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Manuscript/FCI2.tex
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@ -1106,17 +1106,17 @@ $^h${CC3-ext.~from Ref.~\citenum{Sil10c}.}
\section{Theoretical Best Estimates}
Table \ref{Table-tbe} reports our two sets of TBE: a set obtained with the {\AVTZ} basis set and one set with 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. For ES with $\Td$ between $70\%$ and $80\%$,
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. 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, but 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 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 consider such values as reference because
the addition of a second set of diffuse functions only significantly modifies the transition energies for strongly-mixing ES.
the addition of a second set of diffuse functions only significantly modifies the transition energies associated with strongly-mixing ES.
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 cases.
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. Amongst these transition energies, fifteen can be considered as ``unsafe'' and are reported in italics accordingly. This definitely corresponds to a very significant extension of our previous
@ -1132,7 +1132,7 @@ at the {\CCT}/TZVP level for Thiel's set. \cite{Kan14} It should also be pointed
\begin{longtable}{p{3.3cm}lcccccc}
\caption{\small TBE values (in eV) for all considered states alongside their corresponding oscillator strength, $f$, and percentage of single excitations, $\Td$, obtained at the \CCT/{\AVTZ} level.
The composite protocol to generate these TBE is also reported (see footnotes). In the right-most column, we list the TBE values obtained by including an additional correction (obtained at the {\CCT} level) for basis set incompleteness error.
Values displayed in italics are likely to be relatively less accurate. All values are obtained in the FC approximation.} \label{Table-tbe}\\
Values displayed in italics are likely to be relatively less accurate. All values are obtained in the frozen-core approximation.} \label{Table-tbe}\\
\hline
& & & & \mc{2}{l}{TBE/{\AVTZ}} & \mc{2}{l}{TBE/CBS} \\
& State & $f$ & $\Td$ & Value & Protocol$^a$ & Value & Corr. \\
@ -1440,7 +1440,7 @@ CCSDT-3 &126 &0.05 &0.05 &0.07 &0.04 &0.26 &0.00\\
\label{Fig-1}
\end{figure}
Let us analyze the global performances of all methods, starting with the most accurate and computationally demanding models. The relative accuracies of {\CCT} and {\CCSDT}-3 as compared to {\CCSDT} remains an open question in the literature. \cite{Wat13,Dem13}
Let us analyze the global performances of all these methods, starting with the most accurate and computationally demanding 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 the best of our knowledge, the only two previous studies discussing this specific aspect are limited to very small compounds. \cite{Kan17,Loo18a} According to the results gathered in Table \ref{Table-bench}, it appears that {\CCT} has a slight edge over {\CCSDT}-3, although
{\CCSDT}-3 is closer to {\CCSDT} in \titou{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 on smaller derivatives,
\cite{Kan17,Loo18a} but the amplitude of the {\CCSDT}-3 errors is larger in 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
@ -1450,10 +1450,10 @@ To state it more boldly: it appears likely that {\CCT} is even more accurate tha
these methods are combined with large active spaces. This statement seems to hold 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 in CCSDR(3) yields a very small MAE ($0.05$ eV) for a much lighter 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 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}
{\CCSD} provides an interesting case. The calculated MSE ($+0.11$ eV), indicating an overestimation of the transition energies, fits well with several previous recent reports.
{\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}
It is, nonetheless, larger than the one determined for smaller molecules ($+0.05$ eV), \cite{Loo18a} hinting that the performances of {\CCSD} deteriorates for larger compounds.
Moreover, {\CCSD} MAE of $0.13$ 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 K\'ann\'ar and Szalay performed
@ -1463,10 +1463,10 @@ Indeed, as we have shown several times in the present study, {\CASPT} transition
With a single detailed benchmark study to date, \cite{Dut18} the {\STEOM} approach has received relatively little attention and its overall accuracy still needs to be corroborated.
It is noteworthy that {\STEOM} provides a smaller MSE than {\CCSD} and comparable MAE and RMSE. The spread of the error is however slightly larger as evidenced by Figure \ref{Fig-1} and the SDE values
reported in Table \ref{Table-bench}. These trends are the same as for smaller compounds. \cite{Loo18a} For Thiel's set, Dutta and coworkers also reported rather good performance
for {\STEOM} with respect to the {\CCT}/TZVP reference data, though a slightly negative MSE is obtained in their case. \cite{Dut18} This could well be due to the different basis set considered by these two studies. It should be nevertheless stressed that we, here, only consider only ``clean'' {\STEOM} results
for {\STEOM} with respect to the {\CCT}/TZVP reference data, though a slightly negative MSE is obtained in their case. \cite{Dut18} This could well be due to the different basis set considered in these two studies. It should be nevertheless stressed that we, here, only consider ``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 performance of {\STEOM} and {\CCSD}.
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, the performance of the former method clearly deteriorates compared to the two latter which exhibit very
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, the performance of the former method clearly deteriorates compared to the latter two which exhibit very
similar statistical behaviors. These trends were also reported in 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 smaller compounds ($0.22$ eV). \cite{Loo18a} Therefore, in contrast to {\CCSD}, {\CCD} performance 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
@ -1475,7 +1475,7 @@ with the former model, \ie, {\CCD} is less consistent in terms of trends than {\
In Table \ref{Table-bench2}, we report a MAE decomposition for different subsets of ES. Note that, due to implementational limitations, only singlet ES could be computed with CCSDR(3) and CCSDT-3 which explains the lack of data for triplet ES.
A few interesting conclusions emerge from these results. First, the errors for singlet and triplet transitions are rather similar with all models, except for {\CCSD} that
is significantly more effective for triplets. Dutta and coworkers observed the same trend for Thiel's set with MAE of $0.20$ eV and $0.11$ eV for the singlet and triplet ES, respectively. \cite{Dut18}
Turning to the comparison between valence and Rydberg states, we find that {\CCD} provide a better description of the former, whereas {\CCSD} (and higher-order methods) yields the opposite trend.
Turning to the comparison between valence and Rydberg states, we find that {\CCD} provides a better description of the former, whereas {\CCSD} (and higher-order methods) yields the opposite trend.
In fact, {\CCD} has the clear tendency to overestimate valence ES energies (MSE of $+0.10$ eV), and to underestimate Rydberg ES energies (MSE of $-0.17$ eV).
{\CCSD} is found to be 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 perfectly coherent 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}. We believe that it is likely due to the distinct types of Rydberg states considered in these two studies. Indeed, we consider here (relatively) low-lying