\documentclass[aip,jcp,preprint,noshowkeys]{revtex4-1} \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,xspace,float} \usepackage{mathpazo,libertine} \usepackage{natbib} \bibliographystyle{achemso} \AtBeginDocument{\nocite{achemso-control}} \newcommand{\alert}[1]{\textcolor{red}{#1}} \definecolor{darkgreen}{HTML}{009900} \usepackage[normalem]{ulem} \newcommand{\titou}[1]{\textcolor{red}{#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{\fnm}{\footnotemark} \newcommand{\fnt}{\footnotetext} \newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} \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{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} \newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Sorbonne Universit\'e, CNRS, Paris, France} \newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France} \begin{document} \title{A Density-Based Basis-Set Correction For Wave Function Theory} \author{Pierre-Fran\c{c}ois Loos} \email{loos@irsamc.ups-tlse.fr} \affiliation{\LCPQ} \author{Bath\'elemy Pradines} \affiliation{\LCT} \affiliation{\ISCD} \author{Anthony Scemama} \affiliation{\LCPQ} \author{Julien Toulouse} \email{toulouse@lct.jussieu.fr} \affiliation{\LCT} \author{Emmanuel Giner} \email{emmanuel.giner@lct.jussieu.fr} \affiliation{\LCT} \begin{abstract} We report a universal density-based basis-set incompleteness correction that can be applied to any wave function method. The present correction, which appropriately vanishes in the complete basis set (CBS) limit, relies on short-range correlation density functionals (with multi-determinant reference) from range-separated density-functional theory (RS-DFT) to estimate the basis-set incompleteness error. Contrary to conventional RS-DFT schemes which require an \textit{ad hoc} range-separation \textit{parameter} $\mu$, the key ingredient here is a range-separation \textit{function} $\mu(\bf{r})$ that automatically adapts to the spatial non-homogeneity of the basis-set incompleteness error. As illustrative examples, we show how this density-based correction allows us to obtain CCSD(T) atomization and correlation energies near the CBS limit for the G2 set of molecules with compact Gaussian basis sets. \begin{figure}[H] \centering \includegraphics[width=0.5\linewidth]{TOC} \\ \bf TOC Graphic \end{figure} \end{abstract} \maketitle %%%%%%%%%%%%%%%%%%%%%%%% %\section{Introduction} %%%%%%%%%%%%%%%%%%%%%%%% 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.~\onlinecite{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{Theory} %%%%%%%%%%%%%%%%%%%%%%%% 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. Here, we only provide the main working equations. We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation. %Let us assume we have both the energy $\E{\modY}{\Bas}$ and density $\n{\modZ}{\Bas}$ of a $\Ne$-electron system described by two methods $\modY$ and $\modZ$ (potentially identical) in an incomplete basis set $\Bas$. %According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modY}{\Bas}$ and $\n{\modZ}{\Bas}$ are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be approximated as Let us assume \titou{that we have reasonable approximations of the FCI energy and density of a $\Ne$-electron system in an incomplete basis set $\Bas$, say the CCSD(T) energy $\E{\CCSDT}{\Bas}$ and the Hartree-Fock (HF) density $\n{\HF}{\Bas}$. According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, the exact ground-state energy $\E{}{}$ may be approximated as} %we have both the energy \titou{$\E{\CCSDT}{\Bas}$ and density $\n{\HF}{\Bas}$ of a $\Ne$-electron system in an incomplete basis set $\Bas$.} %According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that \titou{$\E{\CCSDT}{\Bas}$ and $\n{\HF}{\Bas}$} are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be approximated as \begin{equation} \label{eq:e0basis} \titou{\E{}{} \approx \E{\CCSDT}{\Bas} + \bE{}{\Bas}[\n{\HF}{\Bas}],} \end{equation} where \begin{equation} \label{eq:E_funcbasis} \bE{}{\Bas}[\n{}{}] = \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}} - \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}} \end{equation} is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i