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Pierre-Francois Loos 2019-05-29 23:31:04 +02:00
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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 2019-05-28 23:01:35 +0200
%% Created for Pierre-Francois Loos at 2019-05-29 23:30:47 +0200
%% Saved with string encoding Unicode (UTF-8)

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Manuscript/Ex-srDFT.tex
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@ -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}