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Manuscript/Ex-srDFT.tex
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@ -178,7 +178,7 @@
\begin{abstract} \begin{abstract}
By combining extrapolated selected configuration interaction (sCI) energies obtained with the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm with the recently proposed short-range density-functional correction for basis-set incompleteness [\href{https://doi.org/10.1063/1.5052714}{Giner \textit{et al.}, \textit{J.~Chem.~Phys.}~\textbf{149}, 194301 (2018)}], we show that one can get chemically accurate vertical and adiabatic excitation energies with, typically, augmented double-$\zeta$ basis sets. By combining extrapolated selected configuration interaction (sCI) energies obtained with the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm with the recently proposed short-range density-functional correction for basis-set incompleteness [\href{https://doi.org/10.1063/1.5052714}{Giner \textit{et al.}, \textit{J.~Chem.~Phys.}~\textbf{149}, 194301 (2018)}], we show that one can get chemically accurate vertical and adiabatic excitation energies with, typically, augmented double-$\zeta$ basis sets.
We illustrate the present approach on various types of excited states (valence, Rydberg, and double excitations) in several small organic molecules (methylene, water, ammonia, carbon dimer and ethylene). We illustrate the present approach on various types of excited states (valence, Rydberg, and double excitations) in several small organic molecules (methylene, water, ammonia, carbon dimer and ethylene).
The present study clearly evidences that special care has to be taken with very diffuse excited states where the present correction \toto{does not} catch the radial incompleteness of the one-electron basis set. The present study clearly evidences that special care has to be taken with very diffuse excited states where the present correction does not catch the radial incompleteness of the one-electron basis set.
\end{abstract} \end{abstract}
\maketitle \maketitle
@ -192,7 +192,7 @@ The overall basis-set incompleteness error can be, qualitatively at least, split
Although for ground-state properties angular incompleteness is by far the main source of error, it is definitely not unusual to have a significant radial incompleteness in the case of excited states (especially for Rydberg states), which can be alleviated by using additional sets of diffuse basis functions (i.e.~augmented basis sets). Although for ground-state properties angular incompleteness is by far the main source of error, it is definitely not unusual to have a significant radial incompleteness in the case of excited states (especially for Rydberg states), which can be alleviated by using additional sets of diffuse basis functions (i.e.~augmented basis sets).
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} 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 \toto{performance} 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} has been much more conflicting. \cite{FliHatKlo-JCP-06, NeiHatKlo-JCP-06} 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 performance 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} has 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 performance of the recently proposed density-based basis set Instead of F12 methods, here we propose to follow a different route and investigate the performance of the recently proposed density-based basis set
incompleteness correction. \cite{GinPraFerAssSavTou-JCP-18} incompleteness correction. \cite{GinPraFerAssSavTou-JCP-18}
@ -237,7 +237,7 @@ is the basis-dependent complementary density functional,
& &
\hWee{} & = \sum_{i<j}^{\Ne} r_{ij}^{-1}, \hWee{} & = \sum_{i<j}^{\Ne} r_{ij}^{-1},
\end{align} \end{align}
are the kinetic and electron-electron repulsion operators, respectively, and $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert spaces spanned by $\Bas$ \titou{and in the CBS limit}, respectively. are the kinetic and electron-electron repulsion operators, respectively, and $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert spaces spanned by $\Bas$ and the complete basis, respectively.
The notation $\wf{}{} \rightsquigarrow \n{}{}$ in Eq.~\eqref{eq:E_funcbasis} states that $\wf{}{}$ yields the one-electron density $\n{}{}$. The notation $\wf{}{} \rightsquigarrow \n{}{}$ in Eq.~\eqref{eq:E_funcbasis} states that $\wf{}{}$ yields the one-electron density $\n{}{}$.
Hence, the CBS excitation energy associated with the $k$th excited state reads Hence, the CBS excitation energy associated with the $k$th excited state reads
@ -414,7 +414,7 @@ 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. 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. 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. 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 densities and on-top pair densities are computed from a very large CIPSI expansion containing up to several \toto{million} of Slater determinants. The one-electron densities and on-top pair densities are computed from a very large CIPSI expansion containing up to several million of Slater determinants.
All the RS-DFT and exFCI calculations have been performed with {\QP}. \cite{QP2} 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} 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 molecular 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 molecular 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.
@ -536,28 +536,28 @@ This trend is quite systematic as we shall see below.
& $0.393$ & $0.393$
& $1.398$ & $1.398$
& $2.516$ \\ & $2.516$ \\
CR-EOMCC (2,3)D\fnm[2]& AVQZ CR-EOMCC (2,3)D\fnm[2]& AV5Z
& $0.412$ & $0.430$
& $1.460$ & $1.464$
& $2.547$ \\ & $2.633$ \\
FCI\fnm[3] & TZ2P FCI\fnm[3] & TZ2P
& $0.483$ & $0.483$
& $1.542$ & $1.542$
& $2.674$ \\ & $2.674$ \\
DMC\fnm[4] & DMC\fnm[4] & CAS(6,6)
& $0.406$ & $0.406$
& $1.416$ & $1.416$
& $2.524$ \\ & $2.524$ \\
Exp.\fnm[5] & Exp.\fnm[5] &
& $0.400$ & $0.406$
& $1.411$ & $1.415$
\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]{Completely-renormalized equation-of-motion coupled cluster (CR-EOMCC) calculations from Refs.~\onlinecite{GouPieWlo-MP-10}.} \fnt[2]{Completely-renormalized equation-of-motion coupled cluster (CR-EOMCC) calculations from Refs.~\onlinecite{GouPieWlo-MP-10}.}
\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]{Experimentally-derived values. See footnotes of Table II from Ref.~\onlinecite{GouPieWlo-MP-10} for additional details.}
\end{table} \end{table}
\end{squeezetable} \end{squeezetable}
%%% %%% %%% %%% %%% %%%
@ -583,10 +583,10 @@ Table \ref{tab:Mol} reports vertical excitation energies for various singlet and
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. 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 effect is even 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 effect is even magnified.
\titou{There is substantial error remaining for AVQZ.} In other words, substantial error remains in these cases even with the largest AVQZ basis set.
In these cases, one really needs doubly 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 behaviors 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 \toto{cannot} catch the radial incompleteness of the one-electron basis set, a feature which is far from being a cusp-related effect. However, these results also clearly evidence that special care has to be taken for very diffuse excited states where the present correction cannot catch the radial incompleteness of the one-electron basis set, a feature which is far from being a cusp-related effect.
%%% TABLE 2 %%% %%% TABLE 2 %%%
@ -839,8 +839,7 @@ See {\SI} for geometries and additional information (including total energies an
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\begin{acknowledgements} \begin{acknowledgements}
PFL would like to thank Denis Jacquemin for numerous discussions on excited states. 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), CALMIP (Toulouse) under allocation 2019-18005 and the Obelix cluster from the \textit{Institut Parisien de Chimie Physique et Th\'eorique}.
\titou{We thank also IP2CT for Obelix cluster.}
\end{acknowledgements} \end{acknowledgements}
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