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Manuscript/Ex-srDFT.bib
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@ -1,13 +1,23 @@
%% 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 11:26:43 +0200 %% Created for Pierre-Francois Loos at 2019-05-30 22:48:58 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{SceCafBenJacLoo-RC-19,
Author = {A. Scemama and M. Caffarel and A. Benali and D. Jacquemin and P. F. Loos},
Date-Added = {2019-05-30 21:47:05 +0200},
Date-Modified = {2019-05-30 21:48:59 +0200},
Doi = {10.1016/j.rechem.2019.100002},
Journal = {Res. Chem.},
Pages = {in press},
Title = {Influence of Pseudopotentials on Excitation Energies From Selected Configuration Interaction and Diffusion Monte Carlo},
Year = {2019}}
@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},
Date-Added = {2019-05-28 22:50:09 +0200}, Date-Added = {2019-05-28 22:50:09 +0200},

43
Manuscript/Ex-srDFT.tex
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@ -323,7 +323,7 @@ It reads
\end{subequations} \end{subequations}
We will refer to this functional as the ``on top'' PBE (PBEot) ECMD functional. We will refer to this functional as the ``on top'' PBE (PBEot) ECMD functional.
More recently, \cite{LooPraSceTouGin-JPCL-19} we have also proposed a simplified version of the PBEot functional where we replaced the on-top pair density by its UEG version, i.e.~$\n{2}{\Bas}(\br{},\br{}) \approx \n{2}{\UEG}(\n{}{}(\br{}),\zeta(\br{}))$, where $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$ with the parametrization of the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}. More recently, \cite{LooPraSceTouGin-JPCL-19} we have also proposed a simplified version of the PBEot functional where we replaced the on-top pair density by its UEG version, i.e.~$\n{2}{\Bas}(\br{},\br{}) \approx \n{2}{\UEG}(\n{}{}(\br{}),\zeta(\br{}))$, where $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$ with the parametrization of the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}.
This computationally-lighter functional will be refered to as PBE. This computationally-lighter functional will be referred to as PBE.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
@ -361,18 +361,19 @@ Methylene is a paradigmatic system in electronic structure theory. \cite{Sch-Sci
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 $2\,^{1}A_1$) 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} 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}
\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 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. 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, 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 energy. 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 use the on-top density) are less impressive, yet they still yeild a significant reduction of the error on the adiabatic 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.
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. Yet, they still yield significant reductions of the basis set incompleteness error.
it is also quite evident that the basis set correction has the tendency of over-correcting the excitation energies by over-stabilizing the excited states compared to the ground state. 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).
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.
%%% TABLE 1 %%% %%% TABLE 1 %%%
\begin{turnpage} \begin{turnpage}
@ -478,7 +479,7 @@ it is also quite evident that the basis set correction has the tendency of over-
\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}.}
\fnt[2]{References \onlinecite{SheLeiVanSch-JCP-98, JenBun-JCP-88}.} \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}.} \fnt[3]{Reference \onlinecite{SheLeiVanSch-JCP-98}.}
\fnt[4]{Diffusion Monte Carlo (DMC) calculations from Ref.~\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}.} \fnt[5]{References \onlinecite{SheLeiVanSch-JCP-98, JenBun-JCP-88}.}
@ -502,7 +503,16 @@ it is also quite evident that the basis set correction has the tendency of over-
\label{sec:H2O-NH3} \label{sec:H2O-NH3}
%======================= %=======================
Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18} and ammonia \cite{SchGoe-JCTC-17, BarDelPerMat-JMS-97, LooSceBloGarCafJac-JCTC-18} are two interesting molecules with Rydberg excited states which are highly sensitive to the radial completeness of the one-electron basis set. 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}.
Table \ref{tab:Mol} reports vertical excitation energies for various singlet and triplet excited states of water and ammonia at various levels of theory.
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.
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.
In these cases, one really needs doubly- or even triply-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.
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.
%%% TABLE 2 %%% %%% TABLE 2 %%%
\begin{squeezetable} \begin{squeezetable}
@ -511,7 +521,8 @@ Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac
Vertical absorption energies $\Eabs$ (in eV) of excited states of ammonia, carbon dimer, water and ethylene for various methods and basis sets. 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.} See the {\SI} for raw data.}
\begin{ruledtabular}{} \label{tab:Mol}
\begin{ruledtabular}
\begin{tabular}{lllddddddddddddd} \begin{tabular}{lllddddddddddddd}
& & & & \mc{12}{c}{Deviation with respect to TBE} & & & & \mc{12}{c}{Deviation with respect to TBE}
\\ \\
@ -658,8 +669,12 @@ Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac
\subsection{Doubly-Excited States of the Carbon Dimer} \subsection{Doubly-Excited States of the Carbon Dimer}
\label{sec:C2} \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 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 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} 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}
The vertical excitation energies associated with these transitions are reported in \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.
\titou{New figure represented $\rsmu{}{}(\br{})$ to understand what's going on here. Maybe it is because strongly correlated?}
%%% FIG 4 %%% %%% FIG 4 %%%
\begin{figure} \begin{figure}
@ -677,7 +692,9 @@ In the carbon dimer, these valence states are of $(\pi,\pi) \ra (\si,\si)$ chara
\label{sec:C2H4} \label{sec:C2H4}
%======================= %=======================
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} Ethylene is the largest molecule we consider here.
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}
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}
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{C2H4} \includegraphics[width=\linewidth]{C2H4}