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Pierre-Francois Loos 2019-05-19 22:05:50 +02:00
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Manuscript/Ex-srDFT.tex
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\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1} \documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable} \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
\usepackage{natbib}
\bibliographystyle{achemso}
\AtBeginDocument{\nocite{achemso-control}}
\usepackage{mathpazo,libertine} \usepackage{mathpazo,libertine}
\usepackage[normalem]{ulem}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\definecolor{darkgreen}{RGB}{0, 180, 0}
\newcommand{\beurk}[1]{\textcolor{darkgreen}{#1}}
\newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}}
\usepackage{hyperref} \usepackage{hyperref}
\hypersetup{ \hypersetup{
@ -16,19 +14,122 @@
urlcolor=blue, urlcolor=blue,
citecolor=blue citecolor=blue
} }
\newcommand{\cdash}{\multicolumn{1}{c}{---}} \newcommand{\alert}[1]{\textcolor{red}{#1}}
\definecolor{darkgreen}{HTML}{009900}
\usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{black}{#1}}
\newcommand{\jt}[1]{\textcolor{purple}{#1}}
\newcommand{\manu}[1]{\textcolor{darkgreen}{#1}}
\newcommand{\toto}[1]{\textcolor{brown}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\trashJT}[1]{\textcolor{purple}{\sout{#1}}}
\newcommand{\trashMG}[1]{\textcolor{darkgreen}{\sout{#1}}}
\newcommand{\trashAS}[1]{\textcolor{brown}{\sout{#1}}}
\newcommand{\MG}[1]{\manu{(\underline{\bf MG}: #1)}}
\newcommand{\JT}[1]{\juju{(\underline{\bf JT}: #1)}}
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
\newcommand{\AS}[1]{\toto{(\underline{\bf TOTO}: #1)}}
\usepackage{hyperref}
\hypersetup{
colorlinks=true,
linkcolor=blue,
filecolor=blue,
urlcolor=blue,
citecolor=blue
}
\newcommand{\mc}{\multicolumn} \newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark} \newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext} \newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} \newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\mr}{\multirow}
\newcommand{\SI}{\textcolor{blue}{supporting information}} \newcommand{\SI}{\textcolor{blue}{supporting information}}
\newcommand{\QP}{\textsc{quantum package}}
% second quantized operators
\newcommand{\ai}[1]{\hat{a}_{#1}}
\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}}
% units
\newcommand{\IneV}[1]{#1 eV}
\newcommand{\InAU}[1]{#1 a.u.}
\newcommand{\InAA}[1]{#1 \AA}
\newcommand{\kcal}{kcal/mol}
% methods
\newcommand{\D}{\text{D}}
\newcommand{\T}{\text{T}}
\newcommand{\Q}{\text{Q}}
\newcommand{\X}{\text{X}}
\newcommand{\UEG}{\text{UEG}}
\newcommand{\HF}{\text{HF}}
\newcommand{\ROHF}{\text{ROHF}}
\newcommand{\LDA}{\text{LDA}}
\newcommand{\PBE}{\text{PBE}}
\newcommand{\FCI}{\text{FCI}}
\newcommand{\CBS}{\text{CBS}}
\newcommand{\exFCI}{\text{exFCI}}
\newcommand{\CCSDT}{\text{CCSD(T)}}
\newcommand{\lr}{\text{lr}}
\newcommand{\sr}{\text{sr}}
\newcommand{\Ne}{N}
\newcommand{\NeUp}{\Ne^{\uparrow}}
\newcommand{\NeDw}{\Ne^{\downarrow}}
\newcommand{\Nb}{N_{\Bas}}
\newcommand{\Ng}{N_\text{grid}}
\newcommand{\nocca}{n_{\text{occ}^{\alpha}}}
\newcommand{\noccb}{n_{\text{occ}^{\beta}}}
\newcommand{\n}[2]{n_{#1}^{#2}}
\newcommand{\Ec}{E_\text{c}}
\newcommand{\E}[2]{E_{#1}^{#2}}
\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
\newcommand{\bEc}[1]{\Bar{E}_\text{c,md}^{#1}}
\newcommand{\e}[2]{\varepsilon_{#1}^{#2}}
\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
\newcommand{\bec}[1]{\Bar{e}^{#1}}
\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
\newcommand{\W}[2]{W_{#1}^{#2}}
\newcommand{\w}[2]{w_{#1}^{#2}}
\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
\newcommand{\V}[2]{V_{#1}^{#2}}
\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
\newcommand{\modY}{Y}
\newcommand{\modZ}{Z}
% basis sets
\newcommand{\Bas}{\mathcal{B}}
\newcommand{\BasFC}{\mathcal{A}}
\newcommand{\FC}{\text{FC}}
\newcommand{\occ}{\text{occ}}
\newcommand{\virt}{\text{virt}}
\newcommand{\val}{\text{val}}
\newcommand{\Cor}{\mathcal{C}}
% operators
\newcommand{\hT}{\Hat{T}}
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
\newcommand{\updw}{\uparrow\downarrow}
\newcommand{\f}[2]{f_{#1}^{#2}}
\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
% coordinates
\newcommand{\br}[1]{\mathbf{r}_{#1}}
\newcommand{\dbr}[1]{d\br{#1}}
\newcommand{\ra}{\rightarrow}
% frozen core
\newcommand{\WFC}[2]{\widetilde{W}_{#1}^{#2}}
\newcommand{\fFC}[2]{\widetilde{f}_{#1}^{#2}}
\newcommand{\rsmuFC}[2]{\widetilde{\mu}_{#1}^{#2}}
\newcommand{\nFC}[2]{\widetilde{n}_{#1}^{#2}}
\newcommand{\br}{\mathbf{r}}
% energies % energies
\newcommand{\EHF}{E_\text{HF}} \newcommand{\EHF}{E_\text{HF}}
\newcommand{\Ec}{E_\text{c}}
\newcommand{\EPT}{E_\text{PT2}} \newcommand{\EPT}{E_\text{PT2}}
\newcommand{\EFCI}{E_\text{FCI}} \newcommand{\EFCI}{E_\text{FCI}}
\newcommand{\EsCI}{E_\text{sCI}} \newcommand{\EsCI}{E_\text{sCI}}
@ -39,19 +140,12 @@
\newcommand{\Eabs}{\Delta E_\text{abs}} \newcommand{\Eabs}{\Delta E_\text{abs}}
\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$} \newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$}
\newcommand{\ra}{\rightarrow}
% units
\newcommand{\IneV}[1]{#1 eV}
\newcommand{\InAU}[1]{#1 a.u.}
\newcommand{\InAA}[1]{#1 \AA}
\newcommand{\pis}{\pi^\star} \newcommand{\pis}{\pi^\star}
\newcommand{\si}{\sigma} \newcommand{\si}{\sigma}
\newcommand{\sis}{\sigma^\star} \newcommand{\sis}{\sigma^\star}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France} \newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}
@ -70,7 +164,7 @@
\affiliation{\LCPQ} \affiliation{\LCPQ}
\begin{abstract} \begin{abstract}
By combining extrapolated selected configuration interaction (sCI) calculations performed with the CIPSI algorithm with the recently proposed short-range density-functional functional correction for basis set incompleteness [\href{https://doi.org/10.1063/1.5052714}{Giner et al., J.~Chem.~Phys.~149, 194301 (2018)}], we show that one can obtain vertical and adiabatic excitation energies with chemical accuracy with a small basis set. By combining extrapolated selected configuration interaction (sCI) calculations performed with the CIPSI algorithm with the recently proposed short-range density-functional functional correction for basis set incompleteness [\href{https://doi.org/10.1063/1.5052714}{Giner et al., \textit{J.~Chem.~Phys.}~\textbf{149}, 194301 (2018)}], we show that one can obtain vertical and adiabatic excitation energies with chemical accuracy with a small basis set.
\end{abstract} \end{abstract}
\maketitle \maketitle
@ -80,18 +174,54 @@ By combining extrapolated selected configuration interaction (sCI) calculations
\label{sec:intro} \label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
One of the most fundamental problem of conventional electronic structure methods is their slow energy convergence with respect to the size of the one-electron basis set. One of the most fundamental problem of conventional electronic structure methods is their slow energy convergence with respect to the size of the one-electron basis set.
This problem was already noticed thirty years ago by Kutzelnigg \cite{Kutzelnigg_1985} who proposed to introduce explicitly the correlation between electrons via the introduction of the interelectronic distance $r_{12} = \abs{\br_1 - \br_2}$ as a basis function. \cite{Kutzelnigg_1991, Termath_1991, Klopper_1991a, Klopper_1991b, Noga_1994}
This yields a prominent improvement of the energy convergence from $O(L^{-3})$ to $O(L^{-7})$ (where $L$ is the maximum angular momentum of the one-electron basis).
This idea was later generalised to more accurate correlation factors $f_{12} \equiv f(r_{12})$. \cite{Persson_1996, Persson_1997, May_2004, Tenno_2004b, Tew_2005, May_2005}
The resulting F12 methods achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Tenno_2012a, Tenno_2012b, Hattig_2012, Kong_2012}
For example, as illustrated by Tew and coworkers, one can obtain, at the CCSD(T) level, quintuple-zeta quality correlation energies with a triple-zeta basis. \cite{Tew_2007b}
In the present study, we rely on the recently proposed short-range density-functional functional correction for basis set incompleteness. \cite{Giner_2018} In the present study, we rely on the recently proposed short-range density-functional functional correction for basis set incompleteness. \cite{GinPraFerAssSavTou-JCP-18}
%Contemporary quantum chemistry has developed in two directions --- wave function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99}
%Although both spring from the same Schr\"odinger equation, each of these philosophies has its own \textit{pros} and \textit{cons}.
%
%WFT is attractive as it exists a well-defined path for systematic improvement as well as powerful tools, such as perturbation theory, to guide the development of new WFT \textit{ans\"atze}.
%The coupled cluster (CC) family of methods is a typical example of the WFT philosophy and is well regarded as the gold standard of quantum chemistry for weakly correlated systems.
%By increasing the excitation degree of the CC expansion, one can systematically converge, for a given basis set, to the exact, full configuration interaction (FCI) limit, although the computational cost associated with such improvement is usually high.
%One of the most fundamental drawbacks of conventional WFT methods is the slow convergence of energies and properties with respect to the size of the one-electron basis set.
%This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kut-TCA-85}
%To palliate this, following Hylleraas' footsteps, \cite{Hyl-ZP-29} Kutzelnigg proposed to introduce explicitly the interelectronic distance $r_{12} = \abs{\br{1} - \br{2}}$ to properly describe the electronic wave function around the coalescence of two electrons. \cite{Kut-TCA-85, KutKlo-JCP-91, NogKut-JCP-94}
%The resulting F12 methods yield a prominent improvement of the energy convergence, and achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18}
%For example, at the CCSD(T) level, one can obtain quintuple-$\zeta$ quality correlation energies with a triple-$\zeta$ basis, \cite{TewKloNeiHat-PCCP-07} although computational overheads are introduced by the large auxiliary basis used to resolve three- and four-electron integrals. \cite{BarLoo-JCP-17}
%To reduce further the computational cost and/or ease the transferability of the F12 correction, approximated and/or universal schemes have recently emerged. \cite{TorVal-JCP-09, KonVal-JCP-10, KonVal-JCP-11, BooCleAlaTew-JCP-2012, IrmHumGru-arXiv-2019, IrmGru-arXiv-2019}
%
%Present-day DFT calculations are almost exclusively done within the so-called Kohn-Sham (KS) formalism, which corresponds to an exact dressed one-electron theory. \cite{KohSha-PR-65}
%The attractiveness of DFT originates from its very favorable accuracy/cost ratio as it often provides reasonably accurate energies and properties at a relatively low computational cost.
%Thanks to this, KS-DFT \cite{HohKoh-PR-64, KohSha-PR-65} has become the workhorse of electronic structure calculations for atoms, molecules and solids. \cite{ParYan-BOOK-89}
%Although there is no clear way on how to systematically improve density-functional approximations, \cite{Bec-JCP-14} climbing Perdew's ladder of DFT is potentially the most satisfactory way forward. \cite{PerSch-AIPCP-01, PerRuzTaoStaScuCso-JCP-05}
%In the context of the present work, one of the interesting feature of density-based methods is their much faster convergence with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15}
%
%Progress toward unifying WFT and DFT are on-going.
%In particular, range-separated DFT (RS-DFT) (see Ref.~\citenum{TouColSav-PRA-04} and references therein) rigorously combines these two approaches via a decomposition of the electron-electron (e-e) interaction into a non-divergent long-range part and a (complementary) short-range part treated with WFT and DFT, respectively.
%As the WFT method is relieved from describing the short-range part of the correlation hole around the e-e coalescence points, the convergence with respect to the one-electron basis set is greatly improved. \cite{FraMusLupTou-JCP-15}
%Therefore, a number of approximate RS-DFT schemes have been developed within single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT approaches.
%Very recently, a major step forward has been taken by some of the present authors thanks to the development of a density-based basis-set correction for WFT methods. \cite{GinPraFerAssSavTou-JCP-18}
%The present work proposes an extension of this new methodological development alongside the first numerical tests on molecular systems.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details} \section{Computational details}
\label{sec:compdetails} \label{sec:compdetails}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
The present basis-set correction relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the incompleteness of the one-electron basis set.
The present methodology is identical to the one described in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} where the main working equation are reported and discussed.
We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.
exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
We refer the interested reader to Refs.~\citenum{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
The one-electron density and on-top density is 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}
The geometries have been extracted from Refs.~\citenum{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJAc-JCTC-19} and have been obtained at the CC3/aug-cc-pVTZ level of theory.
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 FC density-based correction is used consistently with the FC approximation in WFT methods.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
@ -353,9 +483,9 @@ In the present study, we rely on the recently proposed short-range density-funct
\\ \\
\hline \hline
Acetylene & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Sigma_{u}^{-}$ & Val. & 7.10 & 0.10 & 0.00 Acetylene & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Sigma_{u}^{-}$ & Val. & 7.10 & 0.10 & 0.00
& 0.07 & & 0.07 & 0.00
& 0.11 & & 0.11 & 0.00
& 0.11 & & 0.11 & 0.00
\\ \\
& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Delta_{u}$ & Val. & 7.44 & 0.07 & 0.00 & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Delta_{u}$ & Val. & 7.44 & 0.07 & 0.00
& 0.04 & & 0.04 &
@ -363,13 +493,13 @@ In the present study, we rely on the recently proposed short-range density-funct
& 0.11 & & 0.11 &
\\ \\
& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{+}$ & Val. & 5.56 & -0.06 & -0.03 & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{+}$ & Val. & 5.56 & -0.06 & -0.03
& 0.07 & & 0.07 & 0.02
& 0.04 & & 0.04 & 0.00
& 0.02 & & 0.02 & 0.00
\\ \\
& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Delta_{u}$ & Val. & 6.40 & 0.06 & 0.00 & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Delta_{u}$ & Val. & 6.40 & 0.06 & 0.00
& & & 0.1 &
& & & 0.14 &
& & & &
\\ \\
& $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{-}$ & Val. & 7.09 & 0.05 & -0.01 & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{-}$ & Val. & 7.09 & 0.05 & -0.01
@ -457,20 +587,19 @@ In the present study, we rely on the recently proposed short-range density-funct
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting Information} \section*{Supporting Information Available}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
See {\SI} for geometries and additional information (including total energies). See {\SI} for geometries and additional information (including total energies).
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements} \begin{acknowledgements}
This work was performed using HPC resources from The authors would like to thank the \textit{Centre National de la Recherche Scientifique} (CNRS) for funding.
i) GENCI-TGCC (Grant No. 2018-A0040801738), This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
ii) CALMIP (Toulouse) under allocations 2018-0510 and 2018-12158.
\end{acknowledgements} \end{acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{Ex-srDFT} \bibliography{Ex-srDFT,Ex-srDFT-control}
\end{document} \end{document}