From fb85bd2334bf2c92c5225a3da2ef36551e54d237 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Wed, 29 May 2019 23:31:04 +0200 Subject: [PATCH] starting on the results --- Manuscript/Ex-srDFT.bib | 2 +- Manuscript/Ex-srDFT.tex | 57 +++++++++++++++++++++++++++-------------- 2 files changed, 39 insertions(+), 20 deletions(-) diff --git a/Manuscript/Ex-srDFT.bib b/Manuscript/Ex-srDFT.bib index 97da326..8f1e26f 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 2019-05-28 23:01:35 +0200 +%% Created for Pierre-Francois Loos at 2019-05-29 23:30:47 +0200 %% Saved with string encoding Unicode (UTF-8) diff --git a/Manuscript/Ex-srDFT.tex b/Manuscript/Ex-srDFT.tex index ea1eb9a..5a80325 100644 --- a/Manuscript/Ex-srDFT.tex +++ b/Manuscript/Ex-srDFT.tex @@ -98,8 +98,7 @@ \newcommand{\V}[2]{V_{#1}^{#2}} \newcommand{\SO}[2]{\phi_{#1}(\br{#2})} -\newcommand{\modY}{Y} -\newcommand{\modZ}{Z} +\newcommand{\tX}{\text{X}} % basis sets \newcommand{\Bas}{\mathcal{B}} @@ -185,7 +184,7 @@ Although for ground-state properties angular incompleteness is by far the main s Explicitly-correlated F12 methods \cite{Kut-TCA-85, Kut-TCA-85, KutKlo-JCP-91, NogKut-JCP-94} have been specifically designed to efficiently catch angular incompleteness. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} Although they have been extremely successful to speed up convergence of ground-state energies and properties, such as correlation and atomization energies, \cite{TewKloNeiHat-PCCP-07} their performances for excited states \cite{FliHatKlo-JCP-06, NeiHatKlo-JCP-06, HanKoh-JCP-09, Koh-JCP-09, ShiWer-JCP-10, ShiKniWer-JCP-11, ShiWer-JCP-11, ShiWer-MP-13} have been much more conflicting. \cite{FliHatKlo-JCP-06, NeiHatKlo-JCP-06} -Instead of F12 methods, here we propose to follow a different route and investigate the performances of the recently proposed universal density-based basis set +Instead of F12 methods, here we propose to follow a different route and investigate the performance of the recently proposed universal density-based basis set incompleteness correction. \cite{GinPraFerAssSavTou-JCP-18} Contrary to our recent study on atomization and correlation energies, \cite{LooPraSceTouGin-JPCL-19} the present contribution focuses on vertical and adiabatic excitation energies in molecular systems which is a much tougher test for the reasons mentioned above. This density-based correction relies on short-range correlation density functionals (with multideterminant reference) from range-separated density-functional theory \cite{TouColSav-PRA-04, AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15, LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} (RS-DFT) to capture the missing part of the short-range correlation effects, a consequence of the incompleteness of the one-electron basis set. @@ -229,7 +228,7 @@ The notation $\wf{}{} \rightsquigarrow \n{}{}$ in Eq.~\eqref{eq:E_funcbasis} sta Hence, the CBS excitation energy associated with the $k$th excited state reads \begin{equation} - \DE{k}{\CBS} = \E{k}{\CBS} - \E{0}{\CBS} = \DE{k}{\Bas} + \DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}], + \DE{k}{\CBS} = \E{k}{\CBS} - \E{0}{\CBS} \approx \DE{k}{\Bas} + \DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}], \end{equation} where \begin{equation} @@ -344,6 +343,7 @@ Frozen-core calculations are systematically performed and defined as such: a \ce The frozen-core density-based correction is used consistently with the frozen-core approximation in WFT methods. We refer the interested reader to Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} for an explicit derivation of the equations associated with the frozen-core version of the present density-based basis set correction. Compared to the exFCI calculations performed to compute energies and densities, the basis set correction represents, in any case, a marginal computational cost. +In the following, we employ the AVXZ shorthand notations for Dunning's aug-cc-pVXZ basis sets. %%%%%%%%%%%%%%%%%%%%%%%% \section{Results} @@ -355,14 +355,24 @@ Compared to the exFCI calculations performed to compute energies and densities, \label{sec:CH2} %======================= -As a first test of the present basis set correction, we consider the adiabatic transitions of methylene which have been thourhoughly studied in the literature 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} +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} + +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. +We have also computed these adiabatic energies at the exFCI/AV5Z level and used these total energies alongside the quadruple-$\zeta$ ones to extrapolate the excitation energies to the CBS limit with the usual extrapolation formula \cite{HelJorOls-BOOK-02} +\begin{equation} + \E{}{\text{AVXZ}} = \E{}{\CBS} + \frac{\alpha}{(\tX+1/2)^{3}}. +\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} %%% TABLE 1 %%% \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.} - \begin{ruledtabular}{} + \label{tab:CH2} + \begin{ruledtabular} \begin{tabular}{llddddddd} & & \mc{1}{c}{$1\,^{3}B_1$} & \mc{2}{c}{$1\,^{3}B_1 \ra 1\,^{1}A_1$} @@ -392,9 +402,9 @@ As a first test of the present basis set correction, we consider the adiabatic t & -39.03964(3) & 1.392 & -38.99867(1) & 2.507 \\ & CBS & -39.09111 - & -39.07682 & 0.389 - & -39.04000 & 1.391 - & -38.99904 & 2.505 \\ + & -39.07682 & 0.388 + & -39.04000 & 1.390 + & -38.99904 & 2.504 \\ \\ exFCI+LDA & AVDZ & -39.07450(1) & -39.06213(1) & 0.337 @@ -456,10 +466,10 @@ As a first test of the present basis set correction, we consider the adiabatic t & & 1.411 \end{tabular} \end{ruledtabular} - \fnt[1]{Reference \onlinecite{ChiHolAdaOttUmrShaZim-JPCA-18}.} + \fnt[1]{Semistochastic heat-bath CI (SHCI) calculations from Ref.~\onlinecite{ChiHolAdaOttUmrShaZim-JPCA-18}.} \fnt[2]{References \onlinecite{SheLeiVanSch-JCP-98, JenBun-JCP-88}.} \fnt[3]{Reference \onlinecite{SheLeiVanSch-JCP-98}.} - \fnt[4]{Reference \onlinecite{ZimTouZhaMusUmr-JCP-09}.} + \fnt[4]{Diffusion Monte Carlo (DMC) calculations from Ref.~\onlinecite{ZimTouZhaMusUmr-JCP-09}.} \fnt[5]{References \onlinecite{SheLeiVanSch-JCP-98, JenBun-JCP-88}.} \end{table*} \end{squeezetable} @@ -468,7 +478,9 @@ As a first test of the present basis set correction, we consider the adiabatic t %%% 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.} + \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}). + See Table \ref{tab:CH2} for raw data.} \label{fig:CH2} \end{figure} %%% %%% %%% @@ -485,7 +497,8 @@ 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}.} + 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} & & & & \mc{12}{c}{Deviation with respect to TBE} @@ -711,7 +724,9 @@ Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac %%% FIG 2 %%% \begin{figure} \includegraphics[width=\linewidth]{H2O} - \caption{Error in vertical excitation energies (in eV) of water for various basis sets and methods.} + \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}). + See the {\SI} for raw data.} \label{fig:H2O} \end{figure} %%% %%% %%% @@ -719,7 +734,9 @@ Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac %%% FIG 4 %%% \begin{figure} \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. + The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal}). + See the {\SI} for raw data.} \label{fig:NH3} \end{figure} %%% %%% %%% @@ -734,7 +751,9 @@ In the carbon dimer, these valence states are of $(\pi,\pi) \ra (\si,\si)$ chara %%% 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.} + \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}). + See the {\SI} for raw data.} \label{fig:C2} \end{figure} %%% %%% %%% @@ -755,7 +774,9 @@ Ethylene is an interesting molecules as it contains both valence and Rydberg exc \begin{figure} \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. + The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal}). + See the {\SI} for raw data.} \label{fig:C2H4} \end{figure} @@ -784,8 +805,6 @@ This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A004 \end{acknowledgements} %%%%%%%%%%%%%%%%%%%%%%%% - - \bibliography{Ex-srDFT,Ex-srDFT-control} \end{document}