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Manuscript/Ex-srDFT.bib
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%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-05-30 22:48:58 +0200 %% Created for Pierre-Francois Loos at 2019-05-31 09:36:15 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@ -16,7 +16,8 @@
Journal = {Res. Chem.}, Journal = {Res. Chem.},
Pages = {in press}, Pages = {in press},
Title = {Influence of Pseudopotentials on Excitation Energies From Selected Configuration Interaction and Diffusion Monte Carlo}, Title = {Influence of Pseudopotentials on Excitation Energies From Selected Configuration Interaction and Diffusion Monte Carlo},
Year = {2019}} Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1016/j.rechem.2019.100002}}
@article{SheLeiVanSch-JCP-98, @article{SheLeiVanSch-JCP-98,
Author = {C. D. Sherrill and M. L. Leininger and T. J. Van Huis and H. F. Schaefer}, Author = {C. D. Sherrill and M. L. Leininger and T. J. Van Huis and H. F. Schaefer},
@ -523,12 +524,12 @@
@article{LooPraSceTouGin-JPCL-19, @article{LooPraSceTouGin-JPCL-19,
Author = {P. F. Loos and B. Pradines and A. Scemama and J. Toulouse and E. Giner}, Author = {P. F. Loos and B. Pradines and A. Scemama and J. Toulouse and E. Giner},
Date-Added = {2019-05-19 21:51:45 +0200}, Date-Added = {2019-05-19 21:51:45 +0200},
Date-Modified = {2019-05-19 21:54:20 +0200}, Date-Modified = {2019-05-31 09:36:13 +0200},
Doi = {10.1021/acs.jpclett.9b01176}, Doi = {10.1021/acs.jpclett.9b01176},
Journal = {J. Phys. Chem. Lett.}, Journal = {J. Phys. Chem. Lett.},
Number = {xxxx}, Pages = {2931--2937},
Title = {A Density-Based Basis-Set Correction for Wave Function Theory}, Title = {A Density-Based Basis-Set Correction for Wave Function Theory},
Volume = {xx}, Volume = {10},
Year = {2019}, Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.9b01176}} Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.9b01176}}

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Manuscript/Ex-srDFT.tex
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\newcommand{\alert}[1]{\textcolor{red}{#1}} \newcommand{\alert}[1]{\textcolor{red}{#1}}
\definecolor{darkgreen}{HTML}{009900} \definecolor{darkgreen}{HTML}{009900}
\usepackage[normalem]{ulem} \usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{black}{#1}} \newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\jt}[1]{\textcolor{purple}{#1}} \newcommand{\jt}[1]{\textcolor{purple}{#1}}
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@ -360,122 +360,118 @@ In the following, we employ the AVXZ shorthand notations for Dunning's aug-cc-pV
Methylene is a paradigmatic system in electronic structure theory. \cite{Sch-Science-86} Methylene is a paradigmatic system in electronic structure theory. \cite{Sch-Science-86}
Due to its relative small size, its ground and excited states have been thoroughly studied with high-level ab initio methods. \cite{Sch-Science-86, BauTay-JCP-86, JenBun-JCP-88, SheVanYamSch-JMS-97, SheLeiVanSch-JCP-98, AbrShe-JCP-04, AbrShe-CPL-05, ZimTouZhaMusUmr-JCP-09, ChiHolAdaOttUmrShaZim-JPCA-18} Due to its relative small size, its ground and excited states have been thoroughly studied with high-level ab initio methods. \cite{Sch-Science-86, BauTay-JCP-86, JenBun-JCP-88, SheVanYamSch-JMS-97, SheLeiVanSch-JCP-98, AbrShe-JCP-04, AbrShe-CPL-05, ZimTouZhaMusUmr-JCP-09, ChiHolAdaOttUmrShaZim-JPCA-18}
As a first test of the present density-based basis set correction, we consider the four lowest-lying states of methylene ($1\,^{3}B_1$, $1\,^{1}A_1$, $1\,^{1}B_1$ and $2\,^{1}A_1$) and compute the corresponding adiabatic transition energies for various basis sets ranging from AVDZ to AVQZ. As a first test of the present density-based basis set correction, we consider the four lowest-lying states of methylene ($1\,^{3}B_1$, $1\,^{1}A_1$, $1\,^{1}B_1$ and $2\,^{1}A_1$) at their respective equilibrium geometry and compute the corresponding adiabatic transition energies for various basis sets ranging from AVDZ to AVQZ.
We have also computed these adiabatic energies at the exFCI/AV5Z level and used these alongside the quadruple-$\zeta$ ones to extrapolate the excitation energies to the CBS limit with the usual extrapolation formula \cite{HelJorOls-BOOK-02} We have also computed total energies at the exFCI/AV5Z level and used these alongside the quadruple-$\zeta$ ones to extrapolate the total energies to the CBS limit with the usual extrapolation formula \cite{HelJorOls-BOOK-02}
\begin{equation} \begin{equation}
\E{}{\text{AVXZ}} = \E{}{\CBS} + \frac{\alpha}{(\tX+1/2)^{3}}. \E{}{\text{AVXZ}} = \E{}{\CBS} + \frac{\alpha}{(\tX+1/2)^{3}}.
\end{equation} \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} 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 deterministic and stochastic approaches. \cite{ChiHolAdaOttUmrShaZim-JPCA-18, SheLeiVanSch-JCP-98, JenBun-JCP-88, SheLeiVanSch-JCP-98, ZimTouZhaMusUmr-JCP-09}
Total energies for each state can be found in the {\SI}.
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. 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 the triple-$\zeta$ basis onward, the exFCI excitation energies are chemically-accurate though, and drop steadily to the CBS limit when one increases the size of the basis set. From the triple-$\zeta$ basis onward, the exFCI excitation energies are chemically-accurate though, and drop steadily to the CBS limit when one increases the size of the basis set.
Concerning the basis set correction, already at the double-$\zeta$ level, the PBEot correction returns chemically accurate excitation energies. Concerning the basis set correction, already at the double-$\zeta$ level, the PBEot correction returns chemically accurate excitation energies.
The performance of the PBE and LDA functionals (which does not require the computation of the on-top density associated with each state) is less impressive. The performance of the PBE and LDA functionals (which does not require the computation of the on-top density of each state) is less impressive.
Yet, they still yield significant reductions of the basis set incompleteness error. Yet, they still yield significant reductions of the basis set incompleteness error, hence representing a good compromise between computational cost and accuracy.
Note that the results for the PBE functional are not represented in Fig.~\ref{fig:CH2} as they are very similar to the LDA ones (similar considerations apply to the other systems studied here). Note that the results for the PBE functional are not represented in Fig.~\ref{fig:CH2} as they are very similar to the LDA ones (similar considerations apply to the other systems studied below).
It is also quite evident that the basis set correction has the tendency of over-correcting the excitation energies via an over-stabilization of the excited states compared to the ground state. It is also quite evident that, the basis set correction has the tendency of over-correcting the excitation energies via an over-stabilization of the excited states compared to the ground state.
This trend is quite systematic as we shall see below.
%%% TABLE 1 %%% %%% TABLE 1 %%%
\begin{turnpage}
\begin{squeezetable} \begin{squeezetable}
\begin{table*} \begin{table*}
\caption{ \caption{
Total energies $E$ (in hartree) and adiabatic transition energies $\Ead$ (in eV) of excited states of methylene for various methods and basis sets. 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.
The relative difference with respect to the exFCI/CBS result is reported in square brackets.} See the {\SI} for raw data.}
\label{tab:CH2} \label{tab:CH2}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{llddddddd} \begin{tabular}{llddd}
& & \mc{1}{c}{$1\,^{3}B_1$} & & \mc{3}{c}{Transitions} \\
& \mc{2}{c}{$1\,^{3}B_1 \ra 1\,^{1}A_1$} \cline{3-5}
& \mc{2}{c}{$1\,^{3}B_1 \ra 1\,^{1}B_1$} Method & Basis set & \tabc{$1\,^{3}B_1 \ra 1\,^{1}A_1$}
& \mc{2}{c}{$1\,^{3}B_1 \ra 2\,^{1}A_1$} \\ & \tabc{$1\,^{3}B_1 \ra 1\,^{1}B_1$}
\cline{3-3} \cline{4-5} & \tabc{$1\,^{3}B_1 \ra 2\,^{1}A_1$} \\
\cline{6-7} \cline{8-9}
Method & Basis set & \tabc{$E$ (a.u.)}
& \tabc{$E$ (a.u.)} & \tabc{$\Ead$ (eV)}
& \tabc{$E$ (a.u.)} & \tabc{$\Ead$ (eV)}
& \tabc{$E$ (a.u.)} & \tabc{$\Ead$ (eV)} \\
\hline \hline
exFCI & AVDZ & -39.04846(1) exFCI & AVDZ
& -39.03225(1) & 0.441 [+0.053] & 0.441 [+0.053]
& -38.99203(1) & 1.536 [+0.146] & 1.536 [+0.146]
& -38.95076(1) & 2.659 [+0.154] \\ & 2.659 [+0.154] \\
& AVTZ & -39.08064(3) & AVTZ
& -39.06565(2) & 0.408 [+0.020] & 0.408 [+0.020]
& -39.02833(1) & 1.423 [+0.034] & 1.423 [+0.034]
& -38.98709(1) & 2.546 [+0.042] \\ & 2.546 [+0.042] \\
& AVQZ & -39.08854(1) & AVQZ
& -39.07402(2) & 0.395 [+0.007] & 0.395 [+0.007]
& -39.03711(1) & 1.399 [+0.010] & 1.399 [+0.010]
& -38.99607(1) & 2.516 [+0.012] \\ & 2.516 [+0.012] \\
& AV5Z & -39.09079(1) & AV5Z
& -39.07647(1) & 0.390 [+0.001] & 0.390 [+0.001]
& -39.03964(3) & 1.392 [+0.002] & 1.392 [+0.002]
& -38.99867(1) & 2.507 [+0.003] \\ & 2.507 [+0.003] \\
& CBS & -39.09141 & CBS
& -39.07715 & 0.388 & 0.388
& -39.04034 & 1.390 & 1.390
& -38.99939 & 2.504 \\ & 2.504 \\
\\ \\
exFCI+PBEot & AVDZ & -39.06924(1) exFCI+PBEot & AVDZ
& -39.05651(1) & 0.347 [-0.042] & 0.347 [-0.042]
& -39.01777(1) & 1.401 [+0.011] & 1.401 [+0.011]
& -38.97698(1) & 2.511 [+0.007] \\ & 2.511 [+0.007] \\
& AVTZ & -39.08805(3) & AVTZ
& -39.07430(2) & 0.374 [-0.014] & 0.374 [-0.014]
& -39.03742(1) & 1.378 [-0.012] & 1.378 [-0.012]
& -38.99652(1) & 2.491 [-0.013] \\ & 2.491 [-0.013] \\
& AVQZ & -39.09189(1) & AVQZ
& -39.07795(2) & 0.379 [-0.009] & 0.379 [-0.009]
& -39.04124(1) & 1.378 [-0.011] & 1.378 [-0.011]
& -39.00044(1) & 2.489 [-0.016] \\ & 2.489 [-0.016] \\
\\ \\
exFCI+PBE & AVDZ & -39.07282(1) exFCI+PBE & AVDZ
& -39.06150(1) & 0.308 [-0.080] & 0.308 [-0.080]
& -39.02181(1) & 1.388 [-0.002] & 1.388 [-0.002]
& -38.97873(1) & 2.560 [+0.056] \\ & 2.560 [+0.056] \\
& AVTZ & -39.08948(3) & AVTZ
& -39.07639(2) & 0.356 [-0.032] & 0.356 [-0.032]
& -39.03911(1) & 1.371 [-0.019] & 1.371 [-0.019]
& -38.99724(1) & 2.510 [+0.006] \\ & 2.510 [+0.006] \\
& AVQZ & -39.09247(1) & AVQZ
& -39.07885(2) & 0.371 [-0.017] & 0.371 [-0.017]
& -39.04193(1) & 1.375 [-0.015] & 1.375 [-0.015]
& -39.00066(1) & 2.498 [-0.006] \\ & 2.498 [-0.006] \\
\\ \\
exFCI+LDA & AVDZ & -39.07450(1) exFCI+LDA & AVDZ
& -39.06213(1) & 0.337 [-0.051] & 0.337 [-0.051]
& -39.02233(1) & 1.420 [+0.030] & 1.420 [+0.030]
& -38.97946(1) & 2.586 [+0.082] \\ & 2.586 [+0.082] \\
& AVTZ & -39.09099(3) & AVTZ
& -39.07779(2) & 0.359 [-0.029] & 0.359 [-0.029]
& -39.04051(1) & 1.374 [-0.016] & 1.374 [-0.016]
& -38.99859(1) & 2.514 [+0.010] \\ & 2.514 [+0.010] \\
& AVQZ & -39.09319(1) & AVQZ
& -39.07959(2) & 0.370 [-0.018] & 0.370 [-0.018]
& -39.04267(1) & 1.375 [-0.015] & 1.375 [-0.015]
& -39.00135(1) & 2.499 [-0.005] \\ & 2.499 [-0.005] \\
\\ \\
SHCI\fnm[1] & AVQZ & -39.08849(1) SHCI\fnm[1] & AVQZ
& -39.07404(1) & 0.393 & 0.393
& -39.03711(1) & 1.398 & 1.398
& -38.99603(1) & 2.516 \\ & 2.516 \\
CR-EOMCC (2,3)D\fnm[2]& AVQZ & -39.08817 CR-EOMCC (2,3)D\fnm[2]& AVQZ
& -39.07303 & 0.412 & 0.412
& -39.03450 & 1.460 & 1.460
& -38.99457 & 2.547 \\ & 2.547 \\
FCI\fnm[3] & TZ2P & -39.066738 FCI\fnm[3] & TZ2P
& -39.048984 & 0.483 & 0.483
& -39.010059 & 1.542 & 1.542
& -38.968471 & 2.674 \\ & 2.674 \\
DMC\fnm[4] & & DMC\fnm[4] &
& & 0.406 & 0.406
& & 1.416 & 1.416
& & 2.524 \\ & 2.524 \\
Exp.\fnm[5] & & Exp.\fnm[5] &
& & 0.400 & 0.400
& & 1.411 & 1.411
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Semistochastic heat-bath CI (SHCI) calculations from Ref.~\onlinecite{ChiHolAdaOttUmrShaZim-JPCA-18}.} \fnt[1]{Semistochastic heat-bath CI (SHCI) calculations from Ref.~\onlinecite{ChiHolAdaOttUmrShaZim-JPCA-18}.}
@ -485,7 +481,6 @@ It is also quite evident that the basis set correction has the tendency of over-
\fnt[5]{References \onlinecite{SheLeiVanSch-JCP-98, JenBun-JCP-88}.} \fnt[5]{References \onlinecite{SheLeiVanSch-JCP-98, JenBun-JCP-88}.}
\end{table*} \end{table*}
\end{squeezetable} \end{squeezetable}
\end{turnpage}
%%% %%% %%% %%% %%% %%%
%%% FIG 1 %%% %%% FIG 1 %%%
@ -505,12 +500,12 @@ It is also quite evident that the basis set correction has the tendency of over-
For the second test, we consider the water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, SceCafBenJacLoo-RC-19} and ammonia \cite{SchGoe-JCTC-17, BarDelPerMat-JMS-97, LooSceBloGarCafJac-JCTC-18} molecules. For the second test, we consider the water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, SceCafBenJacLoo-RC-19} and ammonia \cite{SchGoe-JCTC-17, BarDelPerMat-JMS-97, LooSceBloGarCafJac-JCTC-18} molecules.
They are both well-studied and possess Rydberg excited states which are highly sensitive to the radial completeness of the one-electron basis set, as evidenced in Ref.~\onlinecite{LooSceBloGarCafJac-JCTC-18}. They are both well-studied and possess Rydberg excited states which are highly sensitive to the radial completeness of the one-electron basis set, as evidenced in Ref.~\onlinecite{LooSceBloGarCafJac-JCTC-18}.
Table \ref{tab:Mol} reports vertical excitation energies for various singlet and triplet excited states of water and ammonia at various levels of theory. Table \ref{tab:Mol} reports vertical excitation energies for various singlet and triplet excited states of water and ammonia at various levels of theory (see the {\SI} for total energies).
The theoretical best estimates (TBEs) have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJac-JCTC-19} and have been obtained on the same geometries. The basis set corrected theoretical best estimates (TBEs) have been extracted from Ref.~\onlinecite{LooSceBloGarCafJac-JCTC-18} and have been obtained on the same geometries.
These results are also depicted in Figs.~\ref{fig:H2O} and \ref{fig:NH3} for \ce{H2O} and \ce{NH3}, respectively. These results are also depicted in Figs.~\ref{fig:H2O} and \ref{fig:NH3} for \ce{H2O} and \ce{NH3}, respectively.
One would have noticed that the basis set effects are particularly strong for the third singlet excited state of water and the third and fourth singlet excited states of ammonia where this is effect is magnified. One would have noticed that the basis set effects are particularly strong for the third singlet excited state of water and the third and fourth singlet excited states of ammonia where this is effect is even magnified.
In these cases, one really needs doubly- or even triply-augmented basis sets to reach radial completeness. In these cases, one really needs doubly-augmented basis sets to reach radial completeness.
The first observation worth reporting is that all three RS-DFT correlation functionals have very similar behavior and they significantly reduce the error on the excitation energies for most of the states. The first observation worth reporting is that all three RS-DFT correlation functionals have very similar behaviors and they significantly reduce the error on the excitation energies for most of the states.
However, these results also clearly evidence that special care has to be taken for very diffuse excited states where the present correction might not be enough to catch the radial incompleteness of the one-electron basis set. However, these results also clearly evidence that special care has to be taken for very diffuse excited states where the present correction might not be enough to catch the radial incompleteness of the one-electron basis set.
@ -539,100 +534,100 @@ However, these results also clearly evidence that special care has to be taken f
& \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVQZ} & \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVQZ}
\\ \\
\hline \hline
Ammonia & $1\,^{1}A_{1} \ra 1\,^{1}A_{2}$ & Ryd. & 6.66 & -0.18 & -0.07 & -0.04 Ammonia & $1\,^{1}A_{1} \ra 1\,^{1}A_{2}$ & Ryd. & 6.66\fnm[1] & -0.18 & -0.07 & -0.04
& -0.04 & -0.02 & -0.01 & -0.04 & -0.02 & -0.01
& -0.07 & -0.03 & -0.02 & -0.07 & -0.03 & -0.02
& -0.07 & -0.03 & -0.02 & -0.07 & -0.03 & -0.02
\\ \\
& $1\,^{1}A_{1} \ra 1\,^{1}E$ & Ryd. & 8.21 & -0.13 & -0.05 & -0.02 & $1\,^{1}A_{1} \ra 1\,^{1}E$ & Ryd. & 8.21\fnm[1] & -0.13 & -0.05 & -0.02
& 0.01 & 0.00 & 0.01 & 0.01 & 0.00 & 0.01
& -0.03 & -0.01 & 0.00 & -0.03 & -0.01 & 0.00
& -0.03 & 0.00 & 0.00 & -0.03 & 0.00 & 0.00
\\ \\
& $1\,^{1}A_{1} \ra 2\,^{1}A_{1}$ & Ryd. & 8.65 & 1.03 & 0.68 & 0.47 & $1\,^{1}A_{1} \ra 2\,^{1}A_{1}$ & Ryd. & 8.65\fnm[1] & 1.03 & 0.68 & 0.47
& 1.17 & 0.73 & 0.50 & 1.17 & 0.73 & 0.50
& 1.12 & 0.72 & 0.49 & 1.12 & 0.72 & 0.49
& 1.11 & 0.71 & 0.49 & 1.11 & 0.71 & 0.49
\\ \\
& $1\,^{1}A_{1} \ra 2\,^{1}A_{2}$ & Ryd. & 8.65 & 1.22 & 0.77 & 0.59 & $1\,^{1}A_{1} \ra 2\,^{1}A_{2}$ & Ryd. & 8.65\fnm[2] & 1.22 & 0.77 & 0.59
& 1.36 & 0.83 & 0.62 & 1.36 & 0.83 & 0.62
& 1.33 & 0.81 & 0.61 & 1.33 & 0.81 & 0.61
& 1.32 & 0.81 & 0.61 & 1.32 & 0.81 & 0.61
\\ \\
& $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & Ryd. & 9.19 & -0.18 & -0.06 & -0.03 & $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & Ryd. & 9.19\fnm[1] & -0.18 & -0.06 & -0.03
& -0.03 & 0.00 & -0.02 & -0.03 & 0.00 & -0.02
& -0.07 & -0.02 & -0.03 & -0.07 & -0.02 & -0.03
& -0.07 & -0.01 & -0.03 & -0.07 & -0.01 & -0.03
\\ \\
\\ \\
Carbon dimer\fnm[1] & $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ & Val. & 2.06 & 0.15 & 0.03 & 0.00 Carbon dimer & $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ & Val. & 2.06\fnm[3] & 0.15 & 0.03 & 0.00
& 0.02 & -0.02 & -0.02 & 0.02 & -0.02 & -0.02
& 0.13 & 0.02 & 0.00 & 0.13 & 0.02 & 0.00
& 0.15 & 0.03 & 0.00 & 0.15 & 0.03 & 0.00
\\ \\
& $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ & Val. & 2.40 & 0.10 & 0.02 & 0.00 & $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ & Val. & 2.40\fnm[3] & 0.10 & 0.02 & 0.00
& 0.02 & -0.03 & -0.02 & 0.02 & -0.03 & -0.02
& 0.09 & 0.01 & 0.00 & 0.09 & 0.01 & 0.00
& 0.11 & 0.02 & 0.00 & 0.11 & 0.02 & 0.00
\\ \\
\\ \\
Water & $1\,^{1}A_1 \ra 1\,^{1}B_1$ & Ryd. & 7.70 & -0.17 & -0.07 & -0.02 Water & $1\,^{1}A_1 \ra 1\,^{1}B_1$ & Ryd. & 7.70\fnm[1] & -0.17 & -0.07 & -0.02
& 0.01 & 0.00 & 0.02 & 0.01 & 0.00 & 0.02
& -0.02 & -0.01 & 0.00 & -0.02 & -0.01 & 0.00
& -0.04 & -0.01 & 0.01 & -0.04 & -0.01 & 0.01
\\ \\
& $1\,^{1}A_1 \ra 1\,^{1}A_2$ & Ryd. & 9.47 & -0.15 & -0.06 & -0.01 & $1\,^{1}A_1 \ra 1\,^{1}A_2$ & Ryd. & 9.47\fnm[1] & -0.15 & -0.06 & -0.01
& 0.03 & 0.01 & 0.03 & 0.03 & 0.01 & 0.03
& 0.00 & 0.00 & 0.02 & 0.00 & 0.00 & 0.02
& -0.03 & 0.00 & 0.00 & -0.03 & 0.00 & 0.00
\\ \\
& $1\,^{1}A_1 \ra 2\,^{1}A_1$ & Ryd. & 9.97 & -0.03 & 0.02 & 0.06 & $1\,^{1}A_1 \ra 2\,^{1}A_1$ & Ryd. & 9.97\fnm[1] & -0.03 & 0.02 & 0.06
& 0.13 & 0.08 & 0.09 & 0.13 & 0.08 & 0.09
& 0.10 & 0.07 & 0.08 & 0.10 & 0.07 & 0.08
& 0.09 & 0.07 & 0.03 & 0.09 & 0.07 & 0.03
\\ \\
& $1\,^{1}A_1 \ra 1\,^{3}B_1$ & Ryd. & 7.33 & -0.19 & -0.08 & -0.03 & $1\,^{1}A_1 \ra 1\,^{3}B_1$ & Ryd. & 7.33\fnm[1] & -0.19 & -0.08 & -0.03
& 0.02 & 0.00 & 0.02 & 0.02 & 0.00 & 0.02
& 0.05 & 0.01 & 0.02 & 0.05 & 0.01 & 0.02
& 0.00 & 0.00 & 0.04 & 0.00 & 0.00 & 0.04
\\ \\
& $1\,^{1}A_1 \ra 1\,^{3}A_2$ & Ryd. & 9.30 & -0.16 & -0.06 & -0.01 & $1\,^{1}A_1 \ra 1\,^{3}A_2$ & Ryd. & 9.30\fnm[1] & -0.16 & -0.06 & -0.01
& 0.04 & 0.02 & 0.04 & 0.04 & 0.02 & 0.04
& 0.07 & 0.03 & 0.04 & 0.07 & 0.03 & 0.04
& 0.03 & 0.03 & 0.04 & 0.03 & 0.03 & 0.04
\\ \\
& $1\,^{1}A_1 \ra 1\,^{3}A_1$ & Ryd. & 9.59 & -0.11 & -0.05 & -0.01 & $1\,^{1}A_1 \ra 1\,^{3}A_1$ & Ryd. & 9.59\fnm[1] & -0.11 & -0.05 & -0.01
& 0.07 & 0.02 & 0.03 & 0.07 & 0.02 & 0.03
& 0.09 & 0.03 & 0.03 & 0.09 & 0.03 & 0.03
& 0.06 & 0.03 & 0.04 & 0.06 & 0.03 & 0.04
\\ \\
\\ \\
Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & Ryd. & 7.43 & -0.12 & -0.04 & Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & Ryd. & 7.43\fnm[3] & -0.12 & -0.04 &
& -0.05 & -0.01 & & -0.05 & -0.01 &
& -0.04 & -0.01 & & -0.04 & -0.01 &
& -0.02 & 0.00 & & -0.02 & 0.00 &
\\ \\
& $1\,^{1}A_{1g} \ra 1\,^{1}B_{1u}$ & Val. & 7.92 & 0.01 & 0.01 & & $1\,^{1}A_{1g} \ra 1\,^{1}B_{1u}$ & Val. & 7.92\fnm[3] & 0.01 & 0.01 &
& 0.00 & 0.00 & & 0.00 & 0.00 &
& 0.06 & 0.03 & & 0.06 & 0.03 &
& 0.06 & 0.03 & & 0.06 & 0.03 &
\\ \\
& $1\,^{1}A_{1g} \ra 1\,^{1}B_{1g}$ & Ryd. & 8.10 & -0.1 & -0.02 & & $1\,^{1}A_{1g} \ra 1\,^{1}B_{1g}$ & Ryd. & 8.10\fnm[3] & -0.1 & -0.02 &
& -0.03 & 0.00 & & -0.03 & 0.00 &
& -0.02 & 0.00 & & -0.02 & 0.00 &
& 0.00 & 0.01 & & 0.00 & 0.01 &
\\ \\
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$ & Val. & 4.54 & 0.01 & 0.00 & & $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$ & Val. & 4.54\fnm[3] & 0.01 & 0.00 &
& 0.07 & 0.03 & & 0.07 & 0.03 &
& 0.10 & 0.04 & & 0.10 & 0.04 &
& 0.08 & 0.04 & & 0.08 & 0.04 &
\\ \\
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{3u}$ & Val. & 7.28 & -0.12 & -0.04 & & $1\,^{1}A_{1g} \ra 1\,^{3}B_{3u}$ & Val. & 7.28\fnm[4] & -0.12 & -0.04 &
& -0.03 & 0.00 & & -0.03 & 0.00 &
& 0.00 & 0.00 & & 0.00 & 0.00 &
& 0.00 & 0.02 & & 0.00 & 0.02 &
\\ \\
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{1g}$ & Val. & 8.00 & -0.07 & -0.01 & & $1\,^{1}A_{1g} \ra 1\,^{3}B_{1g}$ & Val. & 8.00\fnm[4] & -0.07 & -0.01 &
& 0.01 & 0.03 & & 0.01 & 0.03 &
& 0.04 & 0.03 & & 0.04 & 0.03 &
& 0.05 & 0.04 & & 0.05 & 0.04 &
@ -640,7 +635,10 @@ However, these results also clearly evidence that special care has to be taken f
\\ \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Doubly-excited states of $(\pi,\pi) \ra (\si,\si)$ character.} \fnt[1]{exFCI/AVQZ data corrected with the difference between CC3/d-AV5Z and exFCI/AVQZ values. \cite{LooSceBloGarCafJac-JCTC-18}}
\fnt[2]{exFCI/AVTZ data corrected with the difference between CC3/d-AV5Z and exFCI/AVTZ values. \cite{LooSceBloGarCafJac-JCTC-18}}
\fnt[3]{exFCI/AVQZ data from Ref.~\onlinecite{LooBogSceCafJac-JCTC-19}.}
\fnt[4]{exFCI/AVDZ data corrected with the difference between CC3/d-AV5Z and exFCI/AVDZ values. \cite{LooSceBloGarCafJac-JCTC-18}}
\end{table*} \end{table*}
\end{squeezetable} \end{squeezetable}
%%% %%% %%% %%% %%% %%%
@ -655,7 +653,7 @@ However, these results also clearly evidence that special care has to be taken f
\end{figure} \end{figure}
%%% %%% %%% %%% %%% %%%
%%% FIG 4 %%% %%% FIG 3 %%%
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{NH3} \includegraphics[width=\linewidth]{NH3}
\caption{Error in vertical excitation energies (in eV) of ammonia for various basis sets and methods. \caption{Error in vertical excitation energies (in eV) of ammonia for various basis sets and methods.
@ -669,10 +667,10 @@ However, these results also clearly evidence that special care has to be taken f
\subsection{Doubly-Excited States of the Carbon Dimer} \subsection{Doubly-Excited States of the Carbon Dimer}
\label{sec:C2} \label{sec:C2}
%======================= %=======================
In order to have a miscellaneous test set of excitation, in a third time, we propose to study some doubly-excited states of the carbon dimer \ce{C2}, which is a prototype system for strongly correlated and multireference systems. \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, HolUmrSha-JCP-17, VarRoc-PTRSMPES-18} In order to have a miscellaneous test set of excitations, in a third time, we propose to study some doubly-excited states of the carbon dimer \ce{C2}, a prototype system for strongly correlated and multireference systems. \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, HolUmrSha-JCP-17, VarRoc-PTRSMPES-18}
These two valence excitations --- $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ and $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ --- are both of $(\pi,\pi) \ra (\si,\si)$ character. These two valence excitations --- $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ and $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ --- are both of $(\pi,\pi) \ra (\si,\si)$ character.
They have been recently studied with state-of-the-art methods, and have been shown to be ``pure'' doubly-excited states as they do not involve single excitations. \cite{LooBogSceCafJac-JCTC-19} They have been recently studied with state-of-the-art methods, and have been shown to be ``pure'' doubly-excited states as they do not involve single excitations. \cite{LooBogSceCafJac-JCTC-19}
The vertical excitation energies associated with these transitions are reported in \ref{tab:Mol} and represented in Fig.~\ref{fig:C2}. The vertical excitation energies associated with these transitions are reported in Table \ref{tab:Mol} and represented in Fig.~\ref{fig:C2}.
An interesting point here is that one really needs the PBEot to get chemically-accurate absorption energies with the AVDZ atomic basis set. An interesting point here is that one really needs the PBEot to get chemically-accurate absorption energies with the AVDZ atomic basis set.
\titou{New figure represented $\rsmu{}{}(\br{})$ to understand what's going on here. Maybe it is because strongly correlated?} \titou{New figure represented $\rsmu{}{}(\br{})$ to understand what's going on here. Maybe it is because strongly correlated?}
@ -686,16 +684,19 @@ An interesting point here is that one really needs the PBEot to get chemically-a
\end{figure} \end{figure}
%%% %%% %%% %%% %%% %%%
%======================= %=======================
\subsection{Ethylene} \subsection{Ethylene}
\label{sec:C2H4} \label{sec:C2H4}
%======================= %=======================
Ethylene is the largest molecule we consider here. As a final example, we consider the ethylene molecule, yet another system which has been particularly scrutinized theoretically using high-level ab initio methods. \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}
It is an interesting molecule as it contains a mixture of 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} We refer the interested reader to the work of Feller et al.\cite{FelPetDav-JCP-14} for an exhaustive investigation dedicated to the excited states of ethylene using state-of-the-art CI calculations.
The most complete and accurate investigation dedicated to the excited states of ethylene is due to Davidson's group, who performed refined CI calculations. \cite{FelPetDav-JCP-14} In the present context, ethylene is a particularly interesting system as it contains a mixture of valence and Rydberg excited states.
Our basis set corrected vertical excitation energies are gathered in Table \ref{tab:Mol} and depicted in Fig.~\ref{fig:C2H4}.
Except for one particular excitation (the lowest singlet-triplet excitation $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$), the exFCI+PBEot/AVDZ excitation energies are chemically accurate and the errors drop further when one goes to the triple-$\zeta$ basis.
Consistently with the previous examples, the LDA and PBE functionals are slightly less accurate, although they still correct the excitation energies in the right direction.
%%% FIG 5 %%%
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{C2H4} \includegraphics[width=\linewidth]{C2H4}
\caption{Error in vertical excitation energies $\Eabs$ (in eV) of ethylene for various basis sets and methods. \caption{Error in vertical excitation energies $\Eabs$ (in eV) of ethylene for various basis sets and methods.
@ -703,22 +704,26 @@ The most complete and accurate investigation dedicated to the excited states of
See the {\SI} for raw data.} See the {\SI} for raw data.}
\label{fig:C2H4} \label{fig:C2H4}
\end{figure} \end{figure}
%%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion} \section{Conclusion}
\label{sec:ccl} \label{sec:ccl}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
We have shown that by employing the recently proposed density-based basis set correction developed by some of the authors, one can obtain chemically-accurate excitation energies with typically augmented double-$\zeta$ basis sets. We have shown that, by employing the recently proposed density-based basis set correction developed by some of the authors, \cite{GinPraFerAssSavTou-JCP-18} one can obtain chemically-accurate excitation energies with typically augmented double-$\zeta$ basis sets.
This nicely complements our recent investigation on ground-state properties, \cite{LooPraSceTouGin-JPCL-19} which has evidenced that one recovers quintuple-$\zeta$ quality atomization and correlation energies with triple-$\zeta$ basis sets.
The present study clearly shows that, for very diffuse excited states, the present correction relying on short-range correlation functionals from RS-DFT might not be enough to catch the radial incompleteness of the one-electron basis set.
We are currently investigating the performance of the present basis set correction for strongly correlated systems and we hope to report on this in the near future. We are currently investigating the performance of the present basis set correction for strongly correlated systems and we hope to report on this in the near future.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting Information Available} \section*{Supporting Information Available}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
See {\SI} for geometries and additional information (including energetic correction of the various functionals). See {\SI} for geometries and additional information (including total energies and energetic correction of the various functionals).
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements} \begin{acknowledgements}
The authors would like to thank the \textit{Centre National de la Recherche Scientifique} (CNRS) for funding. The authors would like to thank the \textit{Centre National de la Recherche Scientifique} (CNRS) for funding.
PFL would like to thank Denis Jacquemin for numerous discussions on excited states.
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005. This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
\end{acknowledgements} \end{acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%

101
Manuscript/SI/Ex-srDFT-SI.tex
View File

@ -270,10 +270,107 @@ H 0.000000 -0.757532 0.518435
\section{Total energies} \section{Total energies}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
exFCI total energies can be found in the {\SI} of Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJAc-JCTC-19}. The exFCI total energies can be found in the {\SI} of Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJAc-JCTC-19}.
Here, we report the absolute energetic corrections for each state of each molecule obtained with the three short-range correlation functionals considered in the present study (i.e., LDA, PBE and PBEot). Here, we report the absolute energetic corrections for each state of each molecule obtained with the three short-range correlation functionals considered in the present study (i.e., LDA, PBE and PBEot).
Data for methylene can be found in the main text.
%%% TABLE 1 %%%
\begin{squeezetable}
\begin{table*}[h]
\caption{
Total energies (in hartree) 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.}
\label{tab:CH2}
\begin{ruledtabular}
\begin{tabular}{lldddd}
& & \mc{4}{c}{States} \\
\cline{3-6}
Method & Basis set & \tabc{$1\,^{3}B_1$}
& \tabc{$1\,^{1}A_1$}
& \tabc{$1\,^{1}B_1$}
& \tabc{$2\,^{1}A_1$} \\
\hline
exFCI & AVDZ & -39.04846(1)
& -39.03225(1)
& -38.99203(1)
& -38.95076(1) \\
& AVTZ & -39.08064(3)
& -39.06565(2)
& -39.02833(1)
& -38.98709(1) \\
& AVQZ & -39.08854(1)
& -39.07402(2)
& -39.03711(1)
& -38.99607(1) \\
& AV5Z & -39.09079(1)
& -39.07647(1)
& -39.03964(3)
& -38.99867(1) \\
& CBS & -39.09141
& -39.07715
& -39.04034
& -38.99939 \\
\\
exFCI+PBEot & AVDZ & -39.06924(1)
& -39.05651(1)
& -39.01777(1)
& -38.97698(1) \\
& AVTZ & -39.08805(3)
& -39.07430(2)
& -39.03742(1)
& -38.99652(1) \\
& AVQZ & -39.09189(1)
& -39.07795(2)
& -39.04124(1)
& -39.00044(1) \\
\\
exFCI+PBE & AVDZ & -39.07282(1)
& -39.06150(1)
& -39.02181(1)
& -38.97873(1) \\
& AVTZ & -39.08948(3)
& -39.07639(2)
& -39.03911(1)
& -38.99724(1) \\
& AVQZ & -39.09247(1)
& -39.07885(2)
& -39.04193(1)
& -39.00066(1) \\
\\
exFCI+LDA & AVDZ & -39.07450(1)
& -39.06213(1)
& -39.02233(1)
& -38.97946(1) \\
& AVTZ & -39.09099(3)
& -39.07779(2)
& -39.04051(1)
& -38.99859(1) \\
& AVQZ & -39.09319(1)
& -39.07959(2)
& -39.04267(1)
& -39.00135(1) \\
\\
SHCI\fnm[1] & AVQZ & -39.08849(1)
& -39.07404(1)
& -39.03711(1)
& -38.99603(1) \\
CR-EOMCC (2,3)D\fnm[2]& AVQZ& -39.08817
& -39.07303
& -39.03450
& -38.99457 \\
FCI\fnm[3] & TZ2P & -39.066738
& -39.048984
& -39.010059
& -38.968471 \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Semistochastic heat-bath CI (SHCI) calculations from Ref.~\onlinecite{ChiHolAdaOttUmrShaZim-JPCA-18}.}
\fnt[2]{Completely-renormalized equation-of-motion coupled cluster (CR-EOMCC) calculations from Refs.~\onlinecite{SheLeiVanSch-JCP-98, JenBun-JCP-88}.}
\fnt[3]{Reference \onlinecite{SheLeiVanSch-JCP-98}.}
\end{table*}
\end{squeezetable}
%%% %%% %%% %%%
%%% TABLE 2 %%% %%% TABLE 2 %%%
\begin{turnpage} \begin{turnpage}
\begin{squeezetable} \begin{squeezetable}