diff --git a/Manuscript/CH2.pdf b/Manuscript/CH2.pdf index b15f2a7..414ed78 100644 Binary files a/Manuscript/CH2.pdf and b/Manuscript/CH2.pdf differ diff --git a/Manuscript/Ex-srDFT.bib b/Manuscript/Ex-srDFT.bib index 8f1e26f..1de5e6a 100644 --- a/Manuscript/Ex-srDFT.bib +++ b/Manuscript/Ex-srDFT.bib @@ -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}, diff --git a/Manuscript/Ex-srDFT.tex b/Manuscript/Ex-srDFT.tex index 52b32ec..68dec94 100644 --- a/Manuscript/Ex-srDFT.tex +++ b/Manuscript/Ex-srDFT.tex @@ -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}