565 lines
38 KiB
TeX
565 lines
38 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
|
|
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,xspace,wrapfig}
|
|
\usepackage{mathpazo,libertine}
|
|
|
|
\usepackage{natbib}
|
|
\bibliographystyle{achemso}
|
|
\AtBeginDocument{\nocite{achemso-control}}
|
|
|
|
\newcommand{\alert}[1]{\textcolor{red}{#1}}
|
|
\definecolor{darkgreen}{HTML}{009900}
|
|
\usepackage[normalem]{ulem}
|
|
\newcommand{\titou}[1]{\textcolor{red}{#1}}
|
|
\newcommand{\juju}[1]{\textcolor{purple}{#1}}
|
|
\newcommand{\manu}[1]{\textcolor{darkgreen}{#1}}
|
|
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
|
|
\newcommand{\trashJT}[1]{\textcolor{purple}{\sout{#1}}}
|
|
\newcommand{\trashMG}[1]{\textcolor{darkgreen}{\sout{#1}}}
|
|
\newcommand{\MG}[1]{\manu{(\underline{\bf MG}: #1)}}
|
|
\newcommand{\JT}[1]{\juju{(\underline{\bf JT}: #1)}}
|
|
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
|
|
|
|
\usepackage{hyperref}
|
|
\hypersetup{
|
|
colorlinks=true,
|
|
linkcolor=blue,
|
|
filecolor=blue,
|
|
urlcolor=blue,
|
|
citecolor=blue
|
|
}
|
|
\newcommand{\mc}{\multicolumn}
|
|
\newcommand{\fnm}{\footnotemark}
|
|
\newcommand{\fnt}{\footnotetext}
|
|
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
|
|
\newcommand{\SI}{\textcolor{blue}{supporting information}}
|
|
\newcommand{\QP}{\textsc{quantum package}}
|
|
|
|
% second quantized operators
|
|
\newcommand{\ai}[1]{\hat{a}_{#1}}
|
|
\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}}
|
|
|
|
% units
|
|
\newcommand{\IneV}[1]{#1 eV}
|
|
\newcommand{\InAU}[1]{#1 a.u.}
|
|
\newcommand{\InAA}[1]{#1 \AA}
|
|
\newcommand{\kcal}{kcal/mol}
|
|
|
|
% methods
|
|
\newcommand{\D}{\text{D}}
|
|
\newcommand{\T}{\text{T}}
|
|
\newcommand{\Q}{\text{Q}}
|
|
\newcommand{\X}{\text{X}}
|
|
\newcommand{\UEG}{\text{UEG}}
|
|
\newcommand{\HF}{\text{HF}}
|
|
\newcommand{\ROHF}{\text{ROHF}}
|
|
\newcommand{\LDA}{\text{LDA}}
|
|
\newcommand{\PBE}{\text{PBE}}
|
|
\newcommand{\FCI}{\text{FCI}}
|
|
\newcommand{\CBS}{\text{CBS}}
|
|
\newcommand{\exFCI}{\text{exFCI}}
|
|
\newcommand{\CCSDT}{\text{CCSD(T)}}
|
|
\newcommand{\lr}{\text{lr}}
|
|
\newcommand{\sr}{\text{sr}}
|
|
|
|
\newcommand{\Ne}{N}
|
|
\newcommand{\NeUp}{\Ne^{\uparrow}}
|
|
\newcommand{\NeDw}{\Ne^{\downarrow}}
|
|
\newcommand{\Nb}{N_{\Bas}}
|
|
\newcommand{\Ng}{N_\text{grid}}
|
|
\newcommand{\nocca}{n_{\text{occ}^{\alpha}}}
|
|
\newcommand{\noccb}{n_{\text{occ}^{\beta}}}
|
|
|
|
\newcommand{\n}[2]{n_{#1}^{#2}}
|
|
\newcommand{\Ec}{E_\text{c}}
|
|
\newcommand{\E}[2]{E_{#1}^{#2}}
|
|
\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
|
|
\newcommand{\bEc}[1]{\Bar{E}_\text{c,md}^{#1}}
|
|
\newcommand{\e}[2]{\varepsilon_{#1}^{#2}}
|
|
\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
|
|
\newcommand{\bec}[1]{\Bar{e}^{#1}}
|
|
\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
|
|
\newcommand{\W}[2]{W_{#1}^{#2}}
|
|
\newcommand{\w}[2]{w_{#1}^{#2}}
|
|
\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
|
|
\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
|
|
\newcommand{\V}[2]{V_{#1}^{#2}}
|
|
\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
|
|
|
|
\newcommand{\modY}{Y}
|
|
\newcommand{\modZ}{Z}
|
|
|
|
% basis sets
|
|
\newcommand{\Bas}{\mathcal{B}}
|
|
\newcommand{\BasFC}{\mathcal{A}}
|
|
\newcommand{\FC}{\text{FC}}
|
|
\newcommand{\occ}{\text{occ}}
|
|
\newcommand{\virt}{\text{virt}}
|
|
\newcommand{\val}{\text{val}}
|
|
\newcommand{\Cor}{\mathcal{C}}
|
|
|
|
% operators
|
|
\newcommand{\hT}{\Hat{T}}
|
|
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
|
|
\newcommand{\updw}{\uparrow\downarrow}
|
|
\newcommand{\f}[2]{f_{#1}^{#2}}
|
|
\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
|
|
|
|
% coordinates
|
|
\newcommand{\br}[1]{\mathbf{r}_{#1}}
|
|
\newcommand{\dbr}[1]{d\br{#1}}
|
|
|
|
\newcommand{\ra}{\rightarrow}
|
|
\newcommand{\De}{D_\text{e}}
|
|
|
|
\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}[12]{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} $\mu(\bf{r})$ which 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 yields 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}
|
|
DFT's attractiveness originates from its very favorable cost/accuracy 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 written 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 the basis set $\Bas$).
|
|
|
|
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 source of error at this stage lies in the potential approximate nature of the methods $\modY$ and $\modZ$.
|
|
|
|
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 in $\wf{}{\Bas}$ (i.e.~a discontinuous derivative) 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 $\mathbb{R}^3$.
|
|
|
|
The first step of the present basis-set correction consists of 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 $\rsmu{\Bas}{}(\br{})$.
|
|
In the 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}
|
|
\label{eq:int_eq_wee}
|
|
\mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \frac{1}{2}\iint \W{\Bas}{}(\br{1},\br{2}) \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
|
|
\end{equation}
|
|
where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$.
|
|
Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as
|
|
\begin{eqnarray}
|
|
\frac{1}{2}\iint \frac{1}{r_{12}} \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2} = \phantom{xxxxxxxxx}
|
|
\nonumber\\
|
|
\frac{1}{2}\iint \W{\Bas}{}(\br{1},\br{2}) \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
|
|
\end{eqnarray}
|
|
it 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}) = \frac{1}{r_{12}}
|
|
\end{equation}
|
|
for any $(\br{1},\br{2})$ such that $\n{2,\Bas}{}(\br{1},\br{2}) \ne 0$.% and for any $\wf{}{\Bas}$, which guarantees a physically satisfying limit.
|
|
|
|
%=================================================================
|
|
%\subsection{Range-separation function}
|
|
%=================================================================
|
|
|
|
A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons, $\W{\Bas}{}(\br{},{\br{}})$,
|
|
%\begin{equation}
|
|
% \label{eq:wcoal}
|
|
% \W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}),
|
|
%\end{equation}
|
|
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
|
|
%\begin{equation}
|
|
% \w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{\Bas}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{\Bas}{}(\br{2}) r_{12}]}{ r_{12}} }
|
|
%\end{equation}
|
|
\begin{equation}
|
|
\w{}{\lr,\mu}(r_{12}) = \frac{\erf( \mu r_{12})}{r_{12}}
|
|
\end{equation}
|
|
coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}(\br{})}(0) = \W{\Bas}{}(\br{},\br{})$ at any point $\br{}$.
|
|
|
|
%=================================================================
|
|
%\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{multline}
|
|
\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{multline}
|
|
\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 $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
|
|
%\begin{multline}
|
|
% \label{eq:ec_md_mu}
|
|
% \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
|
|
% \\
|
|
% - \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
|
|
%\end{multline}
|
|
%where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
|
|
%\begin{equation}
|
|
%\label{eq:argmin}
|
|
% \wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
|
|
%\end{equation}
|
|
%with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
|
|
%and
|
|
%\begin{equation}
|
|
%\label{eq:erf}
|
|
% \w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}.
|
|
%\end{equation}
|
|
%is the long-range part of the Coulomb operator.
|
|
The ECMD functionals admit, for any $\n{}{}$, the following two limiting forms
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\label{eq:large_mu_ecmd}
|
|
\lim_{\mu \to \infty} \bE{}{\sr}[\n{}{},\rsmu{}{}] & = 0,
|
|
\\
|
|
\label{eq:small_mu_ecmd}
|
|
\lim_{\mu \to 0} \bE{}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
|
|
\end{align}
|
|
\end{subequations}
|
|
where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KS-DFT.
|
|
The choice of the 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 $\wf{}{\Bas} = \wf{}{\rsmu{}{}}$.
|
|
%provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) = \W{\Bas}{}(\br{1},\br{2})$, then $\wf{}{\rsmu{\Bas}{}}$ and $\wf{}{\Bas}$ coincide.
|
|
%The ECMD functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function.
|
|
%This makes them particularly well adapted to the present context where one aims at correcting a general WFT method.
|
|
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{\sigma}{}(\br{})\},\rsmu{\Bas}{}(\br{})) \dbr{},
|
|
\end{equation}
|
|
where $\be{\text{c,md}}{\sr,\LDA}(\{\n{\sigma}{}\},\rsmu{}{})$ is the ECMD correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized in Ref.~\onlinecite{PazMorGorBac-PRB-06} as a function of the spin densities $\{\n{\sigma}{}\}_{\sigma=\uparrow,\downarrow}$ and the range-separation parameter $\mu$.
|
|
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, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional
|
|
\begin{eqnarray}
|
|
\label{eq:def_pbe_tot}
|
|
\bE{\PBE}{\Bas}[\n{}{},\rsmu{\Bas}{}] = \phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxx}
|
|
\nonumber\\
|
|
\int \! \n{}{}(\br{}) \; \be{\text{c,md}}{\sr,\PBE}\qty(\{\n{\sigma}{}(\br{})\},\{\nabla \n{\sigma}{}(\br{})\},\rsmu{\Bas}{}(\br{})) \dbr{},
|
|
\end{eqnarray}
|
|
inspired by the recent functional proposed by some of the authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional~\cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\{\n{\sigma}{}\},\{\nabla \n{\sigma}{}\})$ for $\rsmu{}{}=0$ 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{\sigma}{}\},\{\nabla \n{\sigma}{}\},\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\{\n{\sigma}{}\},\{\nabla \n{\sigma}{}\})}{1 + \beta(\{n_\sigma\},\{\nabla n_\sigma\}, \rsmu{}{})\rsmu{}{3} },
|
|
\\
|
|
\label{eq:beta_cmdpbe}
|
|
\beta(\{n_\sigma\},\{\nabla n_\sigma\},\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\{\n{\sigma}{}\},\{\nabla \n{\sigma}{}\})}{\n{2}{\UEG}(0,\{\n{\sigma}{}\})}.
|
|
\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.~$\n{2,\Bas}{}(\br{},\br{}) \approx \n{2}{\UEG}(0,\{\n{\sigma}{}(\br{})\})$, with $\n{2}{\UEG}(0,\{n_\sigma\}) = 4 \; n_{\uparrow} \; n_{\downarrow} \; g(0,n)$ and the UEG on-top pair-distribution function $g(0,n)$, the $0$ standing for $r_{12}=0$, whose parametrization can be found 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 $\bE{}{\Bas}[\n{\modZ}{}]$ is then equal to $\bE{\LDA}{\Bas}[\n{\modZ}{},\rsmu{\Bas}{}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{},\rsmu{\Bas}{}]$ 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}
|
|
\W{\Bas}{\FC}(\br{1},\br{2}) \! = \!
|
|
\begin{cases}
|
|
\f{\Bas}{\FC}(\br{1},\br{2})/\n{2,\Bas}{\FC}(\br{1},\br{2}),\! & \!\!\! \text{if $\n{2,\Bas}{\FC}(\br{1},\br{2}) \!\ne \! 0$},
|
|
\\
|
|
\infty, \! & \!\!\! \text{otherwise,}
|
|
\end{cases}
|
|
\end{equation}
|
|
with
|
|
\begin{subequations}
|
|
\begin{gather}
|
|
\label{eq:fbasisval}
|
|
\f{\Bas}{\FC}(\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},
|
|
\\
|
|
\n{2,\Bas}{\FC}(\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
|
|
\begin{equation}
|
|
\label{eq:muval}
|
|
\rsmu{\Bas}{\FC}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\FC}(\br{},\br{}).
|
|
\end{equation}
|
|
It is worth noticing that, within the present definition, $\W{\Bas}{\FC}(\br{1},\br{2})$ still tends to the regular Coulomb interaction when $\Bas \to \infty$.
|
|
|
|
Defining $\n{\modZ}{\FC}$ as the FC (i.e.~valence-only) one-electron density obtained with a method $\modZ$, the FC contribution of the complementary functional is then evaluated as $\bE{\LDA}{\Bas}[\n{\modZ}{\FC},\rsmu{\Bas}{\FC}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\FC},\rsmu{\Bas}{\FC}]$.
|
|
|
|
%=================================================================
|
|
%\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.
|
|
Yet embarrassingly parallel, this step scales, in the general case (i.e.~$\wf{}{\Bas}$ is a determinant expansion), as $\Ng \Nb^4$ (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ 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 current 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]{C2_VXZ}
|
|
\hspace{1cm}
|
|
\includegraphics[width=0.30\linewidth]{O2_VXZ}
|
|
\\
|
|
\includegraphics[width=0.30\linewidth]{N2_VXZ}
|
|
\hspace{1cm}
|
|
\includegraphics[width=0.30\linewidth]{F2_VXZ}
|
|
\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]{VDZ}
|
|
\includegraphics[width=\linewidth]{VTZ}
|
|
\includegraphics[width=\linewidth]{VQZ}
|
|
\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 sets (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 family of basis sets.
|
|
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 millions of 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}
|
|
Except for 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 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 refers to the non-frozen MOs.
|
|
The FC density-based correction is used consistently when the FC approximation was applied 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 calculations are converged with a precision of about 0.1 {\kcal} on atomization energies, we can label those 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}
|