T_1$ above -90\%\ but for the two $^1A_1$ transitions that show $\%T_1$ of 88\%\ and 87\%. The agreement between {\CCT} and {\CCSDT} remains nevertheless excellent for the lowest totally symmetric transition. Thiophene is +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\%\ but for the two $^1A_1$ transitions that show $\Td$ of 88\%\ and 87\%. The agreement between {\CCT} and {\CCSDT} remains nevertheless excellent for the lowest totally symmetric transition. Thiophene is also a typical compound in which unambiguous characterization of the nature of the ES is difficult, with \eg, a strong mixing between the second and third singlet $B_2$ ES rendering the assignment of the valence ($\pi \ra \pis$) or Rydberg ($\pi \ra 3p$) character of that transittion uneasy 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, 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). +previously discussed isostructural systems, 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). The data of Table \ref{Table-6} are globally in good agreement with the previously reported values with discrepancies that are however significant for the three highest-lying singlet states. \subsection{Six-atom molecules} @@ -821,7 +821,7 @@ who used {\STEOM}, to study the ES of all these derivatives with a theoretically These three highly-symmetric systems allow to directly perform {\CCSDT}/{\AVTZ} calculations for singlet states without the need for basis set extrapolation. Benzene was studied many times before, \cite{Sob93,Lor95b,Chr96c,Pac96,Del97b,Noo99,Hal02,Li07b,Sch08,Dev08,Sil10b,Sil10c,Li11,Lea12,Kan14,Sch17,Dut18,Sha19,Loo19c} and we report in Tables \ref{Table-7} and S7 estimates obtained for -five singlet and three triplet ES, all characterized by $\%T_1$ exceeding 90\%\ but the lowest singlet (86\%). As can be seen, the two CC approaches are again yielding very consistent transitions energies +five singlet and three triplet ES, all characterized by $\Td$ exceeding 90\%\ but the lowest singlet (86\%). As can be seen, the two CC approaches are again yielding very consistent transitions energies and {\AVTZ} is essentially providing basis set converged transition energies. The present estimates are also very consistent with early {\CCT}\cite{Chr96c} and very recent RASPT2 values. \cite{Sha19} For both the singlet and triplet transitions, our values are slightly larger than available electron impact/multi-photon measurements. \cite{Doe69,Nak80,Joh76,Joh83,Hir91} @@ -936,34 +936,34 @@ $^m${all these doubly ES have a $(n,n \ra \pis, \pis)$ character.} Numerous previous theoretical estimates are available for both pyrazine, \cite{Ful92,Del97b,Web99,Noo99,Li07b,Sch08,Sau09,Sil10c,Woy10,Car10,Lea12,Kan14,Sch17,Dut18} and tetrazine, \cite{Sta96,Del97b,Rub99,Noo99,Ada00,Noo00,Ang09,Sch08,Sau09,Sil10b,Sil10c,Car10,Lea12,Kan14,Sch17,Dut18,Pas18b} -for which the $D_{2h}$ symmetry helps distinguishing the different ES. Our results are collected in Tables \ref{Table-8} and S8. In pyrazine, all transitions are characterized by $\%T_1 > 85\%$, but for $^1B_{1g} (n \ra \pis)$, and the -changes in going from {\CCT} to {\CCSDT} are always trifling but for the highest-lying singlet state considered here. When going from triple-$\zeta$ to quadruple-$\zeta$, the variations do not exceed 0.04 eV, even for the four considered Rydberg +for which the $D_{2h}$ symmetry helps distinguishing the different ES. Our results are collected in Tables \ref{Table-8} and S8. In pyrazine, all transitions are characterized by $\Td > 85\%$, but for $^1B_{1g} (n \ra \pis)$, and the +changes in going from {\CCT} to {\CCSDT} are always trifling but for the highest-lying singlet state considered here. When going from triple-$\zeta$ to quadruple-$\zeta$, the variations do not exceed $0.04$ eV, even for the four considered Rydberg ES. This indicates that one can probably be highly confident in the present estimates. Again, the previous {\CASPT} estimates, \cite{Ful92,Web99,Sch08} appear to be globally too low, whereas the unconventional CASPT3 results that are available, \cite{Woy10} are too large. The same holds for the SAC-CI results. \cite{Li07b} In fact we obtain globally our best match with the {\STEOM} values of Nooijen (but for the highest ES), \cite{Noo99} and recent {\CCT} estimates. \cite{Sch17}. The experimental data we are aware of, \cite{Bol84,Oku90,Wal91,Ste11c} do not report all transitions, but provide globally a similar ranking for the triplet transitions. -For tetrazine, we consider valence ES only, but three transitions present a true double excitation nature ($\%T_1 < 10\%$), for which {\CCT} nor {\CCSDT} can not be viewed as reliable, and the best approach is likely {\NEV}. \cite{Loo19c} -For all other transitions, the $\%T_1$ are in the 80-90\%\ range for singlets and larger than 95\%\ for triplets, and the results of the two CC approaches are very consistent, but for the lowest $^3B_{1u} (\pi \ra \pis)$ excitation. -In all other cases, there is a good consistency between the values we obtained with the two CC models, and the basis set effects are very small beyond {\AVTZ} with maximal variations of 0.02 eV only (Table S8). The present values are +For tetrazine, we consider valence ES only, but three transitions present a true double excitation nature ($\Td < 10\%$), for which {\CCT} nor {\CCSDT} can not be viewed as reliable, and the best approach is likely {\NEV}. \cite{Loo19c} +For all other transitions, the $\Td$ are in the 80-90\%\ range for singlets and larger than 95\%\ for triplets, and the results of the two CC approaches are very consistent, but for the lowest $^3B_{1u} (\pi \ra \pis)$ excitation. +In all other cases, there is a good consistency between the values we obtained with the two CC models, and the basis set effects are very small beyond {\AVTZ} with maximal variations of $0.02$ eV only (Table S8). The present values are almost systematically larger than previous {\CASPT},\cite{Rub99} {\STEOM}, \cite{Noo00} and GVVPT2 \cite{Dev08} estimates, and are globally in agreement with Thiel's {\CCT}/{\AVTZ} values, \cite{Sil10c} although we note variations -of ca. 0.20 eV for some specific transitions like the $B_{2g}$ transitions, likely due to the use of different geometries in that work. The experimental EEL values from Palmer's work, \cite{Pal97} show a reasonable agreement with our estimates. +of ca.~$0.20$ eV for some specific transitions like the $B_{2g}$ transitions, likely due to the use of different geometries in that work. The experimental EEL values from Palmer's work, \cite{Pal97} show a reasonable agreement with our estimates. \subsubsection{Pyridazine, pyridine, pyrimidine, and triazine} Those four azabenzenes, of $C_{2v}$ and $D_{3h}$ symmetry, are also popular molecules for ES calculations. \cite{Pal91,Ful92,Wal92,Lor95,Del97b,Noo97,Noo99,Fis00,Cai00b,Wan01b,Sch08,Sil10b,Sil10c,Car10,Lea12,Kan14,Sch17,Dut18} Our results for pyridazine and pyridine are collected in Tables \ref{Table-9} and S9. For the former compounds, the available wavefunction results \cite{Pal91,Ful92,Del97b,Noo99,Fis00,Sch08,Sil10b,Sil10c,Kan14,Sch17,Dut18} -have all considered the singlet transitions only, at the exception of rather old MRCI, \cite{Pal91} and {\CASPT} investigations. \cite{Fis00} The $\%T_1$ 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 {\AVDZ} {\CCT} and {\CCSDT} energies, but for the $^3B_2 (\pi \ra \pis)$ transition. \hl{BASIS SETS} For the valence singlet +have all considered the singlet transitions only, at the exception of rather old MRCI, \cite{Pal91} and {\CASPT} investigations. \cite{Fis00} The $\Td$ 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 {\AVDZ} {\CCT} and {\CCSDT} energies, but for the $^3B_2 (\pi \ra \pis)$ transition. \hl{BASIS SETS} For the valence singlet ES, we find again a quite good agreement with previous {\STEOM} \cite{Noo99} and CC \cite{Del97b,Sil10c} estimates, but are again significantly higher than {\CASPT} estimates. \cite{Ful92,Sil10c} For the triplets, the present data represents the best published to date. Interestingly, beyond the usually cited experiments, \cite{Inn88,Pal91} there is a very recent experimental EEL analysis for pyridazine, \cite{Lin19} that localized -almost all ES. The transition energies reported in this very recent effort are systematically smaller than our CC estimates, by ca. -0.20 eV, but remarkably show exactly the exact same ranking. +almost all ES. The transition energies reported in this very recent effort are systematically smaller than our CC estimates, by ca.~$-0.20$ eV, but remarkably show exactly the exact same ranking. For pyridine, that has been more thoroughly investigated with wavefunction approaches, \cite{Ful92,Lor95,Del97b,Noo97,Noo99,Cai00b,Wan01b,Sch08,Sil10b,Sil10c,Car10,Kan14,Sch17,Dut18} and for which we could found two - detailed EEL experiments, \cite{Wal90,Lin16} the general trends described for pyridazine pertain: i) large $\%T_1$ and good CC consistency for all transitions listed in Table \ref{Table-9}; ii) \hl{basis}; iii) good agreement with previous + detailed EEL experiments, \cite{Wal90,Lin16} the general trends described for pyridazine pertain: i) large $\Td$ and good CC consistency for all transitions listed in Table \ref{Table-9}; ii) \hl{basis}; iii) good agreement with previous CC estimates; and iv) same ranking of the ES as in the most recent measurements. \cite{Lin16} Beyond those aspects, it is worth to underline that the second $^1B_2 (\pi \ra \pis)$ ES is strongly mixed with a nearby -Rydberg transition that is separated by only 0.03 eV at the {\CCT}/{AVTZ} level, making the analysis particularly challenging for that specific transition. \hl{Keep or not A1 transitiion} +Rydberg transition that is separated by only $0.03$ eV at the {\CCT}/{AVTZ} level, making the analysis particularly challenging for that specific transition. \hl{Keep or not A1 transitiion} \begin{table}[htp] \caption{\small Vertical transition energies (in eV) of pyridazine and pyridine.} @@ -1029,10 +1029,10 @@ $^j${Significant state mixing with a close-lying Rydberg transition, rendering u The results obtained for both pyrimidine and triazine are listed in Tables \ref{Table-10} and S10. For the former derivative previous theoretical \cite{Ful92,Del97b,Ser97b,Noo99,Ohr01,Fis03b,Li07b,Sch08,Sil10b,Sil10c,Car10,Kan14,Sch17,Dut18} and experimental \cite{Bol84,Pal90,Lin15} studies are rather extensive, as it can be viewed as the smallest model of DNA bases. For triazine, which does not have an abelian point group, one finds less theoretical studies, \cite{Ful92,Wal92,Pal95,Del97b,Noo99,Oli05,Sch08,Sil10b,Sil10c,Kan14,Sch17,Dut18}, especially for the triplet transitions. \cite{Wal92,Pal95,Oli05} The experimental data are also less numerous. \cite{Bol84,Wal92} - As in pyridazine and pyridine, all the ES listed in Table \ref{Table-10} show $\%T_1$ larger than 85\%\ for singlet and 95\%\ for triplet, so that {\CCT} and {\CCSDT} are highly coherent, but possibly in one case, the - $^3A_1 (\pi \ra \pis)$ transitions in pyrimidine. The basis set effects are also small, with no variations larger than 0.10 (0.03) eV between double-$\zeta$ and triple-$\zeta$ (triple-$\zeta$ and quadruple-$\zeta$) and only + As in pyridazine and pyridine, all the ES listed in Table \ref{Table-10} show $\Td$ larger than $85\%$ for singlet and $95\%$ for triplet, so that {\CCT} and {\CCSDT} are highly coherent, but possibly in one case, the + $^3A_1 (\pi \ra \pis)$ transitions in pyrimidine. The basis set effects are also small, with no variations larger than $0.10$ ($0.03$) eV between double-$\zeta$ and triple-$\zeta$ (triple-$\zeta$ and quadruple-$\zeta$) and only slightly larger variations for the two Rydberg transitions. For both compounds, the current values are almost systematically larger than most previously published data. For the triplets of triazine, the three lowest ES estimated - by {\CASPT} previously are too low by ca. -0.5 eV. + by {\CASPT} previously are too low by ca.~$-0.5$ eV. s \begin{table}[htp] @@ -1098,15 +1098,15 @@ $^h${CC3-ext.~from Ref.~\citenum{Sil10c}.} \section{Theoretical Best Estimates} -In Table \ref{Table-tbe}, we present our theoretical best estimates as obtained with the {\AVTZ} basis set or further corrected for basis set effects. The details of the approach used to get all the TBE are given as well. -For all states with a dominant single-excitation character, that is when $\%T_1 > 80$\%, we rely on CC results, using the incremental strategy to determine the TBE. For ES with $\%T_1$ between 70\%\ and 80\%, -our previous works indicated that {\CCSDT} tends to overshoot the transition energies by ca. 0.05--0.10 eV, but that the {\NEV} error tends to be slightly larger (on average). \cite{Loo19c} Therefore, if {\CCSDTQ} or {\FCI} +In Table \ref{Table-tbe}, we present our TBE obtained with the {\AVTZ} basis set or further corrected for basis set effects. The details of the approach used to get all the TBE are given as well. +For all states with a dominant single-excitation character (that is when $\Td > 80\%$), we rely on CC results, using the incremental strategy to determine the TBE. For ES with $\Td$ between $70\%$ and $80\%$, +our previous works indicated that {\CCSDT} tends to overshoot the transition energies by roughly $0.05$--$0.10$ eV, but that the {\NEV} error tends to be slightly larger (on average). \cite{Loo19c} Therefore, if {\CCSDTQ} or {\FCI} results are unavailable, it is hard to make the final call. For the other transitions, we relied either on the current or previous FCI or the {\NEV} values as reference. We indicate some transition energies -in italics in Table \ref{Table-tbe} to underline that they are (relatively) less accurate. This is the case when: i) {\NEV} results have to be selected; ii) the affordable CC calculations yield quite large changes from one expansion order to another -despite large $\%T_1$; and iii) there is a very large 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-$\zeta$, +in italics in Table \ref{Table-tbe} to stress that they are (relatively) less accurate. This is the case when: i) {\NEV} results have to be selected; ii) the affordable CC calculations yield quite large changes from one expansion order to another +despite large $\Td$; and iii) there is a very large 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}/{\AVFZ} results. For several compounds, we also provide in the SI, {\CCT}/{\DAVQZ} transition energies. However, we are not using these values as reference. This is because, the addition of a second set of diffuse orbitals tends to modify the computed transition energies significantly only when it induces a more complex state mixing. We also stick to the -frozen-core approximation for two reasons: i) the corrections brought by ``full-correlation'' are generally trifling (typically $\pm$ 0.02 eV) for the compounds under study (see the SI for many examples); and ii) it would be, +frozen-core approximation for two reasons: i) the corrections brought by ``full-correlation'' are generally trifling (typically $\pm 0.02$ eV) for the compounds under study (see the SI for many examples); and ii) it would be, in principle, necessary to add core polarization functions in such ``full'' calculations. Table \ref{Table-tbe} encompasses 238 ES, all at least obtained at the {\CCSDT} level. This set that can be decomposed as follows: 144 singlet and 94 triplet transitions, or 174 valence (99 $\pi \ra \pis$, 71 @@ -1125,12 +1125,12 @@ as well TBE obtained with the same basis set together with the method used to ob that have always been obtained at the {\CCT} level. Values displayed in italics are likelt relatively less accurate. All values are in the FC approximation.} \label{Table-tbe}\\ \hline & & & & \mc{2}{l}{TBE/{\AVTZ}} & \mc{2}{l}{TBE/CBS} \\ - & State & $f$ & $\%T_1$ & Value & Method$^a$ & Value & Corr. \\ + & State & $f$ & $\Td$ & Value & Method$^a$ & Value & Corr. \\ \hline \endfirsthead \hline & & & & \mc{2}{l}{TBE/{\AVTZ}} & \mc{2}{l}{TBE/CBS} \\ - & State & $f$ & $\%T_1$ & Value & Method $^a$ & Value & Corr. \\ + & State & $f$ & $\Td$ & Value & Method $^a$ & Value & Corr. \\ \hline \endhead \hline \mc{7}{r}{{Continued on next page}} \\ @@ -1394,12 +1394,12 @@ Method F: {\FCI}/{\AVDZ} value (from Ref.~\citenum{Loo19c}) corrected by the dif \section{Benchmarks} 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 performances 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} in the SI. As all these approaches are single-reference methods, we have removed from the -benchmark not only the unsafe transition energies (in italics in Table \ref{Table-tbe}), but also the four transitions with a dominant double excitation character ($\%T_1 < 50\%$ listed in Table \ref{Table-tbe}). +{\CCD}, {\STEOM}, {\CCSD}, CCSDR(3), CCSDT-3 and {\CCT}. The complete list of results can be found in Table \hl{SXXX} in the SI. As all these approaches are single-reference methods, we have removed from the +benchmark not only the unsafe transition energies (in italics in Table \ref{Table-tbe}), but also the four transitions with a dominant double excitation character ($\Td < 50\%$ listed in Table \ref{Table-tbe}). Our global results are collected in Table \ref{Table-bench} that presents the MSE, MAE, root mean square deviation (RMS), standard deviation (SD), as well as the positive [\MaxP] and negative [\MaxN] maximum deviations. Figure \ref{Fig-1} shows histograms of the error distributions for all eight methods. Before discussing the obtained results, let us underline two obvious bias of this benchmark: 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 deviation (or difference of deviations) smaller than ca. 0.02--0.03 eV is likely irrelevant. +with these two methods are of the order of $0.01$ and $0.03$ eV, respectively. This means that any deviation (or difference of deviations) smaller than ca.~$0.02$--$0.03$ eV is likely irrelevant. \renewcommand*{\arraystretch}{1.0} \begin{table}[htp] @@ -1430,36 +1430,36 @@ CCSDT-3 &126 &0.05 &0.05 &0.07 &0.04 &0.26 &0.00\\ Let us analyse the global performances of all methods, starting with the most refined models. The relative accuracies of {\CCT} and {\CCSDT}-3 as compared to {\CCSDT} remains an open question in the literature. \cite{Wat13,Dem13} Indeed, to our knowledge, the only two previous reports discussing this specific aspect are limited to tiny compounds. \cite{Kan17,Loo18a} According to the results of Table \ref{Table-bench}, it appears that {\CCT} has the edge, although -{\CCSDT}-3 is closer to {\CCSDT} in formal terms. Indeed, {\CCSDT}-3 seems to provide slightly too large transition energies (MSE of +0.05 eV). These conclusions are qualitatively consistent with the analyses performed for smaller derivatives, +{\CCSDT}-3 is closer to {\CCSDT} in formal terms. Indeed, {\CCSDT}-3 seems to provide slightly too large transition energies (MSE of $+0.05$ eV). These conclusions are qualitatively consistent with the analyses performed for smaller derivatives, \cite{Kan17,Loo18a} but the amplitude of {\CCSDT}-3's errors are larger with the present set. Although the performances of {\CCT} might be unduly inflated by the use of {\CCSDT} and {\CCSDTQ} reference values, it is also clear that this method very rarely fails (Figure \ref{Fig-1}). Consequently, {\CCT} transition energies can be viewed as very solid references for all transitions with a dominant single-excitation character. This conclusion is again consistent with previous analyses performed for smaller compounds, \cite{Kan17,Loo18a} as well as with recent comparisons between theoretical and experimental 0-0 energies that have been performed by some of us on medium-sized molecules.\cite{Loo18b,Loo19a,Sue19} To state it more bluntly: it appears likely that {\CCT} is even more accurate than previously thought. In addition, from all the comparisons made in this work, one can conclude that {\CCT} regularly outperforms {\CASPT} and {\NEV}, even when -these methods are combined with active space chosen by specialists, a statement that seems true as long as the considered ES does not show a strong multiple excitation character, that is, when $\%T_1 < 70\%$. The perturbative -inclusion of triples as made in CCSDR(3) offers a very small MAE (0.05 eV) for a much reduced computational cost as compared to {\CCSDT}. Nevertheless, as with {\CCSDT}-3, the CCSDR(3) transition energies have a clear tendency -of being too large, an error sign likely inherited from the parent {\CCSD} model. This 0.05 eV MAE for CCSDR(3) is rather similar to the one obtained for small compounds when comparing to {\FCI} (0.04 eV), \cite{Loo18a} and is also inline with the -2009 benchmark of Sauer et al. \cite{Sau09} {\CCSD} provides an interesting case. The calculated MSE (+0.11 eV), indicating an overestimation of the transition energies, fits well many previous reports, -\cite{Sch08,Car10,Wat13,Kan14,Jac17b,Kan17,Dut18,Jac18a,Loo18a} but is larger than the one determined for smaller molecules (+0.05 eV), \cite{Loo18a} hinting that the performances of {\CCSD} deteriorates when larger compounds -are considered. The {\CCSD} MAE (0.13 eV) is much smaller than the one reported by Thiel in its original work (0.49 eV) \cite{Sch08} but of the same order of magnitude as in the more recent study of Kannar and Szalay performed -on Thiel's set (0.18 eV for transitions with $\%T_1 > 90\%$ ). \cite{Kan14} In retrospect, the much larger value obtained by Thiel is likely related to the use of {\CASPT} reference values in the 2008 work. Indeed, as we have shown +these methods are combined with active space chosen by specialists, a statement that seems true as long as the considered ES does not show a strong multiple excitation character, that is, when $\Td < 70\%$. The perturbative +inclusion of triples as made in CCSDR(3) offers a very small MAE ($0.05$ eV) for a much reduced computational cost as compared to {\CCSDT}. Nevertheless, as with {\CCSDT}-3, the CCSDR(3) transition energies have a clear tendency +of being too large, an error sign likely inherited from the parent {\CCSD} model. This $0.05$ eV MAE for CCSDR(3) is rather similar to the one obtained for small compounds when comparing to {\FCI} ($0.04$ eV), \cite{Loo18a} and is also inline with the +2009 benchmark of Sauer et al. \cite{Sau09} {\CCSD} provides an interesting case. The calculated MSE ($+0.11$ eV), indicating an overestimation of the transition energies, fits well many previous reports, +\cite{Sch08,Car10,Wat13,Kan14,Jac17b,Kan17,Dut18,Jac18a,Loo18a} but is larger than the one determined for smaller molecules ($+0.05$ eV), \cite{Loo18a} hinting that the performances of {\CCSD} deteriorates when larger compounds +are considered. The {\CCSD} MAE ($0.13$ eV) is much smaller than the one reported by Thiel in its original work ($0.49$ eV) \cite{Sch08} but of the same order of magnitude as in the more recent study of Kannar and Szalay performed +on Thiel's set ($0.18$ eV for transitions with $\Td > 90\%$ ). \cite{Kan14} In retrospect, the much larger value obtained by Thiel is likely related to the use of {\CASPT} reference values in the 2008 work. Indeed, as we have shown in many of the proposed examples, {\CASPT} transitions energies tend to be significantly too low, therefore exacerbating {\CCSD}'s overestimation. The {\STEOM} approach, which received relatively less attention to date -- we are aware of one detailed benchmark \cite{Dut18} only -- provides a smaller MSE than {\CCSD} and comparable MAE and RMS. The spread of the error is however slightly larger as can be seen in Figure \ref{Fig-1} and from the SD values in Table \ref{Table-bench}. These trends are the same as for smaller compounds. \cite{Loo18a} For Thiel's set using {\CCT}/TZVP results as references, Dutta and coworkers also reported a rather good performance of {\STEOM}, though in that case a slightly negative MSE is obtained, \cite{Dut18} which could possibly be due to the different basis sets used. It should be nevertheless stressed that we consider here only ``clean'' {\STEOM} results (see Computational details), therefore removing several difficult cases that are included in the {\CCSD} statistics, \eg, the $A_g$ excitation in butadiene, which can slightly bias the relative accuracies when comparing the two models. Finally, for the three second-order methods, namely CIS(D), {\AD}, and {\CCD}, that are often used as reference to benchmark TD-DFT for ``real-life'' applications, we obtain clearly worse performances for the former approach than for the two latter, that show very -similar statistical behaviors. These trends were also reported in several previous works. \cite{Hat05c,Jac18a,Sch08,Sil10c,Win13,Har14,Jac15b,Kan17,Loo18a} Interestingly, the {\CCD} MAE obtained here, 0.15 eV, is significantly -smaller than the one we found for the smaller compounds (0.22 eV): \cite{Loo18a} in contrast to {\CCSD}, {\CCD} seems to improve with molecular size. As above, Thiel's original MAE for {\CCD} (0.29 eV) was likely too large due +similar statistical behaviors. These trends were also reported in several previous works. \cite{Hat05c,Jac18a,Sch08,Sil10c,Win13,Har14,Jac15b,Kan17,Loo18a} Interestingly, the {\CCD} MAE obtained here, $0.15$ eV, is significantly +smaller than the one we found for the smaller compounds ($0.22$ eV): \cite{Loo18a} in contrast to {\CCSD}, {\CCD} seems to improve with molecular size. As above, Thiel's original MAE for {\CCD} ($0.29$ eV) was likely too large due to the selection of {\CASPT} reference values. \cite{Sch08} As already noticed by Szalay's group, \cite{Kan14,Kan17} although the MSE of {\CCD} is smaller than the one of {\CCSD}, the standard deviation is significantly larger with the former model, \ie, {\CCD} is less consistent in terms of trends than {\CCSD}. In Table \ref{Table-bench2}, we report a decomposition of the MAE for different subsets of ES. Only singlet ES could be determined with both CCSDR(3) and CCSDT-3, which is why no value appears in the triplet column for these two methods. A few interesting conclusions emerge from the displayed data. First, the errors for the singlet and triplet transitions are rather similar with all models, but with {\CCSD} that -is significantly more effective for the triplets. Dutta and coworkers obtained the same conclusions for Thiel's set with MAE of 0.20 eV and 0.11 eV for the singlet and triplet ES, respectively. \cite{Dut18} +is significantly more effective for the triplets. Dutta and coworkers obtained the same conclusions for Thiel's set with MAE of $0.20$ eV and $0.11$ eV for the singlet and triplet ES, respectively. \cite{Dut18} When turning to the comparison between valence and Rydberg states, it is found that {\CCD} actually performs more effectively for the former, whereas {\CCSD} (and higher order methods) yields the opposite trend. -In fact {\CCD} has a tendency to overestimate the energies of the valence ES (MSE of +0.10 eV), but to underestimate their Rydberg counterparts (MSE of -0.17 eV), whereas {\CCSD} is much more consistent -with MSE of 0.12 and 0.09 eV, respectively (see the SI). This relatively poorer performance of {\CCD} as compared to {\CCSD} for Rydberg ES is again consistent with other benchmarks, \cite{Kan17,Dut18} although the MAE -for {\CCD} (0.18 eV) reported in Table \ref{Table-bench2} remains relatively small as compared to the one given in Ref.~\citenum{Kan17}. This is likely a side effect of the consideration of (relatively) low-lying +In fact {\CCD} has a tendency to overestimate the energies of the valence ES (MSE of $+0.10$ eV), but to underestimate their Rydberg counterparts (MSE of $-0.17$ eV), whereas {\CCSD} is much more consistent +with MSE of $0.12$ and $0.09$ eV, respectively (see the SI). This relatively poorer performance of {\CCD} as compared to {\CCSD} for Rydberg ES is again consistent with other benchmarks, \cite{Kan17,Dut18} although the MAE +for {\CCD} ($0.18$ eV) reported in Table \ref{Table-bench2} remains relatively small as compared to the one given in Ref.~\citenum{Kan17}. This is likely a side effect of the consideration of (relatively) low-lying Rydberg transitions in medium-sized molecules in the present work, whereas Kannar and Szalay (mostly) investigated higher-lying Rydberg in smaller compounds. Eventually, CIS(D), {\AD}, {\CCD}, and {\STEOM} better describe $n\ra\pis$ transitions, whereas {\CCSD} seems more suited for $\pi\ra\pis$ transitions; the variations between the two subsets being probably not very significant for the higher-order approaches. The former finding agrees with the results obtained for smaller compounds, \cite{Loo18a} as well for Thiel's set, \cite{Sch08,Kan14} whereas the latter, less expected conclusions, seems to be significantly