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Manuscript/Ex-srDFT.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-05-29 23:30:47 +0200
%% Created for Pierre-Francois Loos at 2019-05-30 11:26:43 +0200
%% Saved with string encoding Unicode (UTF-8)
@ -12712,9 +12712,9 @@
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b00591}}
@article{LooBogSceCafJAc-JCTC-19,
@article{LooBogSceCafJac-JCTC-19,
Author = {Loos, Pierre-Fran{\c c}ois and Boggio-Pasqua, Martial and Scemama, Anthony and Caffarel, Michel and Jacquemin, Denis},
Date-Modified = {2019-04-07 14:02:34 +0200},
Date-Modified = {2019-05-30 11:26:43 +0200},
Doi = {10.1021/acs.jctc.8b01205},
Journal = {J. Chem. Theory Comput.},
Number = {3},

242
Manuscript/Ex-srDFT.tex
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@ -333,11 +333,13 @@ This computationally-lighter functional will be refered to as PBE.
In the present study, we compute the ground- and excited-state energies, one-electron and on-top densities with a selected CI method known as CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively). \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
The total energy of each state is obtained via an efficient extrapolation procedure of the sCI energies designed to reach near-FCI accuracy. \cite{QP2}
These energies will be labeled exFCI in the following.
Using near-FCI excitation energies (within a given basis set) has the indisputable advantage to remove the error inherent to the WFT method.
Indeed, in the present case, the only source of error on the excitation energies is due to basis set incompleteness.
We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details.
The one-electron and on-top densities are computed from a very large CIPSI expansion containing several million determinants.
All the RS-DFT and exFCI calculations have been performed with {\QP}. \cite{QP2}
For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17}
Except for methylene for which FCI/TZVP geometries have been taken from Ref.~\onlinecite{SheLeiVanSch-JCP-98}, the other geometries have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJAc-JCTC-19} and have been obtained at the CC3/aug-cc-pVTZ level of theory.
Except for methylene for which FCI/TZVP geometries have been taken from Ref.~\onlinecite{SheLeiVanSch-JCP-98}, the other geometries have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJac-JCTC-19} and have been obtained at the CC3/aug-cc-pVTZ level of theory.
For the sake of completeness, they are also reported in the {\SI}.
Frozen-core calculations are systematically performed and defined as such: a \ce{He} core is frozen from \ce{Li} to \ce{Ne}, while a \ce{Ne} core is frozen from \ce{Na} to \ce{Ar}.
The frozen-core density-based correction is used consistently with the frozen-core approximation in WFT methods.
@ -365,12 +367,18 @@ We have also computed these adiabatic energies at the exFCI/AV5Z level and used
\end{equation}
These results are illustrated in Fig.~\ref{fig:CH2} and reported in Table \ref{tab:CH2} alongside reference values from the literature obtained with various approaches. \cite{ChiHolAdaOttUmrShaZim-JPCA-18, SheLeiVanSch-JCP-98, JenBun-JCP-88, SheLeiVanSch-JCP-98, ZimTouZhaMusUmr-JCP-09}
Figure \ref{fig:CH2} clearly shows that, for the double-$\zeta$ basis, the exFCI adiabatic energies are far from being chemically accurate with errors as high as 0.015 eV.
From triplet-$\zeta$ onward, the exFCI excitation energies are chemically-accurate though.
%%% TABLE 1 %%%
\begin{turnpage}
\begin{squeezetable}
\begin{table*}
\caption{
Total energies $E$ (in hartree) and adiabatic transition energies $\Ead$ (in eV) of excited states of methylene for various methods and basis sets.}
Total energies $E$ (in hartree) and adiabatic transition energies $\Ead$ (in eV) of excited states of methylene for various methods and basis sets.
The value in parenthesis is an estimate on the last digit of the extrapolation error.
The relative difference with respect to the exFCI/CBS result is reported in square brackets.}
\label{tab:CH2}
\begin{ruledtabular}
\begin{tabular}{llddddddd}
@ -386,64 +394,64 @@ These results are illustrated in Fig.~\ref{fig:CH2} and reported in Table \ref{t
& \tabc{$E$ (a.u.)} & \tabc{$\Ead$ (eV)} \\
\hline
exFCI & AVDZ & -39.04846(1)
& -39.03225(1) & 0.441
& -38.99203(1) & 1.536
& -38.95076(1) & 2.659 \\
& -39.03225(1) & 0.441 [+0.053]
& -38.99203(1) & 1.536 [+0.146]
& -38.95076(1) & 2.659 [+0.154] \\
& AVTZ & -39.08064(3)
& -39.06565(2) & 0.408
& -39.02833(1) & 1.423
& -38.98709(1) & 2.546 \\
& -39.06565(2) & 0.408 [+0.020]
& -39.02833(1) & 1.423 [+0.034]
& -38.98709(1) & 2.546 [+0.042] \\
& AVQZ & -39.08854(1)
& -39.07402(2) & 0.395
& -39.03711(1) & 1.399
& -38.99607(1) & 2.516 \\
& -39.07402(2) & 0.395 [+0.007]
& -39.03711(1) & 1.399 [+0.010]
& -38.99607(1) & 2.516 [+0.012] \\
& AV5Z & -39.09079(1)
& -39.07647(1) & 0.390
& -39.03964(3) & 1.392
& -38.99867(1) & 2.507 \\
& CBS & -39.09111
& -39.07682 & 0.388
& -39.04000 & 1.390
& -38.99904 & 2.504 \\
\\
exFCI+LDA & AVDZ & -39.07450(1)
& -39.06213(1) & 0.337
& -39.02233(1) & 1.420
& -38.97946(1) & 2.586 \\
& AVTZ & -39.09099(3)
& -39.07779(2) & 0.359
& -39.04051(1) & 1.374
& -38.99859(1) & 2.514 \\
& AVQZ & -39.09319(1)
& -39.07959(2) & 0.370
& -39.04267(1) & 1.375
& -39.00135(1) & 2.499 \\
\\
exFCI+PBE & AVDZ & -39.07282(1)
& -39.06150(1) & 0.308
& -39.02181(1) & 1.388
& -38.97873(1) & 2.560 \\
& AVTZ & -39.08948(3)
& -39.07639(2) & 0.356
& -39.03911(1) & 1.371
& -38.99724(1) & 2.510 \\
& AVQZ & -39.09247(1)
& -39.07885(2) & 0.371
& -39.04193(1) & 1.375
& -39.00066(1) & 2.498 \\
& -39.07647(1) & 0.390 [+0.001]
& -39.03964(3) & 1.392 [+0.002]
& -38.99867(1) & 2.507 [+0.003] \\
& CBS & -39.09141
& -39.07715 & 0.388
& -39.04034 & 1.390
& -38.99939 & 2.504 \\
\\
exFCI+PBEot & AVDZ & -39.06924(1)
& -39.05651(1) & 0.347
& -39.01777(1) & 1.401
& -38.97698(1) & 2.511 \\
& -39.05651(1) & 0.347 [-0.042]
& -39.01777(1) & 1.401 [+0.011]
& -38.97698(1) & 2.511 [+0.007] \\
& AVTZ & -39.08805(3)
& -39.07430(2) & 0.374
& -39.03742(1) & 1.378
& -38.99652(1) & 2.491 \\
& -39.07430(2) & 0.374 [-0.014]
& -39.03742(1) & 1.378 [-0.012]
& -38.99652(1) & 2.491 [-0.013] \\
& AVQZ & -39.09189(1)
& -39.07795(2) & 0.379
& -39.04124(1) & 1.378
& -39.00044(1) & 2.489 \\
& -39.07795(2) & 0.379 [-0.009]
& -39.04124(1) & 1.378 [-0.011]
& -39.00044(1) & 2.489 [-0.016] \\
\\
exFCI+PBE & AVDZ & -39.07282(1)
& -39.06150(1) & 0.308 [-0.080]
& -39.02181(1) & 1.388 [-0.002]
& -38.97873(1) & 2.560 [+0.056] \\
& AVTZ & -39.08948(3)
& -39.07639(2) & 0.356 [-0.032]
& -39.03911(1) & 1.371 [-0.019]
& -38.99724(1) & 2.510 [+0.006] \\
& AVQZ & -39.09247(1)
& -39.07885(2) & 0.371 [-0.017]
& -39.04193(1) & 1.375 [-0.015]
& -39.00066(1) & 2.498 [-0.006] \\
\\
exFCI+LDA & AVDZ & -39.07450(1)
& -39.06213(1) & 0.337 [-0.051]
& -39.02233(1) & 1.420 [+0.030]
& -38.97946(1) & 2.586 [+0.082] \\
& AVTZ & -39.09099(3)
& -39.07779(2) & 0.359 [-0.029]
& -39.04051(1) & 1.374 [-0.016]
& -38.99859(1) & 2.514 [+0.010] \\
& AVQZ & -39.09319(1)
& -39.07959(2) & 0.370 [-0.018]
& -39.04267(1) & 1.375 [-0.015]
& -39.00135(1) & 2.499 [-0.005] \\
\\
SHCI\fnm[1] & AVQZ & -39.08849(1)
& -39.07404(1) & 0.393
@ -473,13 +481,14 @@ These results are illustrated in Fig.~\ref{fig:CH2} and reported in Table \ref{t
\fnt[5]{References \onlinecite{SheLeiVanSch-JCP-98, JenBun-JCP-88}.}
\end{table*}
\end{squeezetable}
\end{turnpage}
%%% %%% %%%
%%% FIG 1 %%%
\begin{figure}
\includegraphics[width=\linewidth]{CH2}
\caption{Error in adiabatic excitation energies $\Ead$ (in eV) of methylene for various basis sets and methods.
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal}).
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
See Table \ref{tab:CH2} for raw data.}
\label{fig:CH2}
\end{figure}
@ -497,7 +506,7 @@ Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac
\begin{table*}
\caption{
Vertical absorption energies $\Eabs$ (in eV) of excited states of ammonia, carbon dimer, water and ethylene for various methods and basis sets.
The TBEs have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJAc-JCTC-19} on the same geometries.
The TBEs have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJac-JCTC-19} on the same geometries.
See the {\SI} for raw data.}
\begin{ruledtabular}{}
\begin{tabular}{lllddddddddddddd}
@ -553,33 +562,6 @@ Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac
& 0.11 & 0.02 & 0.00
\\
\\
% Hydrogen chloride& ${}^1\Sigma \ra {}^1\Pi$ & CT\fnm[2] & 7.86 & -0.04 & -0.02 & 0.02
% & 0.13 & 0.06 & 0.06
% & 0.11 & 0.04 & 0.05
% & 0.10 & 0.05 & 0.06
% \\
% \\
% Hydrogen sulfide & $1\,^{1}A_1 \ra 1\,^{1}A_2$ & Ryd. & 6.10 & 0.00 & 0.08 & 0.05
% & 0.15 & 0.12 & 0.07
% & 0.14 & 0.11 & 0.07
% & 0.14 & 0.11 & 0.07
% \\
% & $1\,^{1}A_1 \ra 1\,^{1}B_1$ & Ryd. & 6.29 & 0.00 & -0.05 & 0.00
% & -0.12 & 0.01 & 0.03
% & -0.14 & 0.00 & 0.03
% & -0.14 & 0.01 & 0.03
% \\
% & $1\,^{1}A_1 \ra 1\,^{3}A_2$ & Ryd. & 5.74 & 0.01 & 0.07 & 0.05
% & 0.18 & 0.12 & 0.08
% & 0.20 & 0.13 & 0.08
% & 0.19 & 0.13 & 0.08
% \\
% & $1\,^{1}A_1 \ra 1\,^{3}B_1$ & Ryd. & 5.94 & -0.04 & -0.05 & -0.01
% & 0.07 & 0.02 & 0.03
% & 0.09 & 0.03 & 0.03
% & 0.07 & 0.04 & 0.04
% \\
% \\
Water & $1\,^{1}A_1 \ra 1\,^{1}B_1$ & Ryd. & 7.70 & -0.17 & -0.07 & -0.02
& 0.01 & 0.00 & 0.02
& -0.02 & -0.01 & 0.00
@ -611,32 +593,6 @@ Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac
& 0.06 & 0.03 & 0.04
\\
\\
% Acetylene & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Sigma_{u}^{-}$ & Val. & 7.10 & 0.10 & 0.00
% & 0.07 & 0.00
% & 0.11 & 0.00
% & 0.11 & 0.00
% \\
% & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Delta_{u}$ & Val. & 7.44 & 0.07 & 0.00
% & 0.04 & -0.01
% & 0.12 & 0.02
% & 0.11 & 0.02
% \\
% & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{+}$ & Val. & 5.56 & -0.06 & -0.03
% & 0.07 & 0.02
% & 0.04 & 0.00
% & 0.02 & 0.00
% \\
% & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Delta_{u}$ & Val. & 6.40 & 0.06 & 0.00
% & 0.10 & 0.02
% & 0.14 & 0.03
% & 0.12 & 0.03
% \\
% & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{-}$ & Val. & 7.09 & 0.05 & -0.01
% & 0.08 & 0.00
% & 0.16 & 0.04
% & 0.14 & 0.04
% \\
% \\
Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & Ryd. & 7.43 & -0.12 & -0.04 &
& -0.05 & -0.01 &
& -0.04 & -0.01 &
@ -668,55 +624,9 @@ Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac
& 0.05 & 0.04 &
\\
\\
% Formaldehyde& $1\,^{1}A_{1} \ra 1\,^{1}A_{2}$ & Val. & 3.97 & 0.02 & 0.01 &
% & 0.05 & 0.02 &
% & 0.03 & 0.02 &
% & 0.02 & 0.01 &
% \\
% & $1\,^{1}A_{1} \ra 1\,^{1}B_{2}$ & Ryd. & 7.30 & -0.19 & -0.07 &
% & 0.00 & 0.00 &
% & -0.02 & 0.00 &
% & -0.04 & 0.00 &
% \\
% & $1\,^{1}A_{1} \ra 2\,^{1}B_{2}$ & Ryd. & 8.14 & -0.10 & -0.01 &
% & 0.09 & 0.07 &
% & 0.08 & 0.06 &
% & 0.05 & 0.06 &
% \\
% & $1\,^{1}A_{1} \ra 2\,^{1}A_{1}$ & Ryd. & 8.27 & -0.15 & -0.04 &
% & 0.03 & 0.04 &
% & 0.02 & 0.03 &
% & 0.00 & 0.03 &
% \\
% & $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & Val. & 3.58 & 0.00 & 0.00 &
% & 0.09 & 0.05 &
% & 0.11 & 0.06 &
% & 0.07 & 0.04 &
% \\
% & $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & Val. & 6.07 & 0.03 & 0.01 &
% & 0.13 & 0.04 &
% & 0.15 & 0.05 &
% & 0.11 & 0.04 &
% \\
% & $1\,^{1}A_{1} \ra 1\,^{3}B_{2}$ & Ryd. & 7.14 & -0.19 & -0.08 &
% & 0.01 & 0.01 &
% & 0.02 & 0.01 &
% & -0.01 & 0.00 &
% \\
% & $1\,^{1}A_{1} \ra 2\,^{3}B_{2}$ & Ryd. & 7.96 & -0.09 & -0.02 &
% & 0.13 & 0.08 &
% & 0.14 & 0.08 &
% & 0.10 & 0.07 &
% \\
% & $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & Ryd. & 8.15 & -0.14 & -0.05 &
% & 0.07 & 0.05 &
% & 0.07 & 0.04 &
% & 0.04 & 0.04 &
% \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Doubly-excited states of $(\pi,\pi) \ra (\si,\si)$ character.}
% \fnt[2]{CT stands for charge transfer.}
\end{table*}
\end{squeezetable}
%%% %%% %%%
@ -725,7 +635,7 @@ Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac
\begin{figure}
\includegraphics[width=\linewidth]{H2O}
\caption{Error in vertical excitation energies (in eV) of water for various basis sets and methods.
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal}).
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
See the {\SI} for raw data.}
\label{fig:H2O}
\end{figure}
@ -735,7 +645,7 @@ Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac
\begin{figure}
\includegraphics[width=\linewidth]{NH3}
\caption{Error in vertical excitation energies (in eV) of ammonia for various basis sets and methods.
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal}).
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
See the {\SI} for raw data.}
\label{fig:NH3}
\end{figure}
@ -746,13 +656,13 @@ Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac
\label{sec:C2}
%=======================
It is also interesting to study doubly-excited states. \cite{AbrShe-JCP-04, AbrShe-CPL-05, Var-JCP-08, PurZhaKra-JCP-09, AngCimPas-MP-12, BooCleThoAla-JCP-11, Sha-JCP-15, SokCha-JCP-16, VarRoc-PTRSMPES-18}
In the carbon dimer, these valence states are of $(\pi,\pi) \ra (\si,\si)$ character and they have been recently studied with state-of-the-art methods. \cite{LooBogSceCafJAc-JCTC-19}
In the carbon dimer, these valence states are of $(\pi,\pi) \ra (\si,\si)$ character and they have been recently studied with state-of-the-art methods. \cite{LooBogSceCafJac-JCTC-19}
%%% FIG 4 %%%
\begin{figure}
\includegraphics[width=\linewidth]{C2}
\caption{Error in vertical excitation energies $\Eabs$ (in eV) for two doubly-excited states of the carbon dimer for various basis sets and methods.
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal}).
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
See the {\SI} for raw data.}
\label{fig:C2}
\end{figure}
@ -766,26 +676,14 @@ In the carbon dimer, these valence states are of $(\pi,\pi) \ra (\si,\si)$ chara
Ethylene is an interesting molecules as it contains both valence and Rydberg excited states. \cite{SerMarNebLinRoo-JCP-93, WatGwaBar-JCP-96, WibOliTru-JPCA-02, BarPaiLis-JCP-04, Ang-JCC-08, SchSilSauThi-JCP-08, SilSchSauThi-JCP-10, SilSauSchThi-MP-10, Ang-IJQC-10, DadSmaBooAlaFil-JCTC-12, FelPetDav-JCP-14, ChiHolAdaOttUmrShaZim-JPCA-18}
%\begin{figure}
% \includegraphics[width=\linewidth]{C2H2}
% \caption{Error in vertical excitation energies (in eV) of acetylene for various basis sets and methods.}
% \label{fig:C2H2}
%\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{C2H4}
\caption{Error in vertical excitation energies $\Eabs$ (in eV) of ethylene for various basis sets and methods.
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal}).
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
See the {\SI} for raw data.}
\label{fig:C2H4}
\end{figure}
%\begin{figure}
% \includegraphics[width=\linewidth]{CH2O}
% \caption{Error in vertical excitation energies $\Eabs$ (in eV) of formaldehyde for various basis sets and methods.}
% \label{fig:CH2O}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}
@ -796,7 +694,7 @@ We are currently investigating the performance of the present basis set correcti
%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting Information Available}
%%%%%%%%%%%%%%%%%%%%%%%%
See {\SI} for geometries and additional information (including total energies).
See {\SI} for geometries and additional information (including energetic correction of the various functionals).
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}