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add errors fig

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This commit is contained in:
Pierre-Francois Loos 2020-10-27 09:32:38 +01:00
parent afd1da050f
commit 3251dd129d
3 changed files with 276 additions and 261 deletions
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Data/QUESTDB.nb
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@ -45862,7 +45862,7 @@ Cell[BoxData[{
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93
Manuscript/QUEST_WIREs.tex
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@ -137,7 +137,7 @@ We refer the interested reader to Ref.~\cite{Loos_2019b} where we review the gen
%%% FIGURE 1 %%%
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{fig1/fig1}
\includegraphics[width=0.6\linewidth]{fig1/fig1}
\caption{Composition of each of the five subsets making up the present QUEST dataset of highly-accurate vertical excitation energies.}
\label{fig:scheme}
\end{figure}
@ -207,7 +207,7 @@ The reported oscillator strengths have been computed in the linear-response (LR)
For open-shell molecules, the CCSDT, CCSDTQ, and CCSDTQP calculations performed with MRCC \cite{Rolik_2013,mrcc} do consider an unrestricted Hartree-Fock wave function as reference.
All excited-state calculations are performed, except when explicitly mentioned, in the frozen-core (FC) approximation using large cores for the third-row atoms.
All the SCI calculations are performed within the FC approximation using QUANTUM PACKAGE \cite{Garniron_2019} where the CIPSI algorithm \cite{Huron_1973} is implemented. Details regarding this specific CIPSI implementation can be found in Refs.~\cite{Garniron_2019} and \cite{Scemama_2019}.
All the SCI calculations are performed within the frozen-core approximation using QUANTUM PACKAGE \cite{Garniron_2019} where the CIPSI algorithm \cite{Huron_1973} is implemented. Details regarding this specific CIPSI implementation can be found in Refs.~\cite{Garniron_2019} and \cite{Scemama_2019}.
A state-averaged formalism is employed, i.e., the ground and excited states are described with the same set of determinants, but different CI coefficients.
Our usual protocol \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} consists of performing a preliminary CIPSI calculation using Hartree-Fock orbitals in order to generate a CIPSI wave function with at least $10^7$ determinants.
Natural orbitals are then computed based on this wave function, and a new, larger CIPSI calculation is performed with this new set of orbitals.
@ -315,17 +315,25 @@ Only the last $M>2$ computed energy differences are considered. $M$ is chosen su
If all the values of $P(\mathcal{G})$ are below $0.8$, $M$ is chosen such that $P(\mathcal{G})$ is maximal.
A Python code associated with this procedure is provided in the {\SupInf}.
The singlet and triplet excitation energies obtained at the FCI/6-31+G(d) level are reported in Table \ref{tab:cycles} alongside the computed error bar estimated with the method presented above and the CC3 and CCSDT values from Ref.~\cite{Loos_2020b} computed in the same basis.
For the sake of comparison, we also report the estimated value of the excitation energies obtained via a three-point linear extrapolation considering the three largest SCI wave functions.
In such a case, the error bar is estimated via the difference in excitation energies obtained with the three-point linear extrapolation and the largest variational wave function.
This strategy has been considered in some of our previous works \cite{Loos_2020b,Loos_2020c}.
\alert{Here comes the discussion of the results.}
The singlet and triplet excitation energies obtained at the FCI/6-31+G(d) level are reported in Table \ref{tab:cycles} alongside the computed error bars estimated with the method presented above based on Gaussian random variables.
For the sake of comparison, we also report the CC3 and CCSDT vertical energies from Ref.~\cite{Loos_2020b} computed in the same basis.
The estimated values of the excitation energies obtained via a three-point linear extrapolation considering the three largest CIPSI wave functions are also gathered in Table \ref{tab:cycles}.
In this case, the error bar is estimated via the extrapolation distance, \ie, the difference in excitation energies obtained with the three-point linear extrapolation and the largest CIPSI wave function.
This strategy has been considered in some of our previous works \cite{Loos_2020b,Loos_2020c,Loos_2020e}.
The deviation from the CCSDT excitation energies for the same set of excitations are depicted in Fig.~\ref{fig:errors}, where the red dots correspond to the excitation energies and error bars estimated via the present method, and the blue dots correspond to the excitation energies obtained via a three-point linear fit using the three largest CIPSI wave functions, and error bars estimated via the extrapolation distance.
These results are a good balance between well-behaved and ill-behaved cases.
For example, cyclopentadiene and furan correspond to well-behaved cases where the two flavor of excitation energy estimates are nearly identical and the error bars associated with these two methods overlap nicely.
In these cases, one can observe that our method based on Gaussian random variables provides almost systematically smaller error bars.
Even in less idealistic situations (like in imidazole, pyrrole, and thiophene), the results are very satisfactory.
The six-membered rings correspond to much more challenging cases for SCI methods, and even for these systems the newly-developed method provides realistic error bars.
The present scheme has also been tested on much smaller systems when one can easily tightly converged the CIPSI calculations.
In these cases, the agreement is nearly perfect in every cases.
Some of these results can be found in the {\SupInf}.
%%% TABLE I %%%
\begin{table}
\centering
\caption{Singlet and triplet excitation energies obtained at the CC3, CCSDT, and FCI levels of theory with the 6-31+G* basis set for various five- and six-membered rings.
\caption{Singlet and triplet excitation energies (in eV) obtained at the CC3, CCSDT, and FCI levels of theory with the 6-31+G* basis set for various five- and six-membered rings.
The error bars reported in parenthesis correspond to one standard deviation.}
\label{tab:cycles}
\begin{threeparttable}
@ -333,39 +341,39 @@ The error bars reported in parenthesis correspond to one standard deviation.}
\headrow
\thead{Molecule} & \thead{Transition} & \thead{CC3} & \thead{CCSDT} & \thead{FCI$^a$} & \thead{FCI$^b$}\\
\mc{6}{c}{Five-membered rings} \\
Cyclopentadiene & $^1 B_2 (\pi \ra \pis)$ & 5.79 & 5.80 & 5.80(2) & 5.79(2) \\%& 5.79(7)
& $^3 B_2 (\pi \ra \pis)$ & 3.33 & 3.33 & 3.32(4) & 3.29(7) \\%& 3.29(1)
Furan & $^1A_2(\pi \ra 3s)$ & 6.26 & 6.28 & 6.31(5) & 6.37(1) \\%& 6.37(8)
& $^3B_2(\pi \ra \pis)$ & 4.28 & 4.28 & 4.26(4) & 4.22(7) \\%& 4.22(14)
Imidazole & $^1A''(\pi \ra 3s)$ & 5.77 & 5.77 & 5.78(5) & 5.96(14) \\%& 5.96(31)
& $^3A'(\pi \ra \pis)$ & 4.83 & 4.81 & 4.82(7) & 4.65(22) \\%& 4.65(35)
Pyrrole & $^1A_2(\pi \ra 3s)$ & 5.25 & 5.25 & 5.23(7) & 5.31(1) \\%& 5.31(26)
& $^3B_2(\pi \ra \pis)$ & 4.59 & 4.58 & 4.54(7) & 4.37(23) \\%& 4.37(35)
Thiophene & $^1A_1(\pi \ra \pis)$ & 5.79 & 5.77 & 5.75(8) & 5.73(9) \\%& 5.73(7)
& $^3B_2(\pi \ra \pis)$ & 3.95 & 3.94 & 3.98(1) & 3.99(2) \\%& 3.99(8)
Cyclopentadiene & $^1 B_2 (\pi \ra \pis)$ & 5.79 & 5.80 & 5.80(2) & 5.79(2) \\
& $^3 B_2 (\pi \ra \pis)$ & 3.33 & 3.33 & 3.32(4) & 3.29(7) \\
Furan & $^1A_2(\pi \ra 3s)$ & 6.26 & 6.28 & 6.31(5) & 6.37(1) \\
& $^3B_2(\pi \ra \pis)$ & 4.28 & 4.28 & 4.26(4) & 4.22(7) \\
Imidazole & $^1A''(\pi \ra 3s)$ & 5.77 & 5.77 & 5.78(5) & 5.96(14) \\
& $^3A'(\pi \ra \pis)$ & 4.83 & 4.81 & 4.82(7) & 4.65(22) \\
Pyrrole & $^1A_2(\pi \ra 3s)$ & 5.25 & 5.25 & 5.23(7) & 5.31(1) \\
& $^3B_2(\pi \ra \pis)$ & 4.59 & 4.58 & 4.54(7) & 4.37(23) \\
Thiophene & $^1A_1(\pi \ra \pis)$ & 5.79 & 5.77 & 5.75(8) & 5.73(9) \\
& $^3B_2(\pi \ra \pis)$ & 3.95 & 3.94 & 3.98(1) & 3.99(2) \\
\mc{6}{c}{Six-membered rings} \\
Benzene & $^1B_{2u}(\pi \ra \pis)$ & 5.13 & 5.10 & 5.06(9) & 5.21(7) \\%& 5.21(36)
& $^3B_{1u}(\pi \ra \pis)$ & 4.18 & 4.16 & 4.28(6) & 4.17(7) \\%& 4.17(67)
Cyclopentadienone & $^1A_2(n \ra \pis)$ & 3.03 & 3.03 & 3.08(2) & 3.13(3) \\%& 3.13(8)
& $^3B_2(\pi \ra \pis)$ & 2.30 & 2.32 & 2.37(5) & 2.10(25) \\%& 2.10(45)
Pyrazine & $^1B_{3u}(n \ra \pis)$ & 4.28 & 4.28 & 4.26(9) & 4.10(25) \\%& 4.10(8)
& $^3B_{3u}(n \ra \pis)$ & 3.68 & 3.68 & 3.70(3) & 3.70(1) \\%& 3.70(37)
Tetrazine & $^1B_{3u}(n \ra \pis)$ & 2.53 & 2.54 & 2.56(5) & 5.07(16) \\%& 5.07(77)
& $^3B_{3u}(n \ra \pis)$ & 1.87 & 1.88 & 1.91(3) & 4.04(49) \\%& 4.04(40)
Pyridazine & $^1B_1(n \ra \pis)$ & 3.95 & 3.95 & 3.97(10)& 3.60(43) \\%& 3.60(26)
& $^3B_1(n \ra \pis)$ & 3.27 & 3.26 & 3.27(15)& 3.46(14) \\%& 3.46(1.61)
Pyridine & $^1B_1(n \ra \pis)$ & 5.12 & 5.10 & 5.15(12)& 4.90(24) \\%& 4.90(1.34)
& $^3A_1(\pi \ra \pis)$ & 4.33 & 4.31 & 4.42(85)& 3.68(1.05) \\%& 3.68(0.65)
Pyrimidine & $^1B_1(n \ra \pis)$ & 4.58 & 4.57 & 4.64(11)& 2.54(5) \\%& 2.54(13)
& $^3B_1(n \ra \pis)$ & 4.20 & 4.20 & 4.55(37)& 2.18(27) \\%& 2.18(29)
Triazine & $^1A_1''(n \ra \pis)$ & 4.85 & 4.84 & 4.77(13)& 5.12(51) \\%& 5.12(13)
& $^3A_2''(n \ra \pis)$ & 4.40 & 4.40 & 4.45(39)& 4.73(6) \\%& 4.73(1.07)
Benzene & $^1B_{2u}(\pi \ra \pis)$ & 5.13 & 5.10 & 5.06(9) & 5.21(7) \\
& $^3B_{1u}(\pi \ra \pis)$ & 4.18 & 4.16 & 4.28(6) & 4.17(7) \\
Cyclopentadienone & $^1A_2(n \ra \pis)$ & 3.03 & 3.03 & 3.08(2) & 3.13(3) \\
& $^3B_2(\pi \ra \pis)$ & 2.30 & 2.32 & 2.37(5) & 2.10(25) \\
Pyrazine & $^1B_{3u}(n \ra \pis)$ & 4.28 & 4.28 & 4.26(9) & 4.10(25) \\
& $^3B_{3u}(n \ra \pis)$ & 3.68 & 3.68 & 3.70(3) & 3.70(1) \\
Tetrazine & $^1B_{3u}(n \ra \pis)$ & 2.53 & 2.54 & 2.56(5) & 5.07(16) \\
& $^3B_{3u}(n \ra \pis)$ & 1.87 & 1.88 & 1.91(3) & 4.04(49) \\
Pyridazine & $^1B_1(n \ra \pis)$ & 3.95 & 3.95 & 3.97(10)& 3.60(43) \\
& $^3B_1(n \ra \pis)$ & 3.27 & 3.26 & 3.27(15)& 3.46(14) \\
Pyridine & $^1B_1(n \ra \pis)$ & 5.12 & 5.10 & 5.15(12)& 4.90(24) \\
& $^3A_1(\pi \ra \pis)$ & 4.33 & 4.31 & 4.42(85)& 3.68(1.05) \\
Pyrimidine & $^1B_1(n \ra \pis)$ & 4.58 & 4.57 & 4.64(11)& 2.54(5) \\
& $^3B_1(n \ra \pis)$ & 4.20 & 4.20 & 4.55(37)& 2.18(27) \\
Triazine & $^1A_1''(n \ra \pis)$ & 4.85 & 4.84 & 4.77(13)& 5.12(51) \\
& $^3A_2''(n \ra \pis)$ & 4.40 & 4.40 & 4.45(39)& 4.73(6) \\
%\hiderowcolors
\hline % Please only put a hline at the end of the table
\end{tabular}
\begin{tablenotes}
\item $^a$ Excitation energies and error bars estimated via the present method (see Sec.~\ref{sec:error}).
\item $^b$ Excitation energies obtained via a three-point linear fit using the three largest variational wave functions, and error bars estimated via the extrapolation distance, \ie, the difference in excitation energies obtained with the three-point linear extrapolation and the largest variational wave function.
\item $^b$ Excitation energies obtained via a three-point linear fit using the three largest CIPSI variational wave functions, and error bars estimated via the extrapolation distance, \ie, the difference in excitation energies obtained with the three-point linear extrapolation and the largest CIPSI wave function.
\end{tablenotes}
\end{threeparttable}
\end{table}
@ -373,10 +381,10 @@ Triazine & $^1A_1''(n \ra \pis)$ & 4.85 & 4.84 & 4.77(13)& 5.12(51) \\%& 5.1
%%% FIGURE 2 %%%
\begin{figure}
\centering
\label{fig:errors}
\includegraphics[width=0.5\linewidth]{errors}
\caption{Deviation from the CCSDT excitation energies of singlet and triplet excitation energies of five- and six-membered rings obtained at the FCI/6-31+G* level of theory. Red dots: excitation energies and error bars estimated via the present method (see Sec.~\ref{sec:error}). Blue dots: excitation energies obtained via a three-point linear fit using the three largest variational wave functions, and error bars estimated via the extrapolation distance, \ie, the difference in excitation energies obtained with the three-point linear extrapolation and the largest variational wave function.
\includegraphics[width=0.6\linewidth]{errors}
\caption{Deviation from the CCSDT excitation energies for singlet and triplet excitation energies (in eV) of five- and six-membered rings obtained at the FCI/6-31+G* level of theory. Red dots: excitation energies and error bars estimated via the present method (see Sec.~\ref{sec:error}). Blue dots: excitation energies obtained via a three-point linear fit using the three largest CIPSI wave functions, and error bars estimated via the extrapolation distance, \ie, the difference in excitation energies obtained with the three-point linear extrapolation and the largest CIPSI wave function.
The error bars corresponds to one standard deviation.}
\label{fig:errors}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -405,7 +413,7 @@ Throughout the present article, we report several statistical indicators: the me
%=======================
The QUEST\#1 benchmark set \cite{Loos_2018a} consists of 110 vertical excitation energies (as well as oscillator strengths) from 18 molecules with sizes ranging from one to three non-hydrogen atoms (water, hydrogen sulfide, ammonia, hydrogen chloride, dinitrogen, carbon monoxide, acetylene, ethylene, formaldehyde, methanimine, thioformaldehyde, acetaldehyde, cyclopropene, diazomethane, formamide, ketene, nitrosomethane, and the smallest
streptocyanine). For this set, we provided two sets of TBEs: i) one obtained within the frozen-core approximation and the aug-cc-pVTZ basis set, and ii) another one including further corrections for basis set incompleteness and ``all electron'' effects.
For the former set, we systematically selected FCI/aug-cc-pVTZ values to define our TBEs except in very few cases.
For the former set, we systematically employed FCI/aug-cc-pVTZ values to define our TBEs except in very few cases.
For the latter set, both the ``all electron'' correlation and the basis set corrections were systematically obtained at the CC3 level of theory and with the d-aug-cc-pV5Z basis for the nine smallest molecules, and slightly more compact basis sets for the larger compounds.
Our TBE/aug-cc-pVTZ reference excitation energies were employed to benchmark a series of popular excited-state wave function methods partially or fully accounting for double and triple excitations, namely CIS(D), CC2, CCSD, STEOM-CCSD, CCSDR(3), CCSDT-3, CC3, ADC(2), and ADC(3).
Our main conclusions were that i) ADC(2) and CC2 show strong similarities in terms of accuracy, ii) STEOM-CCSD is, on average, as accurate as CCSD, the latter overestimating transition energies, iii) CC3 is extremely accurate (with a mean absolute error of only $\sim 0.03$ eV) and that although slightly less accurate than CC3, CCSDT-3 could be used as a reliable reference for benchmark studies, and iv) ADC(3) was found to be significantly less accurate than CC3 by overcorrecting ADC(2) excitation energies.
@ -452,6 +460,7 @@ Likewise, the excitation energies obtained with CCSD are much less satisfying fo
The QUEST\#5 subset is composed by additional accurate excitation energies that we have produced for the present article.
This new set gathers 13 new systems composed by small molecules as well as larger molecules (aza-naphthalene, benzoquinone, cyclopentadienone, cyclopentadienethione, diazirine, hexatriene, maleimide, naphthalene, nitroxyl, octatetraene, streptocyanine-C3, streptocyanine-C5, and thioacrolein).
For these new transitions, we report at least CCSDT/aug-cc-pVTZ vertical energies.
The interested reader will find in the {\SupInf} a detailed discussion for each of these molecules in which comparisons are made with literature data.
%\begin{table}[bt]
@ -584,7 +593,7 @@ The interested reader will find in the {\SupInf} a detailed discussion for each
\label{sec:TBE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We discuss in this section the generation of the TBEs obtained with the aug-cc-pVTZ basis as well as oscillator strengths for most transitions.
An exhaustive list of TBEs can be found in {\SupInf}.
An exhaustive list of TBEs can be found in the {\SupInf}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Benchmarks}
@ -609,7 +618,7 @@ All quantities are given in eV. ``Count'' refers to the number of transitions co
& \thead{SOS-ADC(2)[TM]} & \thead{SOS-CC2[TM]} & \thead{SCS-CC2[TM]} & \thead{SOS-ADC(2) [QC]} & \thead{ADC(2)} & \thead{ADC(3)} & \thead{ADC(2.5)} \\
Count & & 429 & 431 & 427 & 360 & 431 & 259 & 251 & 431 & 430 & 430 & 430 & 430 & 426 & 423 & 423 \\
Max(+) & & 1.06 & 0.63 & 0.80 & 0.59 & 0.80 & 0.43 & 0.26 & 0.19 & 0.87 & 0.84 & 0.76 & 0.73 & 0.64 & 0.60 & 0.24 \\
Min($-$) & & -0.69 & -0.71 & -0.38 & -0.56 & -0.25 & -0.07 & -0.07 & -0.09 & -0.29 & -0.24 & -0.92 & -0.46 & -0.76 & -0.79 & -0.34 \\
Max($-$) & & -0.69 & -0.71 & -0.38 & -0.56 & -0.25 & -0.07 & -0.07 & -0.09 & -0.29 & -0.24 & -0.92 & -0.46 & -0.76 & -0.79 & -0.34 \\
MSE & & 0.13 & 0.02 & 0.18 & -0.01 & 0.10 & 0.04 & 0.04 & 0.00 & 0.18 & 0.21 & 0.15 & 0.02 & -0.01 & -0.12 & -0.06 \\
& singlet & 0.10 & -0.02 & 0.22 & 0.03 & 0.14 & 0.04 & 0.04 & 0.00 & 0.18 & 0.20 & 0.13 & 0.00 & -0.04 & -0.08 & -0.06 \\
& triplet & 0.19 & 0.08 & 0.14 & -0.07 & 0.03 & & & 0.00 & 0.19 & 0.22 & 0.17 & 0.04 & 0.04 & -0.18 & -0.07 \\

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