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\newcommand{\SI}{\textcolor{blue}{supporting information}}
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\newcommand{\QP}{\textsc{quantum package}}
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% second quantized operators
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\newcommand{\ai}[1]{\hat{a}_{#1}}
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\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}}
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% units
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\newcommand{\IneV}[1]{#1 eV}
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\newcommand{\InAU}[1]{#1 a.u.}
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\newcommand{\InAA}[1]{#1 \AA}
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\newcommand{\kcal}{kcal/mol}
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% methods
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\newcommand{\D}{\text{D}}
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\newcommand{\T}{\text{T}}
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\newcommand{\Q}{\text{Q}}
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\newcommand{\X}{\text{X}}
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\newcommand{\UEG}{\text{UEG}}
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\newcommand{\HF}{\text{HF}}
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\newcommand{\ROHF}{\text{ROHF}}
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\newcommand{\LDA}{\text{LDA}}
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\newcommand{\PBE}{\text{PBE}}
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\newcommand{\FCI}{\text{FCI}}
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\newcommand{\CBS}{\text{CBS}}
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\newcommand{\exFCI}{\text{exFCI}}
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\newcommand{\CCSDT}{\text{CCSD(T)}}
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\newcommand{\lr}{\text{lr}}
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\newcommand{\sr}{\text{sr}}
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\newcommand{\Ne}{N}
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\newcommand{\NeUp}{\Ne^{\uparrow}}
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\newcommand{\NeDw}{\Ne^{\downarrow}}
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\newcommand{\Nb}{N_{\Bas}}
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\newcommand{\Ng}{N_\text{grid}}
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\newcommand{\nocca}{n_{\text{occ}^{\alpha}}}
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\newcommand{\noccb}{n_{\text{occ}^{\beta}}}
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\newcommand{\n}[2]{n_{#1}^{#2}}
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\E}[2]{E_{#1}^{#2}}
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\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
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\newcommand{\bEc}[1]{\Bar{E}_\text{c,md}^{#1}}
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\newcommand{\e}[2]{\varepsilon_{#1}^{#2}}
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\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
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\newcommand{\bec}[1]{\Bar{e}^{#1}}
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\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
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\newcommand{\W}[2]{W_{#1}^{#2}}
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\newcommand{\w}[2]{w_{#1}^{#2}}
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\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
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\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
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\newcommand{\V}[2]{V_{#1}^{#2}}
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\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
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\newcommand{\modY}{Y}
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\newcommand{\modZ}{Z}
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% basis sets
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\newcommand{\Bas}{\mathcal{B}}
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\newcommand{\BasFC}{\mathcal{A}}
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\newcommand{\FC}{\text{FC}}
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\newcommand{\occ}{\text{occ}}
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\newcommand{\virt}{\text{virt}}
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\newcommand{\val}{\text{val}}
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\newcommand{\Cor}{\mathcal{C}}
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% operators
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\newcommand{\hT}{\Hat{T}}
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\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
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\newcommand{\updw}{\uparrow\downarrow}
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\newcommand{\f}[2]{f_{#1}^{#2}}
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\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
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% coordinates
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\newcommand{\br}[1]{\mathbf{r}_{#1}}
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\newcommand{\dbr}[1]{d\br{#1}}
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\newcommand{\ra}{\rightarrow}
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% frozen core
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\newcommand{\WFC}[2]{\widetilde{W}_{#1}^{#2}}
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\newcommand{\fFC}[2]{\widetilde{f}_{#1}^{#2}}
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\newcommand{\rsmuFC}[2]{\widetilde{\mu}_{#1}^{#2}}
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\newcommand{\nFC}[2]{\widetilde{n}_{#1}^{#2}}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Sorbonne Universit\'e, CNRS, Paris, France}
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\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France}
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\begin{document}
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\title{A Density-Based Basis-Set Correction For Wave Function Theory}
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\author{Pierre-Fran\c{c}ois Loos}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\author{Bath\'elemy Pradines}
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\affiliation{\LCT}
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\affiliation{\ISCD}
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\author{Anthony Scemama}
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\affiliation{\LCPQ}
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\author{Julien Toulouse}
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\email{toulouse@lct.jussieu.fr}
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\affiliation{\LCT}
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\author{Emmanuel Giner}
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\email{emmanuel.giner@lct.jussieu.fr}
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\affiliation{\LCT}
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\begin{abstract}
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\begin{wrapfigure}[7]{o}[-1.2cm]{0.4\linewidth}
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\centering
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\includegraphics[width=\linewidth]{TOC}
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\end{wrapfigure}
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We report a universal density-based basis-set incompleteness correction that can be applied to any wave function method.
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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.
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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.
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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.
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\end{abstract}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{Introduction}
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%%%%%%%%%%%%%%%%%%%%%%%%
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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}
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Although both spring from the same Schr\"odinger equation, each of these philosophies has its own \textit{pros} and \textit{cons}.
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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}.
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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.
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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.
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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.
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This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kut-TCA-85}
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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}
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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}
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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}
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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}
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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}
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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.
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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}
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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}
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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}
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Progress toward unifying WFT and DFT are on-going.
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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.
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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}
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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.
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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}
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The present work proposes an extension of this new methodological development alongside the first numerical tests on molecular systems.
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%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{Theory}
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%%%%%%%%%%%%%%%%%%%%%%%%
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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.
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Here, we only provide the main working equations.
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We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.
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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 $\Bas$.
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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
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\begin{equation}
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\label{eq:e0basis}
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\E{}{}
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\approx \E{\modY}{\Bas}
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+ \bE{}{\Bas}[\n{\modZ}{\Bas}],
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\end{equation}
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where
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\begin{equation}
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\label{eq:E_funcbasis}
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\bE{}{\Bas}[\n{}{}]
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= \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}}
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- \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
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\end{equation}
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is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
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In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set (CBS), respectively.
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Both wave functions yield the same target density $\n{}{}$ (assumed to be representable in $\Bas$).
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Importantly, in the CBS limit (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$.
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This implies that
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\begin{equation}
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\label{eq:limitfunc}
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\lim_{\Bas \to \infty} \qty( \E{\modY}{\Bas} + \bE{}{\Bas}[\n{\modZ}{\Bas}] ) = \E{\modY}{} \approx E,
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\end{equation}
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where $\E{\modY}{}$ is the energy associated with the method $\modY$ in the CBS limit.
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In the case where $\modY = \FCI$ in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
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Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only sources of error at this stage lie in the potential approximate nature of the methods $\modY$ and $\modZ$, and the lack of self-consistency in the present scheme.
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The functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
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Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct
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for the lack of cusp (i.e.~discontinuous derivative) in $\wf{}{\Bas}$ at the e-e coalescence points, a universal condition of exact wave functions.
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Because the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent two-electron interaction at coalescence.
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Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ by a short-range density functional which is complementary to a non-divergent long-range interaction.
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Contrary to the conventional RS-DFT scheme which requires a range-separation \textit{parameter} $\rsmu{}{}$, here we use a range-separation \textit{function} $\rsmu{}{\Bas}(\br{})$ that automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.
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% https://english.stackexchange.com/questions/61600/consist-in-vs-consist-of
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The first step of the present basis-set correction consists in obtaining an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$.
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In a second step, we shall link $\W{}{\Bas}(\br{1},\br{2})$ to $\rsmu{}{\Bas}(\br{})$.
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As a final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{}{\Bas}(\br{})$ as range-separation function.
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%=================================================================
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%\subsection{Effective Coulomb operator}
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%=================================================================
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We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18}
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\begin{equation}
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\label{eq:def_weebasis}
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\W{}{\Bas}(\br{1},\br{2}) =
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\begin{cases}
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\f{}{\Bas}(\br{1},\br{2})/\n{2}{\Bas}(\br{1},\br{2}), & \text{if $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$,}
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\\
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\infty, & \text{otherwise,}
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\end{cases}
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\end{equation}
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where
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\begin{equation}
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\label{eq:n2basis}
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\n{2}{\Bas}(\br{1},\br{2})
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= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
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\end{equation}
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and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\uparrow}\ai{q_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO),
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\begin{equation}
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\label{eq:fbasis}
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\f{}{\Bas}(\br{1},\br{2})
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= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
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\end{equation}
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and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals.
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With such a definition, $\W{}{\Bas}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
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\begin{equation}
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\iint \frac{ \n{2}{\Bas}(\br{1},\br{2})}{r_{12}} \dbr{1} \dbr{2} =
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\iint \W{}{\Bas}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{equation}
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which intuitively motivates $\W{}{\Bas}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
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Note that the divergence condition of $\W{}{\Bas}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems are free of correction as the present approach must only correct the basis-set incompleteness error originating from the e-e cusp.
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As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{}{\Bas}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational, nor rotational invariant if $\Bas$ does not have such symmetries.
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Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
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\begin{equation}
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\label{eq:lim_W}
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\lim_{\Bas \to \infty}\W{}{\Bas}(\br{1},\br{2}) = r_{12}^{-1},
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\end{equation}
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for any $(\br{1},\br{2})$ such that $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$.
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%=================================================================
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%\subsection{Range-separation function}
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%=================================================================
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A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons, $\W{}{\Bas}(\br{},{\br{}})$,
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which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{\Bas}(\br{},\br{})$ is non vanishing.
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Because $\W{}{\Bas}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be naturally linked to RS-DFT which employs a non-divergent long-range interaction operator.
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Although this choice is not unique, we choose here the range-separation function
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\begin{equation}
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\label{eq:mu_of_r}
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\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{}),
|
||||
\end{equation}
|
||||
such that the long-range interaction of RS-DFT, $\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$, coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any $\br{}$.
|
||||
|
||||
%=================================================================
|
||||
%\subsection{Short-range correlation functionals}
|
||||
%=================================================================
|
||||
Once $\rsmu{}{\Bas}(\br{})$ is defined, it can be used within RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$.
|
||||
As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as correlation energy with multi-determinantal reference (ECMD) whose general definition reads \cite{TouGorSav-TCA-05}
|
||||
\begin{equation}
|
||||
\label{eq:ec_md_mu}
|
||||
\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]
|
||||
= \min_{\wf{}{} \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
|
||||
- \mel*{\wf{}{\rsmu{}{}}}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}},
|
||||
\end{equation}
|
||||
where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization
|
||||
\begin{equation}
|
||||
\label{eq:argmin}
|
||||
\wf{}{\rsmu{}{}} = \arg \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
|
||||
\end{equation}
|
||||
with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
|
||||
The ECMD functionals admit, for any $\n{}{}$, the following two limiting forms
|
||||
\begin{align}
|
||||
\label{eq:large_mu_ecmd}
|
||||
\lim_{\mu \to \infty} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
|
||||
&
|
||||
% \label{eq:small_mu_ecmd}
|
||||
\lim_{\mu \to 0} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
|
||||
\end{align}
|
||||
where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KS-DFT.
|
||||
The choice of ECMD in the present scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [Eq.~\eqref{eq:E_funcbasis}] and the ECMD functional [Eq.~\eqref{eq:ec_md_mu}].
|
||||
Indeed, the two functionals coincide if $\wf{}{\Bas} = \wf{}{\rsmu{}{}}$.
|
||||
Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by ECMD functionals evaluated with the range-separation function $\rsmu{}{\Bas}(\br{})$.
|
||||
|
||||
The local-density approximation (LDA) of the ECMD complementary functional is defined as
|
||||
\begin{equation}
|
||||
\label{eq:def_lda_tot}
|
||||
\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
|
||||
\end{equation}
|
||||
where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
|
||||
The short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$.
|
||||
In order to correct such a defect, inspired by the recent functional proposed by some of the authors~\cite{FerGinTou-JCP-18}, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional
|
||||
\begin{equation}
|
||||
\label{eq:def_pbe_tot}
|
||||
\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}] =
|
||||
\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
|
||||
\end{equation}
|
||||
where $s=\abs{\nabla \n{}{}}/\n{}{4/3}$ is the reduced density gradient.
|
||||
$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$, at $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
|
||||
\begin{subequations}
|
||||
\begin{gather}
|
||||
\label{eq:epsilon_cmdpbe}
|
||||
\be{\text{c,md}}{\sr,\PBE}(\n{}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta(\n{}{},s,\zeta) \rsmu{}{3} },
|
||||
\\
|
||||
\label{eq:beta_cmdpbe}
|
||||
\beta(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\UEG}(\n{}{},\zeta)}.
|
||||
\end{gather}
|
||||
\end{subequations}
|
||||
The difference between the ECMD functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe}-\eqref{eq:beta_cmdpbe} is that we approximate here 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.~\onlinecite{GorSav-PRA-06}.
|
||||
This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{\Bas}(\br{},\br{})$.
|
||||
|
||||
Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{\Bas}]$ is approximated by $\bE{\LDA}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
|
||||
|
||||
%=================================================================
|
||||
%\subsection{Frozen-core approximation}
|
||||
%=================================================================
|
||||
|
||||
As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a subset of MOs.
|
||||
We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ and $\BasFC$ are the sets of core and active MOs, respectively) and define the FC version of the effective interaction as
|
||||
\begin{equation}
|
||||
\label{eq:WFC}
|
||||
\WFC{}{\Bas}(\br{1},\br{2}) =
|
||||
\begin{cases}
|
||||
\fFC{}{\Bas}(\br{1},\br{2})/\nFC{2}{\Bas}(\br{1},\br{2}), & \text{if $\nFC{2}{\Bas}(\br{1},\br{2}) \ne 0$},
|
||||
\\
|
||||
\infty, & \text{otherwise,}
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
with
|
||||
\begin{subequations}
|
||||
\begin{gather}
|
||||
\label{eq:fbasisval}
|
||||
\fFC{}{\Bas}(\br{1},\br{2})
|
||||
= \sum_{pq \in \Bas} \sum_{rstu \in \BasFC} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
|
||||
\\
|
||||
\nFC{2}{\Bas}(\br{1},\br{2})
|
||||
= \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
|
||||
\end{gather}
|
||||
\end{subequations}
|
||||
and the corresponding FC range-separation function $\rsmuFC{}{\Bas}(\br{}) = (\sqrt{\pi}/2) \WFC{}{\Bas}(\br{},\br{})$.
|
||||
It is noteworthy that, within the present definition, $\WFC{}{\Bas}(\br{1},\br{2})$ still tends to the regular Coulomb interaction as $\Bas \to \infty$.
|
||||
|
||||
Defining $\nFC{\modZ}{\Bas}$ as the FC (i.e.~valence-only) one-electron density obtained with a method $\modZ$ in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\LDA}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$.
|
||||
|
||||
%=================================================================
|
||||
%\subsection{Computational considerations}
|
||||
%=================================================================
|
||||
The most computationally intensive task of the present approach is the evaluation of $\W{}{\Bas}(\br{},\br{})$ at each quadrature grid point.
|
||||
In the general case (i.e.~$\wf{}{\Bas}$ is a multi-determinant expansion), we compute this embarrassingly parallel step in $\order*{\Ng \Nb^4}$ computational cost with a memory requirement of $\order*{ \Ng \Nb^2}$, where $\Nb$ is the number of basis functions in $\Bas$.
|
||||
The computational cost can be reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ with no memory footprint when $\wf{}{\Bas}$ is a single Slater determinant.
|
||||
As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, this choice for $\wf{}{\Bas}$ already provides, for weakly correlated systems, a quantitative representation of the incompleteness of $\Bas$.
|
||||
Hence, we will stick to this choice throughout the present study.
|
||||
In our current implementation, the computational bottleneck is the four-index transformation to get the two-electron integrals in the MO basis which appear in Eqs.~\eqref{eq:n2basis} and \eqref{eq:fbasis}.
|
||||
Nevertheless, this step usually has to be performed for most correlated WFT calculations.
|
||||
Modern integral decomposition techniques (such as density fitting \cite{Whi-JCP-73}) or atomic-orbital-based algorithms could be employed to significantly speed up this step.
|
||||
|
||||
To conclude this section, we point out that, thanks to the definitions \eqref{eq:def_weebasis} and \eqref{eq:mu_of_r} as well as the properties \eqref{eq:lim_W} and \eqref{eq:large_mu_ecmd}, independently of the DFT functional, the present basis-set correction
|
||||
i) can be applied to any WFT method that provides an energy and a density,
|
||||
ii) does not correct one-electron systems, and
|
||||
iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a given WFT method.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\section{Results}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%% FIGURE 1 %%%
|
||||
\begin{figure*}
|
||||
\includegraphics[width=0.30\linewidth]{fig1a}
|
||||
\hspace{1cm}
|
||||
\includegraphics[width=0.30\linewidth]{fig1b}
|
||||
\\
|
||||
\includegraphics[width=0.30\linewidth]{fig1c}
|
||||
\hspace{1cm}
|
||||
\includegraphics[width=0.30\linewidth]{fig1d}
|
||||
\caption{
|
||||
Deviation (in \kcal) from CBS atomization energies of \ce{C2} (top left), \ce{O2} (top right), \ce{N2} (bottom left) and \ce{F2} (bottom right) obtained with various methods and basis sets.
|
||||
The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
|
||||
See {\SI} for raw data.
|
||||
\label{fig:diatomics}}
|
||||
\end{figure*}
|
||||
|
||||
%%% TABLE II %%%
|
||||
\begin{table}
|
||||
\caption{
|
||||
Statistical analysis (in \kcal) of the G2 atomization energies depicted in Fig.~\ref{fig:G2_Ec}.
|
||||
Mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS reference atomization energies.
|
||||
CA corresponds to the number of cases (out of 55) obtained with chemical accuracy.
|
||||
See {\SI} for raw data.
|
||||
\label{tab:stats}}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{ldddd}
|
||||
Method & \tabc{MAD} & \tabc{RMSD} & \tabc{MAX} & \tabc{CA} \\
|
||||
\hline
|
||||
CCSD(T)/cc-pVDZ & 14.29 & 16.21 & 36.95 & 2 \\
|
||||
CCSD(T)/cc-pVTZ & 6.06 & 6.84 & 14.25 & 2 \\
|
||||
CCSD(T)/cc-pVQZ & 2.50 & 2.86 & 6.75 & 9 \\
|
||||
CCSD(T)/cc-pV5Z & 1.28 & 1.46 & 3.46 & 21 \\
|
||||
\\
|
||||
CCSD(T)+LDA/cc-pVDZ & 3.24 & 3.67 & 8.13 & 7 \\
|
||||
CCSD(T)+LDA/cc-pVTZ & 1.19 & 1.49 & 4.67 & 27 \\
|
||||
CCSD(T)+LDA/cc-pVQZ & 0.33 & 0.44 & 1.32 & 53 \\
|
||||
\\
|
||||
CCSD(T)+PBE/cc-pVDZ & 1.96 & 2.59 & 7.33 & 19 \\
|
||||
CCSD(T)+PBE/cc-pVTZ & 0.85 & 1.11 & 2.64 & 36 \\
|
||||
CCSD(T)+PBE/cc-pVQZ & 0.31 & 0.42 & 1.16 & 53 \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
|
||||
%%% FIGURE 2 %%%
|
||||
\begin{figure*}
|
||||
\includegraphics[width=\linewidth]{fig2a}
|
||||
\includegraphics[width=\linewidth]{fig2b}
|
||||
\includegraphics[width=\linewidth]{fig2c}
|
||||
\caption{
|
||||
Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
|
||||
The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
|
||||
See {\SI} for raw data.
|
||||
\label{fig:G2_Ec}}
|
||||
\end{figure*}
|
||||
|
||||
We begin our investigation of the performance of the basis-set correction by computing the atomization energies of \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} obtained with Dunning's cc-pVXZ basis (X $=$ D, T, Q and 5).
|
||||
\ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2 set \cite{CurRagTruPop-JCP-91} (see below), whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11}
|
||||
In a second time, we compute the atomization energies of the entire G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ basis set family.
|
||||
This molecular set has been intensively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08, Gro-JCP-09, FelPet-JCP-09, NemTowNee-JCP-10, FelPetHil-JCP-11, HauKlo-JCP-12, PetTouUmr-JCP-12, FelPet-JCP-13, KesSylKohTewMar-JCP-18}) and can be considered as a representative set of small organic and inorganic molecules.
|
||||
As a method $\modY$ we employ either CCSD(T) or exFCI.
|
||||
Here, 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.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
|
||||
In the case of the CCSD(T) calculations, we have $\modZ = \ROHF$ as we use the restricted open-shell HF (ROHF) one-electron density to compute the complementary basis-set correction energy.
|
||||
In the case of exFCI, the one-electron density is computed from a very large CIPSI expansion containing several million determinants.
|
||||
CCSD(T) energies are computed with Gaussian09 using standard threshold values, \cite{g09} while RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}
|
||||
For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17}
|
||||
Apart from the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-09} and have been obtained at the B3LYP/6-31G(2df,p) level of theory.
|
||||
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}.
|
||||
In the context of the basis-set correction, the set of active MOs, $\BasFC$, involved in the definition of the effective interaction [see Eq.~\eqref{eq:WFC}] refers to the non-frozen MOs.
|
||||
The FC density-based correction is used consistently with the FC approximation in WFT methods.
|
||||
To estimate the CBS limit of each method, following Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98}, we perform a two-point X$^{-3}$ extrapolation of the correlation energies using the quadruple- and quintuple-$\zeta$ data that we add up to the HF energies obtained in the largest (i.e.~quintuple-$\zeta$) basis.
|
||||
|
||||
As the exFCI atomization energies are converged with a precision of about 0.1 {\kcal}, we can label these as near FCI.
|
||||
Hence, they will be our references for \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2}.
|
||||
The results for these diatomic molecules are reported in Fig.~\ref{fig:diatomics}.
|
||||
The corresponding numerical data can be found in the {\SI}.
|
||||
As one can see, the convergence of the exFCI atomization energies is, as expected, slow with respect to the basis set: chemical accuracy (error below 1 {\kcal}) is barely reached for \ce{C2}, \ce{O2} and \ce{F2} even with the cc-pV5Z basis set, and the atomization energies are consistently underestimated.
|
||||
A similar trend holds for CCSD(T).
|
||||
Regarding the effect of the basis-set correction, several general observations can be made for both exFCI and CCSD(T).
|
||||
First, in a given basis set, the basis-set correction systematically improves the atomization energies (both at the LDA and PBE levels).
|
||||
A small overestimation can occur compared to the CBS value by a few tenths of a {\kcal} (the largest deviation being 0.6 {\kcal} for \ce{N2} at the CCSD(T)+PBE/cc-pV5Z level).
|
||||
Nevertheless, the deviation observed for the largest basis set is typically within the CBS extrapolation error, which is highly satisfactory knowing the marginal computational cost of the present correction.
|
||||
In most cases, the basis-set corrected triple-$\zeta$ atomization energies are on par with the uncorrected quintuple-$\zeta$ ones.
|
||||
Importantly, the sensitivity with respect to the RS-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better.
|
||||
However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}.
|
||||
Such weak sensitivity to the density-functional approximation when reaching large basis sets shows the robustness of the approach.
|
||||
|
||||
As a second set of numerical examples, we compute the error (with respect to the CBS values) of the atomization energies from the G2 test set with $\modY=\CCSDT$, $\modZ=\ROHF$ and the cc-pVXZ basis sets.
|
||||
Here, all atomization energies have been computed with the same near-CBS HF/cc-pV5Z energies; only the correlation energy contribution varies from one method to the other.
|
||||
Investigating the convergence of correlation energies (or difference of such quantities) is commonly done to appreciate the performance of basis-set corrections aiming at correcting two-electron effects. \cite{Tenno-CPL-04, TewKloNeiHat-PCCP-07, IrmGru-arXiv-2019}
|
||||
The ``plain'' CCSD(T) atomization energies as well as the corrected CCSD(T)+LDA and CCSD(T)+PBE values are depicted in Fig.~\ref{fig:G2_Ec}.
|
||||
The raw data can be found in the {\SI}.
|
||||
A statistical analysis of these data is also provided in Table \ref{tab:stats}, where we report the mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS atomization energies.
|
||||
Note that the MAD of our CCSD(T)/CBS atomization energies is only 0.37 {\kcal} compared to the values extracted from Ref.~\onlinecite{HauKlo-JCP-12} which corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
|
||||
From double-$\zeta$ to quintuple-$\zeta$ basis, the MAD associated with the CCSD(T) atomization energies goes down slowly from 14.29 to 1.28 {\kcal}.
|
||||
For a commonly used basis like cc-pVTZ, the MAD of CCSD(T) is still 6.06 {\kcal}.
|
||||
Applying the basis-set correction drastically reduces the basis-set incompleteness error.
|
||||
Already at the CCSD(T)+LDA/cc-pVDZ and CCSD(T)+PBE/cc-pVDZ levels, the MAD is reduced to 3.24 and 1.96 {\kcal}.
|
||||
With the triple-$\zeta$ basis, the MAD of CCSD(T)+PBE/cc-pVTZ is already below 1 {\kcal} with 36 cases (out of 55) where we achieve chemical accuracy.
|
||||
CCSD(T)+LDA/cc-pVQZ and CCSD(T)+PBE/cc-pVQZ return MAD of 0.33 and 0.31 kcal/mol (respectively) while CCSD(T)/cc-pVQZ still yields a fairly large MAD of 2.50 {\kcal}.
|
||||
|
||||
Therefore, similar to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis-set correction provides significant basis-set reduction and recovers quintuple-$\zeta$ quality atomization and correlation energies with triple-$\zeta$ basis sets for a much cheaper computational cost.
|
||||
Encouraged by these promising results, we are currently pursuing various avenues toward basis-set reduction for strongly correlated systems and electronically excited states.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section*{Supporting information}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
See {\SI} for raw data associated with the atomization energies of the four diatomic molecules and the G2 set.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{acknowledgements}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
The authors would like to thank the \emph{Centre National de la Recherche Scientifique} (CNRS) and the \emph{Institut des Sciences du Calcul et des Donn\'ees} for funding.
|
||||
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\end{acknowledgements}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\bibliography{G2-srDFT,G2-srDFT-control}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\end{document}
|
||||
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\newcommand{\SI}{\textcolor{blue}{supporting information}}
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\newcommand{\QP}{\textsc{quantum package}}
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% second quantized operators
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\newcommand{\ai}[1]{\hat{a}_{#1}}
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\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}}
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||||
% units
|
||||
\newcommand{\IneV}[1]{#1 eV}
|
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\newcommand{\InAU}[1]{#1 a.u.}
|
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\newcommand{\InAA}[1]{#1 \AA}
|
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\newcommand{\kcal}{kcal/mol}
|
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||||
% methods
|
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\newcommand{\D}{\text{D}}
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\newcommand{\T}{\text{T}}
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\newcommand{\Q}{\text{Q}}
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\newcommand{\X}{\text{X}}
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\newcommand{\UEG}{\text{UEG}}
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\newcommand{\HF}{\text{HF}}
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\newcommand{\LDA}{\text{LDA}}
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\newcommand{\PBE}{\text{PBE}}
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\newcommand{\FCI}{\text{FCI}}
|
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\newcommand{\CBS}{\text{CBS}}
|
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\newcommand{\exFCI}{\text{exFCI}}
|
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\newcommand{\CCSDT}{\text{CCSD(T)}}
|
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\newcommand{\lr}{\text{lr}}
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\newcommand{\sr}{\text{sr}}
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\newcommand{\Ne}{N}
|
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\newcommand{\NeUp}{\Ne^{\uparrow}}
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\newcommand{\NeDw}{\Ne^{\downarrow}}
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\newcommand{\Nb}{N_{\Bas}}
|
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\newcommand{\Ng}{N_\text{grid}}
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\newcommand{\nocca}{n_{\text{occ}^{\alpha}}}
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\newcommand{\noccb}{n_{\text{occ}^{\beta}}}
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\newcommand{\n}[2]{n_{#1}^{#2}}
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\E}[2]{E_{#1}^{#2}}
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\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
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\newcommand{\bEc}[1]{\Bar{E}_\text{c,md}^{#1}}
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\newcommand{\e}[2]{\varepsilon_{#1}^{#2}}
|
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\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
|
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\newcommand{\bec}[1]{\Bar{e}^{#1}}
|
||||
\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
|
||||
\newcommand{\W}[2]{W_{#1}^{#2}}
|
||||
\newcommand{\w}[2]{w_{#1}^{#2}}
|
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\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
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||||
\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}}
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||||
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||||
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||||
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|
||||
|
||||
% operators
|
||||
\newcommand{\hT}{\Hat{T}}
|
||||
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
|
||||
\newcommand{\updw}{\uparrow\downarrow}
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||||
\newcommand{\f}[2]{f_{#1}^{#2}}
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||||
\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
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||||
|
||||
% coordinates
|
||||
\newcommand{\br}[1]{\mathbf{r}_{#1}}
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\newcommand{\dbr}[1]{d\br{#1}}
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|
||||
% frozen core
|
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\newcommand{\WFC}[2]{\widetilde{W}_{#1}^{#2}}
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\newcommand{\fFC}[2]{\widetilde{f}_{#1}^{#2}}
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\newcommand{\rsmuFC}[2]{\widetilde{\mu}_{#1}^{#2}}
|
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\newcommand{\nFC}[2]{\widetilde{n}_{#1}^{#2}}
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|
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|
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\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}
|
||||
\begin{wrapfigure}[7]{o}[-1.2cm]{0.4\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{TOC}
|
||||
\end{wrapfigure}
|
||||
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.
|
||||
\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
|
||||
\begin{equation}
|
||||
\label{eq:e0basis}
|
||||
\E{}{}
|
||||
\approx \E{\modY}{\Bas}
|
||||
+ \bE{}{\Bas}[\n{\modZ}{\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 $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
|
||||
In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set (CBS), respectively.
|
||||
Both wave functions yield the same target density $\n{}{}$ (assumed to be representable in $\Bas$).
|
||||
Importantly, in the CBS limit (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$.
|
||||
This implies that
|
||||
\begin{equation}
|
||||
\label{eq:limitfunc}
|
||||
\lim_{\Bas \to \infty} \qty( \E{\modY}{\Bas} + \bE{}{\Bas}[\n{\modZ}{\Bas}] ) = \E{\modY}{} \approx E,
|
||||
\end{equation}
|
||||
where $\E{\modY}{}$ is the energy associated with the method $\modY$ in the CBS limit.
|
||||
In the case where $\modY = \FCI$ in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
|
||||
Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only sources of error at this stage lie in the potential approximate nature of the methods $\modY$ and $\modZ$, and the lack of self-consistency in the present scheme.
|
||||
|
||||
The functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
|
||||
Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct
|
||||
for the lack of cusp (i.e.~discontinuous derivative) in $\wf{}{\Bas}$ at the e-e coalescence points, a universal condition of exact wave functions.
|
||||
Because the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent two-electron interaction at coalescence.
|
||||
Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ by a short-range density functional which is complementary to a non-divergent long-range interaction.
|
||||
Contrary to the conventional RS-DFT scheme which requires a range-separation \textit{parameter} $\rsmu{}{}$, here we use a range-separation \textit{function} $\rsmu{}{\Bas}(\br{})$ that automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.
|
||||
|
||||
% https://english.stackexchange.com/questions/61600/consist-in-vs-consist-of
|
||||
The first step of the present basis-set correction consists in obtaining an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$.
|
||||
In a second step, we shall link $\W{}{\Bas}(\br{1},\br{2})$ to $\rsmu{}{\Bas}(\br{})$.
|
||||
As a final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{}{\Bas}(\br{})$ as range-separation function.
|
||||
|
||||
%=================================================================
|
||||
%\subsection{Effective Coulomb operator}
|
||||
%=================================================================
|
||||
We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18}
|
||||
\begin{equation}
|
||||
\label{eq:def_weebasis}
|
||||
\W{}{\Bas}(\br{1},\br{2}) =
|
||||
\begin{cases}
|
||||
\f{}{\Bas}(\br{1},\br{2})/\n{2}{\Bas}(\br{1},\br{2}), & \text{if $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$,}
|
||||
\\
|
||||
\infty, & \text{otherwise,}
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
where
|
||||
\begin{equation}
|
||||
\label{eq:n2basis}
|
||||
\n{2}{\Bas}(\br{1},\br{2})
|
||||
= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
|
||||
\end{equation}
|
||||
and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\uparrow}\ai{q_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO),
|
||||
\begin{equation}
|
||||
\label{eq:fbasis}
|
||||
\f{}{\Bas}(\br{1},\br{2})
|
||||
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
|
||||
\end{equation}
|
||||
and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals.
|
||||
With such a definition, $\W{}{\Bas}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
|
||||
\begin{equation}
|
||||
\iint \frac{ \n{2}{\Bas}(\br{1},\br{2})}{r_{12}} \dbr{1} \dbr{2} =
|
||||
\iint \W{}{\Bas}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},
|
||||
\end{equation}
|
||||
which intuitively motivates $\W{}{\Bas}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
|
||||
Note that the divergence condition of $\W{}{\Bas}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems are free of correction as the present approach must only correct the basis-set incompleteness error originating from the e-e cusp.
|
||||
As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{}{\Bas}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational, nor rotational invariant if $\Bas$ does not have such symmetries.
|
||||
Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
|
||||
\begin{equation}
|
||||
\label{eq:lim_W}
|
||||
\lim_{\Bas \to \infty}\W{}{\Bas}(\br{1},\br{2}) = r_{12}^{-1},
|
||||
\end{equation}
|
||||
for any $(\br{1},\br{2})$ such that $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$.
|
||||
|
||||
%=================================================================
|
||||
%\subsection{Range-separation function}
|
||||
%=================================================================
|
||||
|
||||
A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons, $\W{}{\Bas}(\br{},{\br{}})$,
|
||||
which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{\Bas}(\br{},\br{})$ is non vanishing.
|
||||
Because $\W{}{\Bas}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be naturally linked to RS-DFT which employs a non-divergent long-range interaction operator.
|
||||
Although this choice is not unique, we choose here the range-separation function
|
||||
\begin{equation}
|
||||
\label{eq:mu_of_r}
|
||||
\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{}),
|
||||
\end{equation}
|
||||
such that the long-range interaction of RS-DFT, $\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$, coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any $\br{}$.
|
||||
|
||||
%=================================================================
|
||||
%\subsection{Short-range correlation functionals}
|
||||
%=================================================================
|
||||
Once $\rsmu{}{\Bas}(\br{})$ is defined, it can be used within RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$.
|
||||
As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as correlation energy with multi-determinantal reference (ECMD) whose general definition reads \cite{TouGorSav-TCA-05}
|
||||
\begin{equation}
|
||||
\label{eq:ec_md_mu}
|
||||
\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]
|
||||
= \min_{\wf{}{} \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
|
||||
- \mel*{\wf{}{\rsmu{}{}}}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}},
|
||||
\end{equation}
|
||||
where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization
|
||||
\begin{equation}
|
||||
\label{eq:argmin}
|
||||
\wf{}{\rsmu{}{}} = \arg \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
|
||||
\end{equation}
|
||||
with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
|
||||
The ECMD functionals admit, for any $\n{}{}$, the following two limiting forms
|
||||
\begin{align}
|
||||
\label{eq:large_mu_ecmd}
|
||||
\lim_{\mu \to \infty} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
|
||||
&
|
||||
% \label{eq:small_mu_ecmd}
|
||||
\lim_{\mu \to 0} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
|
||||
\end{align}
|
||||
where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KS-DFT.
|
||||
The choice of ECMD in the present scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [Eq.~\eqref{eq:E_funcbasis}] and the ECMD functional [Eq.~\eqref{eq:ec_md_mu}].
|
||||
Indeed, the two functionals coincide if $\wf{}{\Bas} = \wf{}{\rsmu{}{}}$.
|
||||
Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by ECMD functionals evaluated with the range-separation function $\rsmu{}{\Bas}(\br{})$.
|
||||
|
||||
The local-density approximation (LDA) of the ECMD complementary functional is defined as
|
||||
\begin{equation}
|
||||
\label{eq:def_lda_tot}
|
||||
\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
|
||||
\end{equation}
|
||||
where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
|
||||
The short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$.
|
||||
In order to correct such a defect, inspired by the recent functional proposed by some of the authors~\cite{FerGinTou-JCP-18}, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional
|
||||
\begin{equation}
|
||||
\label{eq:def_pbe_tot}
|
||||
\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}] =
|
||||
\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
|
||||
\end{equation}
|
||||
where $s=\abs{\nabla \n{}{}}/\n{}{4/3}$ is the reduced density gradient.
|
||||
$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$, at $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
|
||||
\begin{subequations}
|
||||
\begin{gather}
|
||||
\label{eq:epsilon_cmdpbe}
|
||||
\be{\text{c,md}}{\sr,\PBE}(\n{}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta(\n{}{},s,\zeta) \rsmu{}{3} },
|
||||
\\
|
||||
\label{eq:beta_cmdpbe}
|
||||
\beta(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\UEG}(\n{}{},\zeta)}.
|
||||
\end{gather}
|
||||
\end{subequations}
|
||||
The difference between the ECMD functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe}-\eqref{eq:beta_cmdpbe} is that we approximate here 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.~\onlinecite{GorSav-PRA-06}.
|
||||
This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{\Bas}(\br{},\br{})$.
|
||||
|
||||
Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{\Bas}]$ is approximated by $\bE{\LDA}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
|
||||
|
||||
%=================================================================
|
||||
%\subsection{Frozen-core approximation}
|
||||
%=================================================================
|
||||
|
||||
As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a subset of MOs.
|
||||
We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ and $\BasFC$ are the sets of core and active MOs, respectively) and define the FC version of the effective interaction as
|
||||
\begin{equation}
|
||||
\label{eq:WFC}
|
||||
\WFC{}{\Bas}(\br{1},\br{2}) =
|
||||
\begin{cases}
|
||||
\fFC{}{\Bas}(\br{1},\br{2})/\nFC{2}{\Bas}(\br{1},\br{2}), & \text{if $\nFC{2}{\Bas}(\br{1},\br{2}) \ne 0$},
|
||||
\\
|
||||
\infty, & \text{otherwise,}
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
with
|
||||
\begin{subequations}
|
||||
\begin{gather}
|
||||
\label{eq:fbasisval}
|
||||
\fFC{}{\Bas}(\br{1},\br{2})
|
||||
= \sum_{pq \in \Bas} \sum_{rstu \in \BasFC} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
|
||||
\\
|
||||
\nFC{2}{\Bas}(\br{1},\br{2})
|
||||
= \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
|
||||
\end{gather}
|
||||
\end{subequations}
|
||||
and the corresponding FC range-separation function $\rsmuFC{}{\Bas}(\br{}) = (\sqrt{\pi}/2) \WFC{}{\Bas}(\br{},\br{})$.
|
||||
It is noteworthy that, within the present definition, $\WFC{}{\Bas}(\br{1},\br{2})$ still tends to the regular Coulomb interaction as $\Bas \to \infty$.
|
||||
|
||||
Defining $\nFC{\modZ}{\Bas}$ as the FC (i.e.~valence-only) one-electron density obtained with a method $\modZ$ in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\LDA}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$.
|
||||
|
||||
%=================================================================
|
||||
%\subsection{Computational considerations}
|
||||
%=================================================================
|
||||
The most computationally intensive task of the present approach is the evaluation of $\W{}{\Bas}(\br{},\br{})$ at each quadrature grid point.
|
||||
In the general case (i.e.~$\wf{}{\Bas}$ is a multi-determinant expansion), we compute this embarrassingly parallel step in $\order*{\Ng \Nb^4}$ computational cost with a memory requirement of $\order*{ \Ng \Nb^2}$, where $\Nb$ is the number of basis functions in $\Bas$.
|
||||
The computational cost can be reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ with no memory footprint when $\wf{}{\Bas}$ is a single Slater determinant.
|
||||
As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, this choice for $\wf{}{\Bas}$ already provides, for weakly correlated systems, a quantitative representation of the incompleteness of $\Bas$.
|
||||
Hence, we will stick to this choice throughout the present study.
|
||||
In our current implementation, the computational bottleneck is the four-index transformation to get the two-electron integrals in the MO basis which appear in Eqs.~\eqref{eq:n2basis} and \eqref{eq:fbasis}.
|
||||
Nevertheless, this step usually has to be performed for most correlated WFT calculations.
|
||||
Modern integral decomposition techniques (such as density fitting \cite{Whi-JCP-73}) or atomic-orbital-based algorithms could be employed to significantly speed up this step.
|
||||
|
||||
To conclude this section, we point out that, thanks to the definitions \eqref{eq:def_weebasis} and \eqref{eq:mu_of_r} as well as the properties \eqref{eq:lim_W} and \eqref{eq:large_mu_ecmd}, independently of the DFT functional, the present basis-set correction
|
||||
i) can be applied to any WFT method that provides an energy and a density,
|
||||
ii) does not correct one-electron systems, and
|
||||
iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a given WFT method.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\section{Results}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%% FIGURE 1 %%%
|
||||
\begin{figure*}
|
||||
\includegraphics[width=0.30\linewidth]{fig1a}
|
||||
\hspace{1cm}
|
||||
\includegraphics[width=0.30\linewidth]{fig1b}
|
||||
\\
|
||||
\includegraphics[width=0.30\linewidth]{fig1c}
|
||||
\hspace{1cm}
|
||||
\includegraphics[width=0.30\linewidth]{fig1d}
|
||||
\caption{
|
||||
Deviation (in \kcal) from CBS atomization energies of \ce{C2} (top left), \ce{O2} (top right), \ce{N2} (bottom left) and \ce{F2} (bottom right) obtained with various methods and basis sets.
|
||||
The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
|
||||
See {\SI} for raw data.
|
||||
\label{fig:diatomics}}
|
||||
\end{figure*}
|
||||
|
||||
%%% TABLE II %%%
|
||||
\begin{table}
|
||||
\caption{
|
||||
Statistical analysis (in \kcal) of the G2 atomization energies depicted in Fig.~\ref{fig:G2_Ec}.
|
||||
Mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS reference atomization energies.
|
||||
CA corresponds to the number of cases (out of 55) obtained with chemical accuracy.
|
||||
See {\SI} for raw data.
|
||||
\label{tab:stats}}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{ldddd}
|
||||
Method & \tabc{MAD} & \tabc{RMSD} & \tabc{MAX} & \tabc{CA} \\
|
||||
\hline
|
||||
CCSD(T)/cc-pVDZ & 14.29 & 16.21 & 36.95 & 2 \\
|
||||
CCSD(T)/cc-pVTZ & 6.06 & 6.84 & 14.25 & 2 \\
|
||||
CCSD(T)/cc-pVQZ & 2.50 & 2.86 & 6.75 & 9 \\
|
||||
CCSD(T)/cc-pV5Z & 1.28 & 1.46 & 3.46 & 21 \\
|
||||
\\
|
||||
CCSD(T)+LDA/cc-pVDZ & 3.24 & 3.67 & 8.13 & 7 \\
|
||||
CCSD(T)+LDA/cc-pVTZ & 1.19 & 1.49 & 4.67 & 27 \\
|
||||
CCSD(T)+LDA/cc-pVQZ & 0.33 & 0.44 & 1.32 & 53 \\
|
||||
\\
|
||||
CCSD(T)+PBE/cc-pVDZ & 1.96 & 2.59 & 7.33 & 19 \\
|
||||
CCSD(T)+PBE/cc-pVTZ & 0.85 & 1.11 & 2.64 & 36 \\
|
||||
CCSD(T)+PBE/cc-pVQZ & 0.31 & 0.42 & 1.16 & 53 \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
|
||||
%%% FIGURE 2 %%%
|
||||
\begin{figure*}
|
||||
\includegraphics[width=\linewidth]{fig2a}
|
||||
\includegraphics[width=\linewidth]{fig2b}
|
||||
\includegraphics[width=\linewidth]{fig2c}
|
||||
\caption{
|
||||
Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
|
||||
The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
|
||||
See {\SI} for raw data.
|
||||
\label{fig:G2_Ec}}
|
||||
\end{figure*}
|
||||
|
||||
We begin our investigation of the performance of the basis-set correction by computing the atomization energies of \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} obtained with Dunning's cc-pVXZ basis (X $=$ D, T, Q and 5).
|
||||
\ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2 set \cite{CurRagTruPop-JCP-91} (see below), whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11}
|
||||
In a second time, we compute the atomization energies of the entire G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ basis set family.
|
||||
This molecular set has been intensively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08, Gro-JCP-09, FelPet-JCP-09, NemTowNee-JCP-10, FelPetHil-JCP-11, HauKlo-JCP-12, PetTouUmr-JCP-12, FelPet-JCP-13, KesSylKohTewMar-JCP-18}) and can be considered as a representative set of small organic and inorganic molecules.
|
||||
As a method $\modY$ we employ either CCSD(T) or exFCI.
|
||||
Here, 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.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
|
||||
In the case of the CCSD(T) calculations, we have $\modZ = \ROHF$ as we use the restricted open-shell HF (ROHF) one-electron density to compute the complementary basis-set correction energy.
|
||||
In the case of exFCI, the one-electron density is computed from a very large CIPSI expansion containing several million determinants.
|
||||
CCSD(T) energies are computed with Gaussian09 using standard threshold values, \cite{g09} while RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}
|
||||
For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17}
|
||||
Apart from the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-09} and have been obtained at the B3LYP/6-31G(2df,p) level of theory.
|
||||
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}.
|
||||
In the context of the basis-set correction, the set of active MOs, $\BasFC$, involved in the definition of the effective interaction [see Eq.~\eqref{eq:WFC}] refers to the non-frozen MOs.
|
||||
The FC density-based correction is used consistently with the FC approximation in WFT methods.
|
||||
To estimate the CBS limit of each method, following Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98}, we perform a two-point X$^{-3}$ extrapolation of the correlation energies using the quadruple- and quintuple-$\zeta$ data that we add up to the HF energies obtained in the largest (i.e.~quintuple-$\zeta$) basis.
|
||||
|
||||
As the exFCI atomization energies are converged with a precision of about 0.1 {\kcal}, we can label these as near FCI.
|
||||
Hence, they will be our references for \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2}.
|
||||
The results for these diatomic molecules are reported in Fig.~\ref{fig:diatomics}.
|
||||
The corresponding numerical data can be found in the {\SI}.
|
||||
As one can see, the convergence of the exFCI atomization energies is, as expected, slow with respect to the basis set: chemical accuracy (error below 1 {\kcal}) is barely reached for \ce{C2}, \ce{O2} and \ce{F2} even with the cc-pV5Z basis set, and the atomization energies are consistently underestimated.
|
||||
A similar trend holds for CCSD(T).
|
||||
Regarding the effect of the basis-set correction, several general observations can be made for both exFCI and CCSD(T).
|
||||
First, in a given basis set, the basis-set correction systematically improves the atomization energies (both at the LDA and PBE levels).
|
||||
A small overestimation can occur compared to the CBS value by a few tenths of a {\kcal} (the largest deviation being 0.6 {\kcal} for \ce{N2} at the CCSD(T)+PBE/cc-pV5Z level).
|
||||
Nevertheless, the deviation observed for the largest basis set is typically within the CBS extrapolation error, which is highly satisfactory knowing the marginal computational cost of the present correction.
|
||||
In most cases, the basis-set corrected triple-$\zeta$ atomization energies are on par with the uncorrected quintuple-$\zeta$ ones.
|
||||
Importantly, the sensitivity with respect to the RS-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better.
|
||||
However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}.
|
||||
Such weak sensitivity to the density-functional approximation when reaching large basis sets shows the robustness of the approach.
|
||||
|
||||
As a second set of numerical examples, we compute the error (with respect to the CBS values) of the atomization energies from the G2 test set with $\modY=\CCSDT$, $\modZ=\ROHF$ and the cc-pVXZ basis sets.
|
||||
Here, all atomization energies have been computed with the same near-CBS HF/cc-pV5Z energies; only the correlation energy contribution varies from one method to the other.
|
||||
Investigating the convergence of correlation energies (or difference of such quantities) is commonly done to appreciate the performance of basis-set corrections aiming at correcting two-electron effects. \cite{Tenno-CPL-04, TewKloNeiHat-PCCP-07, IrmGru-arXiv-2019}
|
||||
The ``plain'' CCSD(T) atomization energies as well as the corrected CCSD(T)+LDA and CCSD(T)+PBE values are depicted in Fig.~\ref{fig:G2_Ec}.
|
||||
The raw data can be found in the {\SI}.
|
||||
A statistical analysis of these data is also provided in Table \ref{tab:stats}, where we report the mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS atomization energies.
|
||||
Note that the MAD of our CCSD(T)/CBS atomization energies is only 0.37 {\kcal} compared to the values extracted from Ref.~\onlinecite{HauKlo-JCP-12} which corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
|
||||
From double-$\zeta$ to quintuple-$\zeta$ basis, the MAD associated with the CCSD(T) atomization energies goes down slowly from 14.29 to 1.28 {\kcal}.
|
||||
For a commonly used basis like cc-pVTZ, the MAD of CCSD(T) is still 6.06 {\kcal}.
|
||||
Applying the basis-set correction drastically reduces the basis-set incompleteness error.
|
||||
Already at the CCSD(T)+LDA/cc-pVDZ and CCSD(T)+PBE/cc-pVDZ levels, the MAD is reduced to 3.24 and 1.96 {\kcal}.
|
||||
With the triple-$\zeta$ basis, the MAD of CCSD(T)+PBE/cc-pVTZ is already below 1 {\kcal} with 36 cases (out of 55) where we achieve chemical accuracy.
|
||||
CCSD(T)+LDA/cc-pVQZ and CCSD(T)+PBE/cc-pVQZ return MAD of 0.33 and 0.31 kcal/mol (respectively) while CCSD(T)/cc-pVQZ still yields a fairly large MAD of 2.50 {\kcal}.
|
||||
|
||||
Therefore, similar to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis-set correction provides significant basis-set reduction and recovers quintuple-$\zeta$ quality atomization and correlation energies with triple-$\zeta$ basis sets for a much cheaper computational cost.
|
||||
Encouraged by these promising results, we are currently pursuing various avenues toward basis-set reduction for strongly correlated systems and electronically excited states.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section*{Supporting information}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
See {\SI} for raw data associated with the atomization energies of the four diatomic molecules and the G2 set.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{acknowledgements}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
The authors would like to thank the \emph{Centre National de la Recherche Scientifique} (CNRS) and the \emph{Institut des Sciences du Calcul et des Donn\'ees} for funding.
|
||||
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\end{acknowledgements}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\bibliography{G2-srDFT,G2-srDFT-control}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,336 @@
|
|||
\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
|
||||
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|
||||
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|
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||||
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|
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|
||||
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|
||||
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|
||||
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||||
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|
||||
|
||||
% second quantized operators
|
||||
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|
||||
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|
||||
|
||||
% units
|
||||
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|
||||
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|
||||
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|
||||
\newcommand{\kcal}{kcal/mol}
|
||||
|
||||
% methods
|
||||
\newcommand{\D}{\text{D}}
|
||||
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|
||||
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|
||||
\newcommand{\X}{\text{X}}
|
||||
\newcommand{\UEG}{\text{UEG}}
|
||||
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|
||||
\newcommand{\LDA}{\text{LDA}}
|
||||
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|
||||
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|
||||
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|
||||
\newcommand{\exFCI}{\text{exFCI}}
|
||||
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|
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|
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||||
|
||||
% basis sets
|
||||
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||||
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||||
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||||
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|
||||
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||||
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\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{Supplementary Information for ``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}
|
||||
\end{abstract}
|
||||
|
||||
\maketitle
|
||||
|
||||
%%% TABLE I %%%
|
||||
\begin{table*}
|
||||
\caption{
|
||||
\label{tab:diatomics}
|
||||
Frozen-core atomization energies (in {\kcal}) of \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} computed with various methods and basis sets.
|
||||
The deviations with respect to the corresponding CBS values are reported in parenthesis.
|
||||
See main text for more details.
|
||||
}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{llddddd}
|
||||
& & \mc{4}{c}{Dunning's basis set cc-pVXZ}
|
||||
\\
|
||||
\cline{3-6}
|
||||
Molecule & Method & \tabc{$\X = \D$} & \tabc{$\X = \T$} & \tabc{$\X = \Q$} & \tabc{$\X = 5$} & \tabc{CBS}
|
||||
\\
|
||||
\hline
|
||||
\ce{C2} & exFCI & 132.0 (-13.7 ) & 140.3 (-5.4 ) & 143.6 (-2.1 ) & 144.7 (-1.0 ) & 145.7 \\
|
||||
& exFCI+LDA & 141.3 (-4.4 ) & 145.1 (-0.6 ) & 146.4 (+0.7 ) & 146.3 (+0.6 ) & \\
|
||||
& exFCI+PBE & 145.7 (+0.0 ) & 145.7 (+0.0 ) & 146.3 (+0.6 ) & 146.2 (+0.5 ) & \\
|
||||
& CCSD(T) & 129.2 (-16.2 ) & 139.1 (-6.3 ) & 143.0 (-2.4 ) & 144.2 (-1.2 ) & 145.4 \\
|
||||
& CCSD(T)+LDA & 139.1 (-6.3 ) & 143.7 (-1.7 ) & 145.9 (+0.5 ) & 145.9 (+0.5 ) & \\
|
||||
& CCSD(T)+PBE & 142.8 (-2.6 ) & 144.2 (-1.2 ) & 145.9 (+0.5 ) & 145.8 (+0.4 ) & \\ \\
|
||||
% \ce{C2} & exFCI\fnm[2] & 131.0 (-16.1 ) & 141.5 (-5.6 ) & 145.1 (-2.0 ) & 146.1 (-1.0 ) & 147.1 \\
|
||||
% (cc-pCVXZ) & exFCI+LDA\fnm[2] & 141.4 (-5.7 ) & 146.7 (-0.4 ) & 147.8 (+0.7 ) & 147.6 (+0.5 ) & \\
|
||||
% & exFCI+PBE\fnm[2] & 145.1 (-2.0 ) & 147.0 (-0.1 ) & 147.7 (+0.6 ) & 147.5 (+0.4 ) & \\ \\
|
||||
\ce{N2} & exFCI & 201.1 (-26.7 ) & 217.1 (-10.7 ) & 223.5 (-4.3 ) & 225.7 (-2.1 ) & 227.8 \\
|
||||
& exFCI+LDA & 217.9 (-9.9 ) & 225.9 (-1.9 ) & 228.0 (+0.2 ) & 228.6 (+0.8 ) & \\
|
||||
& exFCI+PBE & 227.7 (-0.1 ) & 227.8 (+0.0 ) & 228.3 (+0.5 ) & 228.5 (+0.7 ) & \\
|
||||
& CCSD(T) & 199.9 (-27.3 ) & 216.3 (-10.9 ) & 222.8 (-4.4 ) & 225.0 (-2.2 ) & 227.2 \\
|
||||
& CCSD(T)+LDA & 216.3 (-10.9 ) & 224.8 (-2.4 ) & 227.2 (-0.0 ) & 227.8 (+0.6 ) & \\
|
||||
& CCSD(T)+PBE & 225.9 (-1.3 ) & 226.7 (-0.5 ) & 227.5 (+0.3 ) & 227.8 (+0.6 ) & \\ \\
|
||||
% \ce{N2} & exFCI\fnm[2] & 202.2 (-26.6 ) & 218.5 (-10.3 ) & 224.4 (-4.4 ) & 226.6 (-2.2 ) & 228.8 \\
|
||||
% (cc-pCVXZ) & exFCI+LDA\fnm[2] & 218.0 (-10.8 ) & 226.8 (-2.0 ) & 229.1 (+0.3 ) & 229.4 (+0.6 ) & \\
|
||||
% & exFCI+PBE\fnm[2] & 226.4 (-2.4 ) & 228.2 (-0.6 ) & 229.1 (+0.3 ) & 229.2 (+0.4 ) & \\ \\
|
||||
\ce{O2} & exFCI & 105.2 (-14.8 ) & 114.5 (-5.5 ) & 118.0 (-2.0 ) & 119.1 (-0.9 ) & 120.0 \\
|
||||
& exFCI+LDA & 112.4 (-7.6 ) & 118.4 (-1.6 ) & 120.2 (+0.2 ) & 120.4 (+0.4 ) & \\
|
||||
& exFCI+PBE & 117.2 (-2.8 ) & 119.4 (-0.6 ) & 120.3 (+0.3 ) & 120.4 (+0.4 ) & \\
|
||||
& CCSD(T) & 103.9 (-16.1 ) & 113.6 (-6.0 ) & 117.1 (-2.5 ) & 118.6 (-1.0 ) & 119.6 \\
|
||||
& CCSD(T)+LDA & 110.6 (-9.0 ) & 117.2 (-2.4 ) & 119.2 (-0.4 ) & 119.8 (+0.2 ) & \\
|
||||
& CCSD(T)+PBE & 115.1 (-4.5 ) & 118.0 (-1.6 ) & 119.3 (-0.3 ) & 119.8 (+0.2 ) & \\ \\
|
||||
\ce{F2} & exFCI & 26.7 (-12.3 ) & 35.1 (-3.9 ) & 37.1 (-1.9 ) & 38.0 (-1.0 ) & 39.0 \\
|
||||
& exFCI+LDA & 30.4 (-8.6 ) & 37.2 (-1.8 ) & 38.4 (-0.6 ) & 38.9 (-0.1 ) & \\
|
||||
& exFCI+PBE & 33.1 (-5.9 ) & 37.9 (-1.1 ) & 38.5 (-0.5 ) & 38.9 (-0.1 ) & \\
|
||||
& CCSD(T) & 25.7 (-12.5 ) & 34.4 (-3.8 ) & 36.5 (-1.7 ) & 37.4 (-0.8 ) & 38.2 \\
|
||||
& CCSD(T)+LDA & 29.2 (-9.0 ) & 36.5 (-1.7 ) & 37.2 (-1.0 ) & 38.2 (+0.0 ) & \\
|
||||
& CCSD(T)+PBE & 31.5 (-6.7 ) & 37.1 (-1.1 ) & 37.8 (-0.4 ) & 38.2 (+0.0 ) & \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
% \fnt[1]{ calculations. Only valence orbitals are taken into account in the basis set correction.}
|
||||
% \fnt[2]{``Full'' calculation, i.e., all electrons are correlated. All spinorbitals are taken into account in the basis set correction.}
|
||||
\end{table*}
|
||||
|
||||
|
||||
\begin{turnpage}
|
||||
\begin{squeezetable}
|
||||
\begin{table}
|
||||
\caption{
|
||||
\label{tab:AE}
|
||||
Deviation from the reference CCSD(T)/CBS atomization energies (in {\kcal}) for various methods and basis sets.
|
||||
See main text for more details.}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{lddddddddddd}
|
||||
&
|
||||
& \mc{10}{c}{Deviation from CBS atomization energies} \\
|
||||
\cline{3-12}
|
||||
& & \mc{4}{c}{CCSD(T)} & \mc{3}{c}{CCSD(T)+LDA} & \mc{3}{c}{CCSD(T)+PBE} \\
|
||||
\cline{3-6} \cline{7-9} \cline{10-12}
|
||||
Molecule & \tabc{CCSD(T)/CBS}
|
||||
& \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z}
|
||||
& \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ}
|
||||
& \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ}
|
||||
\\
|
||||
\hline
|
||||
\ce{BeH} & 50.12 & -3.38 & -1.01 & -0.36 & -0.19 & -0.75 & -0.03 & 0.04 & -0.75 & -0.03 & 0.04 \\
|
||||
\ce{C2H2} & 403.00 & -22.99 & -8.69 & -3.43 & -1.76 & -5.12 & -1.08 & 0.37 & -5.12 & -1.08 & 0.37 \\
|
||||
\ce{C2H4} & 561.69 & -27.16 & -9.44 & -3.59 & -1.84 & -4.88 & -0.56 & 0.58 & -4.88 & -0.56 & 0.58 \\
|
||||
\ce{C2H6} & 710.81 & -32.62 & -10.65 & -3.97 & -2.03 & -5.28 & -0.20 & 0.76 & -5.28 & -0.20 & 0.76 \\
|
||||
\ce{CH} & 83.89 & -5.96 & -1.97 & -0.75 & -0.39 & -0.81 & -0.03 & 0.15 & -0.81 & -0.03 & 0.15 \\
|
||||
\ce{CH2 ^1A_1} & 180.61 & -10.99 & -3.57 & -1.34 & -0.68 & -1.55 & 0.00 & 0.29 & -1.55 & 0.00 & 0.29 \\
|
||||
\ce{CH2 ^3B_1} & 189.94 & -8.82 & -2.78 & -1.0 & -0.51 & -1.87 & -0.38 & 0.00 & -1.87 & -0.38 & -0.00 \\
|
||||
\ce{CH3} & 306.81 & -14.33 & -4.47 & -1.62 & -0.83 & -2.33 & -0.08 & 0.27 & -2.33 & -0.08 & 0.27 \\
|
||||
\ce{CH3Cl} & 395.02 & -20.77 & -8.34 & -3.35 & -1.72 & -4.38 & -1.60 & -0.16 & -4.38 & -1.60 & -0.16 \\
|
||||
\ce{CH4} & 419.19 & -18.35 & -5.64 & -2.01 & -1.03 & -2.55 & 0.18 & 0.51 & -2.55 & 0.18 & 0.51 \\
|
||||
\ce{CN} & 179.32 & -14.98 & -7.26 & -3.18 & -1.63 & -2.16 & -0.82 & 0.20 & -2.16 & -0.82 & 0.20 \\
|
||||
\ce{CO} & 258.64 & -11.01 & -5.94 & -2.60 & -1.33 & -0.93 & -0.61 & 0.44 & -0.93 & -0.61 & 0.44 \\
|
||||
\ce{CO2} & 388.29 & -20.79 & -10.95 & -4.67 & -2.39 & -3.19 & -1.69 & 0.60 & -3.19 & -1.69 & 0.60 \\
|
||||
\ce{CS} & 170.82 & -10.72 & -6.25 & -2.88 & -1.47 & -1.94 & -1.58 & -0.16 & -1.94 & -1.58 & -0.16 \\
|
||||
\ce{Cl2} & 59.33 & -8.26 & -5.23 & -2.49 & -1.28 & -3.63 & -2.46 & -0.90 & -3.63 & -2.46 & -0.90 \\
|
||||
\ce{ClF} & 62.43 & -8.51 & -4.83 & -2.33 & -1.19 & -4.02 & -2.23 & -0.81 & -4.02 & -2.23 & -0.81 \\
|
||||
\ce{ClO} & 64.35 & -11.86 & -6.35 & -3.16 & -1.62 & -5.99 & -3.08 & -1.25 & -5.99 & -3.08 & -1.25 \\
|
||||
\ce{F2} & 38.24 & -6.62 & -3.52 & -1.72 & -0.88 & -3.17 & -1.54 & -0.48 & -3.17 & -1.54 & -0.48 \\
|
||||
\ce{H2CO} & 373.18 & -19.31 & -8.07 & -3.31 & -1.70 & -2.84 & -0.70 & 0.50 & -2.84 & -0.70 & 0.50 \\
|
||||
\ce{H2O} & 232.78 & -15.21 & -5.83 & -2.37 & -1.21 & -4.03 & -1.07 & -0.14 & -4.03 & -1.07 & -0.14 \\
|
||||
\ce{H2O2} & 268.77 & -22.78 & -9.44 & -4.08 & -2.09 & -6.64 & -2.06 & -0.30 & -6.64 & -2.06 & -0.30 \\
|
||||
\ce{H2S} & 183.36 & -10.66 & -4.33 & -1.55 & -0.79 & -2.50 & -1.16 & -0.12 & -2.50 & -1.16 & -0.12 \\
|
||||
\ce{H3COH} & 512.25 & -28.32 & -10.40 & -4.10 & -2.10 & -5.95 & -1.17 & 0.31 & -5.95 & -1.17 & 0.31 \\
|
||||
\ce{H3CSH} & 473.92 & -25.84 & -10.02 & -3.88 & -1.98 & -5.49 & -1.79 & -0.01 & -5.49 & -1.79 & -0.01 \\
|
||||
\ce{HCN} & 311.54 & -20.86 & -9.06 & -3.80 & -1.95 & -3.59 & -0.85 & 0.37 & -3.59 & -0.85 & 0.37 \\
|
||||
\ce{HCO} & 277.92 & -15.04 & -6.87 & -2.84 & -1.46 & -2.38 & -0.89 & 0.37 & -2.38 & -0.89 & 0.37 \\
|
||||
\ce{HCl} & 107.31 & -5.64 & -2.46 & -0.93 & -0.48 & -1.37 & -0.81 & -0.22 & -1.37 & -0.81 & -0.22 \\
|
||||
\ce{HF} & 141.67 & -8.45 & -3.48 & -1.42 & -0.73 & -2.70 & -0.96 & -0.23 & -2.70 & -0.96 & -0.23 \\
|
||||
\ce{HOCl} & 165.85 & -15.82 & -7.34 & -3.28 & -1.68 & -5.39 & -2.29 & -0.62 & -5.39 & -2.29 & -0.62 \\
|
||||
\ce{Li2} & 24.10 & -0.78 & -0.30 & -0.07 & -0.04 & 0.39 & 0.13 & 0.1 & 0.39 & 0.13 & 0.1 \\
|
||||
\ce{LiF} & 138.33 & -10.78 & -4.56 & -1.95 & -1.0 & -3.30 & -0.91 & 0.02 & -3.30 & -0.91 & 0.02 \\
|
||||
\ce{LiH} & 57.70 & -4.15 & -1.05 & -0.38 & -0.20 & -0.71 & 0.12 & 0.11 & -0.71 & 0.12 & 0.11 \\
|
||||
\ce{N2} & 227.18 & -18.66 & -9.41 & -4.20 & -2.15 & -2.26 & -0.90 & 0.20 & -2.26 & -0.90 & 0.20 \\
|
||||
\ce{N2H4} & 437.39 & -36.95 & -14.25 & -5.83 & -2.99 & -8.13 & -1.60 & -0.01 & -8.13 & -1.60 & -0.01 \\
|
||||
\ce{NH} & 82.86 & -8.10 & -3.02 & -1.24 & -0.63 & -1.51 & -0.22 & -0.06 & -1.51 & -0.22 & -0.06 \\
|
||||
\ce{NH2} & 182.14 & -15.31 & -5.61 & -2.26 & -1.16 & -3.16 & -0.50 & -0.03 & -3.16 & -0.50 & -0.03 \\
|
||||
\ce{NH3} & 297.46 & -21.21 & -7.62 & -3.01 & -1.54 & -4.42 & -0.62 & 0.07 & -4.42 & -0.62 & 0.07 \\
|
||||
\ce{NO} & 151.73 & -13.87 & -7.26 & -3.25 & -1.66 & -2.20 & -1.10 & 0.08 & -2.20 & -1.10 & 0.08 \\
|
||||
\ce{Na2} & 16.44 & -0.71 & -0.14 & -0.06 & -0.03 & 0.20 & 0.15 & 0.08 & 0.20 & 0.15 & 0.08 \\
|
||||
\ce{NaCl} & 99.61 & -11.48 & -5.41 & -2.26 & -1.16 & -4.34 & -2.13 & -0.52 & -4.34 & -2.13 & -0.52 \\
|
||||
\ce{O2} & 119.64 & -9.86 & -5.34 & -2.40 & -1.23 & -3.08 & -1.72 & -0.30 & -3.08 & -1.72 & -0.30 \\
|
||||
\ce{OH} & 107.08 & -8.37 & -3.27 & -1.34 & -0.69 & -2.23 & -0.63 & -0.08 & -2.23 & -0.63 & -0.08 \\
|
||||
\ce{P2} & 115.68 & -18.05 & -9.34 & -4.00 & -2.05 & -3.96 & -1.89 & 0.26 & -3.96 & -1.89 & 0.26 \\
|
||||
\ce{PH2} & 153.91 & -11.50 & -4.32 & -1.51 & -0.77 & -2.56 & -0.82 & 0.16 & -2.56 & -0.82 & 0.16 \\
|
||||
\ce{PH3} & 241.49 & -15.43 & -5.80 & -2.03 & -1.04 & -3.12 & -1.03 & 0.20 & -3.12 & -1.03 & 0.20 \\
|
||||
\ce{S2} & 103.07 & -12.76 & -7.34 & -3.28 & -1.68 & -5.69 & -3.18 & -0.95 & -5.69 & -3.18 & -0.95 \\
|
||||
\ce{SO} & 125.12 & -10.77 & -6.14 & -2.94 & -1.51 & -3.33 & -2.10 & -0.59 & -3.33 & -2.10 & -0.59 \\
|
||||
\ce{SO2} & 258.09 & -22.35 & -13.90 & -6.75 & -3.46 & -5.16 & -4.67 & -1.32 & -5.16 & -4.67 & -1.32 \\
|
||||
\ce{Si2} & 75.74 & -10.01 & -4.47 & -1.82 & -0.93 & -4.19 & -1.49 & -0.17 & -4.19 & -1.49 & -0.17 \\
|
||||
\ce{Si2H6} & 535.58 & -27.32 & -10.13 & -3.46 & -1.77 & -5.59 & -1.86 & 0.30 & -5.59 & -1.86 & 0.30 \\
|
||||
\ce{SiH2 ^1A_1} & 153.66 & -8.72 & -2.98 & -0.98 & -0.50 & -3.51 & -1.46 & -0.34 & -3.51 & -1.46 & -0.34 \\
|
||||
\ce{SiH2 ^3B_1} & 133.17 & -6.68 & -2.40 & -0.73 & -0.37 & -3.02 & -1.57 & -0.49 & -3.02 & -1.57 & -0.49 \\
|
||||
\ce{SiH3} & 227.99 & -10.53 & -3.72 & -1.15 & -0.59 & -1.82 & -0.74 & 0.12 & -1.82 & -0.74 & 0.12 \\
|
||||
\ce{SiH4} & 324.59 & -14.10 & -4.88 & -1.52 & -0.78 & -1.91 & -0.58 & 0.33 & -1.91 & -0.58 & 0.33 \\
|
||||
\ce{SiO} & 191.48 & -11.34 & -6.72 & -3.15 & -1.61 & -0.76 & -1.18 & 0.12 & -0.76 & -1.18 & 0.12 \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
\end{squeezetable}
|
||||
\end{turnpage}
|
||||
|
||||
\bibliography{../G2_srDFT}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
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|
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|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
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|
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|
||||
|
||||
|
||||
|
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|
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|
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|
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|
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|
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|
||||
|
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|
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|
||||
|
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|
||||
|
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|
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|
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|
||||
|
||||
|
||||
|
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|
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|
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|
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|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
\end{document}
|
||||
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|
|
@ -155,371 +155,4 @@ Contrary to conventional RS-DFT schemes which require an \textit{ad hoc} range-s
|
|||
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.
|
||||
\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}
|
||||
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
|
||||
\begin{equation}
|
||||
\label{eq:e0basis}
|
||||
\E{}{}
|
||||
\approx \E{\modY}{\Bas}
|
||||
+ \bE{}{\Bas}[\n{\modZ}{\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 $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
|
||||
In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set (CBS), respectively.
|
||||
Both wave functions yield the same target density $\n{}{}$ (assumed to be representable in $\Bas$).
|
||||
Importantly, in the CBS limit (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$.
|
||||
This implies that
|
||||
\begin{equation}
|
||||
\label{eq:limitfunc}
|
||||
\lim_{\Bas \to \infty} \qty( \E{\modY}{\Bas} + \bE{}{\Bas}[\n{\modZ}{\Bas}] ) = \E{\modY}{} \approx E,
|
||||
\end{equation}
|
||||
where $\E{\modY}{}$ is the energy associated with the method $\modY$ in the CBS limit.
|
||||
In the case where $\modY = \FCI$ in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
|
||||
Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only sources of error at this stage lie in the potential approximate nature of the methods $\modY$ and $\modZ$, and the lack of self-consistency in the present scheme.
|
||||
|
||||
The functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
|
||||
Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct
|
||||
for the lack of cusp (i.e.~discontinuous derivative) in $\wf{}{\Bas}$ at the e-e coalescence points, a universal condition of exact wave functions.
|
||||
Because the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent two-electron interaction at coalescence.
|
||||
Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ by a short-range density functional which is complementary to a non-divergent long-range interaction.
|
||||
Contrary to the conventional RS-DFT scheme which requires a range-separation \textit{parameter} $\rsmu{}{}$, here we use a range-separation \textit{function} $\rsmu{}{\Bas}(\br{})$ that automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.
|
||||
|
||||
% https://english.stackexchange.com/questions/61600/consist-in-vs-consist-of
|
||||
The first step of the present basis-set correction consists in obtaining an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$.
|
||||
In a second step, we shall link $\W{}{\Bas}(\br{1},\br{2})$ to $\rsmu{}{\Bas}(\br{})$.
|
||||
As a final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{}{\Bas}(\br{})$ as range-separation function.
|
||||
|
||||
%=================================================================
|
||||
%\subsection{Effective Coulomb operator}
|
||||
%=================================================================
|
||||
We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18}
|
||||
\begin{equation}
|
||||
\label{eq:def_weebasis}
|
||||
\W{}{\Bas}(\br{1},\br{2}) =
|
||||
\begin{cases}
|
||||
\f{}{\Bas}(\br{1},\br{2})/\n{2}{\Bas}(\br{1},\br{2}), & \text{if $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$,}
|
||||
\\
|
||||
\infty, & \text{otherwise,}
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
where
|
||||
\begin{equation}
|
||||
\label{eq:n2basis}
|
||||
\n{2}{\Bas}(\br{1},\br{2})
|
||||
= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
|
||||
\end{equation}
|
||||
and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\uparrow}\ai{q_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO),
|
||||
\begin{equation}
|
||||
\label{eq:fbasis}
|
||||
\f{}{\Bas}(\br{1},\br{2})
|
||||
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
|
||||
\end{equation}
|
||||
and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals.
|
||||
With such a definition, $\W{}{\Bas}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
|
||||
\begin{equation}
|
||||
\iint \frac{ \n{2}{\Bas}(\br{1},\br{2})}{r_{12}} \dbr{1} \dbr{2} =
|
||||
\iint \W{}{\Bas}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},
|
||||
\end{equation}
|
||||
which intuitively motivates $\W{}{\Bas}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
|
||||
Note that the divergence condition of $\W{}{\Bas}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems are free of correction as the present approach must only correct the basis-set incompleteness error originating from the e-e cusp.
|
||||
As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{}{\Bas}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational, nor rotational invariant if $\Bas$ does not have such symmetries.
|
||||
Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
|
||||
\begin{equation}
|
||||
\label{eq:lim_W}
|
||||
\lim_{\Bas \to \infty}\W{}{\Bas}(\br{1},\br{2}) = r_{12}^{-1},
|
||||
\end{equation}
|
||||
for any $(\br{1},\br{2})$ such that $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$.
|
||||
|
||||
%=================================================================
|
||||
%\subsection{Range-separation function}
|
||||
%=================================================================
|
||||
|
||||
A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons, $\W{}{\Bas}(\br{},{\br{}})$,
|
||||
which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{\Bas}(\br{},\br{})$ is non vanishing.
|
||||
Because $\W{}{\Bas}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be naturally linked to RS-DFT which employs a non-divergent long-range interaction operator.
|
||||
Although this choice is not unique, we choose here the range-separation function
|
||||
\begin{equation}
|
||||
\label{eq:mu_of_r}
|
||||
\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{}),
|
||||
\end{equation}
|
||||
such that the long-range interaction of RS-DFT, $\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$, coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any $\br{}$.
|
||||
|
||||
%=================================================================
|
||||
%\subsection{Short-range correlation functionals}
|
||||
%=================================================================
|
||||
Once $\rsmu{}{\Bas}(\br{})$ is defined, it can be used within RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$.
|
||||
As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as correlation energy with multi-determinantal reference (ECMD) whose general definition reads \cite{TouGorSav-TCA-05}
|
||||
\begin{equation}
|
||||
\label{eq:ec_md_mu}
|
||||
\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]
|
||||
= \min_{\wf{}{} \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
|
||||
- \mel*{\wf{}{\rsmu{}{}}}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}},
|
||||
\end{equation}
|
||||
where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization
|
||||
\begin{equation}
|
||||
\label{eq:argmin}
|
||||
\wf{}{\rsmu{}{}} = \arg \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
|
||||
\end{equation}
|
||||
with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
|
||||
The ECMD functionals admit, for any $\n{}{}$, the following two limiting forms
|
||||
\begin{align}
|
||||
\label{eq:large_mu_ecmd}
|
||||
\lim_{\mu \to \infty} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
|
||||
&
|
||||
% \label{eq:small_mu_ecmd}
|
||||
\lim_{\mu \to 0} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
|
||||
\end{align}
|
||||
where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KS-DFT.
|
||||
The choice of ECMD in the present scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [Eq.~\eqref{eq:E_funcbasis}] and the ECMD functional [Eq.~\eqref{eq:ec_md_mu}].
|
||||
Indeed, the two functionals coincide if $\wf{}{\Bas} = \wf{}{\rsmu{}{}}$.
|
||||
Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by ECMD functionals evaluated with the range-separation function $\rsmu{}{\Bas}(\br{})$.
|
||||
|
||||
The local-density approximation (LDA) of the ECMD complementary functional is defined as
|
||||
\begin{equation}
|
||||
\label{eq:def_lda_tot}
|
||||
\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
|
||||
\end{equation}
|
||||
where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
|
||||
The short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$.
|
||||
In order to correct such a defect, inspired by the recent functional proposed by some of the authors~\cite{FerGinTou-JCP-18}, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional
|
||||
\begin{equation}
|
||||
\label{eq:def_pbe_tot}
|
||||
\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}] =
|
||||
\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
|
||||
\end{equation}
|
||||
where $s=\abs{\nabla \n{}{}}/\n{}{4/3}$ is the reduced density gradient.
|
||||
$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$, at $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
|
||||
\begin{subequations}
|
||||
\begin{gather}
|
||||
\label{eq:epsilon_cmdpbe}
|
||||
\be{\text{c,md}}{\sr,\PBE}(\n{}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta(\n{}{},s,\zeta) \rsmu{}{3} },
|
||||
\\
|
||||
\label{eq:beta_cmdpbe}
|
||||
\beta(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\UEG}(\n{}{},\zeta)}.
|
||||
\end{gather}
|
||||
\end{subequations}
|
||||
The difference between the ECMD functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe}-\eqref{eq:beta_cmdpbe} is that we approximate here 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.~\onlinecite{GorSav-PRA-06}.
|
||||
This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{\Bas}(\br{},\br{})$.
|
||||
|
||||
Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{\Bas}]$ is approximated by $\bE{\LDA}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
|
||||
|
||||
%=================================================================
|
||||
%\subsection{Frozen-core approximation}
|
||||
%=================================================================
|
||||
|
||||
As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a subset of MOs.
|
||||
We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ and $\BasFC$ are the sets of core and active MOs, respectively) and define the FC version of the effective interaction as
|
||||
\begin{equation}
|
||||
\label{eq:WFC}
|
||||
\WFC{}{\Bas}(\br{1},\br{2}) =
|
||||
\begin{cases}
|
||||
\fFC{}{\Bas}(\br{1},\br{2})/\nFC{2}{\Bas}(\br{1},\br{2}), & \text{if $\nFC{2}{\Bas}(\br{1},\br{2}) \ne 0$},
|
||||
\\
|
||||
\infty, & \text{otherwise,}
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
with
|
||||
\begin{subequations}
|
||||
\begin{gather}
|
||||
\label{eq:fbasisval}
|
||||
\fFC{}{\Bas}(\br{1},\br{2})
|
||||
= \sum_{pq \in \Bas} \sum_{rstu \in \BasFC} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
|
||||
\\
|
||||
\nFC{2}{\Bas}(\br{1},\br{2})
|
||||
= \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
|
||||
\end{gather}
|
||||
\end{subequations}
|
||||
and the corresponding FC range-separation function $\rsmuFC{}{\Bas}(\br{}) = (\sqrt{\pi}/2) \WFC{}{\Bas}(\br{},\br{})$.
|
||||
It is noteworthy that, within the present definition, $\WFC{}{\Bas}(\br{1},\br{2})$ still tends to the regular Coulomb interaction as $\Bas \to \infty$.
|
||||
|
||||
Defining $\nFC{\modZ}{\Bas}$ as the FC (i.e.~valence-only) one-electron density obtained with a method $\modZ$ in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\LDA}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$.
|
||||
|
||||
%=================================================================
|
||||
%\subsection{Computational considerations}
|
||||
%=================================================================
|
||||
The most computationally intensive task of the present approach is the evaluation of $\W{}{\Bas}(\br{},\br{})$ at each quadrature grid point.
|
||||
In the general case (i.e.~$\wf{}{\Bas}$ is a multi-determinant expansion), we compute this embarrassingly parallel step in $\order*{\Ng \Nb^4}$ computational cost with a memory requirement of $\order*{ \Ng \Nb^2}$, where $\Nb$ is the number of basis functions in $\Bas$.
|
||||
The computational cost can be reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ with no memory footprint when $\wf{}{\Bas}$ is a single Slater determinant.
|
||||
As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, this choice for $\wf{}{\Bas}$ already provides, for weakly correlated systems, a quantitative representation of the incompleteness of $\Bas$.
|
||||
Hence, we will stick to this choice throughout the present study.
|
||||
In our current implementation, the computational bottleneck is the four-index transformation to get the two-electron integrals in the MO basis which appear in Eqs.~\eqref{eq:n2basis} and \eqref{eq:fbasis}.
|
||||
Nevertheless, this step usually has to be performed for most correlated WFT calculations.
|
||||
Modern integral decomposition techniques (such as density fitting \cite{Whi-JCP-73}) or atomic-orbital-based algorithms could be employed to significantly speed up this step.
|
||||
|
||||
To conclude this section, we point out that, thanks to the definitions \eqref{eq:def_weebasis} and \eqref{eq:mu_of_r} as well as the properties \eqref{eq:lim_W} and \eqref{eq:large_mu_ecmd}, independently of the DFT functional, the present basis-set correction
|
||||
i) can be applied to any WFT method that provides an energy and a density,
|
||||
ii) does not correct one-electron systems, and
|
||||
iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a given WFT method.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\section{Results}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%% FIGURE 1 %%%
|
||||
\begin{figure*}
|
||||
\includegraphics[width=0.30\linewidth]{fig1a}
|
||||
\hspace{1cm}
|
||||
\includegraphics[width=0.30\linewidth]{fig1b}
|
||||
\\
|
||||
\includegraphics[width=0.30\linewidth]{fig1c}
|
||||
\hspace{1cm}
|
||||
\includegraphics[width=0.30\linewidth]{fig1d}
|
||||
\caption{
|
||||
Deviation (in \kcal) from CBS atomization energies of \ce{C2} (top left), \ce{O2} (top right), \ce{N2} (bottom left) and \ce{F2} (bottom right) obtained with various methods and basis sets.
|
||||
The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
|
||||
See {\SI} for raw data.
|
||||
\label{fig:diatomics}}
|
||||
\end{figure*}
|
||||
|
||||
%%% TABLE II %%%
|
||||
\begin{table}
|
||||
\caption{
|
||||
Statistical analysis (in \kcal) of the G2 atomization energies depicted in Fig.~\ref{fig:G2_Ec}.
|
||||
Mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS reference atomization energies.
|
||||
CA corresponds to the number of cases (out of 55) obtained with chemical accuracy.
|
||||
See {\SI} for raw data.
|
||||
\label{tab:stats}}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{ldddd}
|
||||
Method & \tabc{MAD} & \tabc{RMSD} & \tabc{MAX} & \tabc{CA} \\
|
||||
\hline
|
||||
CCSD(T)/cc-pVDZ & 14.29 & 16.21 & 36.95 & 2 \\
|
||||
CCSD(T)/cc-pVTZ & 6.06 & 6.84 & 14.25 & 2 \\
|
||||
CCSD(T)/cc-pVQZ & 2.50 & 2.86 & 6.75 & 9 \\
|
||||
CCSD(T)/cc-pV5Z & 1.28 & 1.46 & 3.46 & 21 \\
|
||||
\\
|
||||
CCSD(T)+LDA/cc-pVDZ & 3.24 & 3.67 & 8.13 & 7 \\
|
||||
CCSD(T)+LDA/cc-pVTZ & 1.19 & 1.49 & 4.67 & 27 \\
|
||||
CCSD(T)+LDA/cc-pVQZ & 0.33 & 0.44 & 1.32 & 53 \\
|
||||
\\
|
||||
CCSD(T)+PBE/cc-pVDZ & 1.96 & 2.59 & 7.33 & 19 \\
|
||||
CCSD(T)+PBE/cc-pVTZ & 0.85 & 1.11 & 2.64 & 36 \\
|
||||
CCSD(T)+PBE/cc-pVQZ & 0.31 & 0.42 & 1.16 & 53 \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
|
||||
%%% FIGURE 2 %%%
|
||||
\begin{figure*}
|
||||
\includegraphics[width=\linewidth]{fig2a}
|
||||
\includegraphics[width=\linewidth]{fig2b}
|
||||
\includegraphics[width=\linewidth]{fig2c}
|
||||
\caption{
|
||||
Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
|
||||
The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
|
||||
See {\SI} for raw data.
|
||||
\label{fig:G2_Ec}}
|
||||
\end{figure*}
|
||||
|
||||
We begin our investigation of the performance of the basis-set correction by computing the atomization energies of \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} obtained with Dunning's cc-pVXZ basis (X $=$ D, T, Q and 5).
|
||||
\ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2 set \cite{CurRagTruPop-JCP-91} (see below), whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11}
|
||||
In a second time, we compute the atomization energies of the entire G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ basis set family.
|
||||
This molecular set has been intensively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08, Gro-JCP-09, FelPet-JCP-09, NemTowNee-JCP-10, FelPetHil-JCP-11, HauKlo-JCP-12, PetTouUmr-JCP-12, FelPet-JCP-13, KesSylKohTewMar-JCP-18}) and can be considered as a representative set of small organic and inorganic molecules.
|
||||
As a method $\modY$ we employ either CCSD(T) or exFCI.
|
||||
Here, 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.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
|
||||
In the case of the CCSD(T) calculations, we have $\modZ = \ROHF$ as we use the restricted open-shell HF (ROHF) one-electron density to compute the complementary basis-set correction energy.
|
||||
In the case of exFCI, the one-electron density is computed from a very large CIPSI expansion containing several million determinants.
|
||||
CCSD(T) energies are computed with Gaussian09 using standard threshold values, \cite{g09} while RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}
|
||||
For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17}
|
||||
Apart from the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-09} and have been obtained at the B3LYP/6-31G(2df,p) level of theory.
|
||||
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}.
|
||||
In the context of the basis-set correction, the set of active MOs, $\BasFC$, involved in the definition of the effective interaction [see Eq.~\eqref{eq:WFC}] refers to the non-frozen MOs.
|
||||
The FC density-based correction is used consistently with the FC approximation in WFT methods.
|
||||
To estimate the CBS limit of each method, following Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98}, we perform a two-point X$^{-3}$ extrapolation of the correlation energies using the quadruple- and quintuple-$\zeta$ data that we add up to the HF energies obtained in the largest (i.e.~quintuple-$\zeta$) basis.
|
||||
|
||||
As the exFCI atomization energies are converged with a precision of about 0.1 {\kcal}, we can label these as near FCI.
|
||||
Hence, they will be our references for \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2}.
|
||||
The results for these diatomic molecules are reported in Fig.~\ref{fig:diatomics}.
|
||||
The corresponding numerical data can be found in the {\SI}.
|
||||
As one can see, the convergence of the exFCI atomization energies is, as expected, slow with respect to the basis set: chemical accuracy (error below 1 {\kcal}) is barely reached for \ce{C2}, \ce{O2} and \ce{F2} even with the cc-pV5Z basis set, and the atomization energies are consistently underestimated.
|
||||
A similar trend holds for CCSD(T).
|
||||
Regarding the effect of the basis-set correction, several general observations can be made for both exFCI and CCSD(T).
|
||||
First, in a given basis set, the basis-set correction systematically improves the atomization energies (both at the LDA and PBE levels).
|
||||
A small overestimation can occur compared to the CBS value by a few tenths of a {\kcal} (the largest deviation being 0.6 {\kcal} for \ce{N2} at the CCSD(T)+PBE/cc-pV5Z level).
|
||||
Nevertheless, the deviation observed for the largest basis set is typically within the CBS extrapolation error, which is highly satisfactory knowing the marginal computational cost of the present correction.
|
||||
In most cases, the basis-set corrected triple-$\zeta$ atomization energies are on par with the uncorrected quintuple-$\zeta$ ones.
|
||||
Importantly, the sensitivity with respect to the RS-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better.
|
||||
However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}.
|
||||
Such weak sensitivity to the density-functional approximation when reaching large basis sets shows the robustness of the approach.
|
||||
|
||||
As a second set of numerical examples, we compute the error (with respect to the CBS values) of the atomization energies from the G2 test set with $\modY=\CCSDT$, $\modZ=\ROHF$ and the cc-pVXZ basis sets.
|
||||
Here, all atomization energies have been computed with the same near-CBS HF/cc-pV5Z energies; only the correlation energy contribution varies from one method to the other.
|
||||
Investigating the convergence of correlation energies (or difference of such quantities) is commonly done to appreciate the performance of basis-set corrections aiming at correcting two-electron effects. \cite{Tenno-CPL-04, TewKloNeiHat-PCCP-07, IrmGru-arXiv-2019}
|
||||
The ``plain'' CCSD(T) atomization energies as well as the corrected CCSD(T)+LDA and CCSD(T)+PBE values are depicted in Fig.~\ref{fig:G2_Ec}.
|
||||
The raw data can be found in the {\SI}.
|
||||
A statistical analysis of these data is also provided in Table \ref{tab:stats}, where we report the mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS atomization energies.
|
||||
Note that the MAD of our CCSD(T)/CBS atomization energies is only 0.37 {\kcal} compared to the values extracted from Ref.~\onlinecite{HauKlo-JCP-12} which corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
|
||||
From double-$\zeta$ to quintuple-$\zeta$ basis, the MAD associated with the CCSD(T) atomization energies goes down slowly from 14.29 to 1.28 {\kcal}.
|
||||
For a commonly used basis like cc-pVTZ, the MAD of CCSD(T) is still 6.06 {\kcal}.
|
||||
Applying the basis-set correction drastically reduces the basis-set incompleteness error.
|
||||
Already at the CCSD(T)+LDA/cc-pVDZ and CCSD(T)+PBE/cc-pVDZ levels, the MAD is reduced to 3.24 and 1.96 {\kcal}.
|
||||
With the triple-$\zeta$ basis, the MAD of CCSD(T)+PBE/cc-pVTZ is already below 1 {\kcal} with 36 cases (out of 55) where we achieve chemical accuracy.
|
||||
CCSD(T)+LDA/cc-pVQZ and CCSD(T)+PBE/cc-pVQZ return MAD of 0.33 and 0.31 kcal/mol (respectively) while CCSD(T)/cc-pVQZ still yields a fairly large MAD of 2.50 {\kcal}.
|
||||
|
||||
Therefore, similar to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis-set correction provides significant basis-set reduction and recovers quintuple-$\zeta$ quality atomization and correlation energies with triple-$\zeta$ basis sets for a much cheaper computational cost.
|
||||
Encouraged by these promising results, we are currently pursuing various avenues toward basis-set reduction for strongly correlated systems and electronically excited states.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section*{Supporting information}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
See {\SI} for raw data associated with the atomization energies of the four diatomic molecules and the G2 set.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{acknowledgements}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
The authors would like to thank the \emph{Centre National de la Recherche Scientifique} (CNRS) and the \emph{Institut des Sciences du Calcul et des Donn\'ees} for funding.
|
||||
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\end{acknowledgements}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\bibliography{G2-srDFT,G2-srDFT-control}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\end{document}
|
||||
\maketitle
|
||||
Loading…
Reference in New Issue