a bunch of modifications
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@ -7832,6 +7832,7 @@
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@article{MusReiAngTou-JCP-15,
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Author = {B. Mussard and P. Reinhardt and J. G. \'Angy\'an and J. Toulouse},
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title = {Spin-unrestricted random-phase approximation with range separation: Benchmark on atomization energies and reaction barrier heights},
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Journal = {J. Chem. Phys.},
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Note = {Erratum: J. Chem. Phys. {\bf 142}, 219901 (2015)},
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Pages = {154123},
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@ -73,7 +73,7 @@
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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}^{#1}}
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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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@ -118,7 +118,7 @@
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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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\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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@ -141,7 +141,7 @@
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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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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} $\mu(\bf{r})$ which 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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@ -154,7 +154,7 @@ As illustrative examples, we show how this density-based correction allows us to
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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 attractive WFT \textit{ans\"atze}.
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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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@ -165,23 +165,23 @@ For example, at the CCSD(T) level, one can obtain quintuple-$\zeta$ quality corr
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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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DFT's attractiveness originates from its very favorable cost/efficiency ratio as it can provide accurate energies and properties at a relatively low computational cost.
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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.
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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 the Jacob's ladder of DFT is potentially the most satisfactory way forward. \cite{PerSch-AIPCP-01, PerRuzTaoStaScuCso-JCP-05}
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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 smooth long-range part and a (complementary) short-range part treated with WFT and DFT, respectively.
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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} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, FerGinTou-JCP-18} WFT approaches.
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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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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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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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@ -201,8 +201,8 @@ where
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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 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 $\n{}{}$.
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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 the basis set $\Bas$).
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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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@ -210,7 +210,7 @@ This implies that
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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 complete basis set.
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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 source of error at this stage lies in the potential approximate nature of the methods $\modY$ and $\modZ$.
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@ -218,12 +218,12 @@ The functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it d
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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 in $\wf{}{\Bas}$ (i.e.~a discontinuous derivative) 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 $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals where one deals with smooth long-range interactions.
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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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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 $\Bas$.
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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 $\Bas$.
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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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In the final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{\Bas}{}(\br{})$ as range separation.
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In the 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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@ -265,12 +265,6 @@ it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate fo
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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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A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons
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\begin{equation}
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\label{eq:wcoal}
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\W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}),
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\end{equation}
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which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{}(\br{})$ is non vanishing.
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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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@ -282,34 +276,45 @@ for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$.% and for any
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%\subsection{Range-separation function}
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%=================================================================
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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 smooth long-range operators.
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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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%\begin{equation}
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% \label{eq:wcoal}
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% \W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}),
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%\end{equation}
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which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{}(\br{},\br{})$ 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{}) ,
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\rsmu{\Bas}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{},\br{}),
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\end{equation}
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such that the long-range interaction
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such that the long-range interaction of RS-DFT
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%\begin{equation}
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% \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}} }
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%\end{equation}
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\begin{equation}
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\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}} }
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\w{}{\lr,\mu}(r_{12}) = \frac{\erf( \mu r_{12})}{r_{12}}
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\end{equation}
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coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{})$.
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coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}(\br{})}(0) = \W{\Bas}{}(\br{},\br{})$ at any point $\br{}$.
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%=================================================================
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%\subsection{Short-range correlation functionals}
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%=================================================================
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Once $\rsmu{\Bas}{}(\br{})$ is defined, it can be used within RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$.
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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}
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\begin{multline}
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%\begin{multline}
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\begin{equation}
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\label{eq:ec_md_mu}
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\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}]
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= \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
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\\
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\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]
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= \min_{\wf{}{} \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
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% \\
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- \mel*{\wf{}{\rsmu{}{}}}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}},
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\end{multline}
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%\end{multline}
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\end{equation}
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where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization
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\begin{equation}
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\label{eq:argmin}
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\wf{}{\rsmu{}{}} = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
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\wf{}{\rsmu{}{}} = \arg \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
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\end{equation}
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with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
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%\begin{multline}
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@ -330,56 +335,58 @@ with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
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% \w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}.
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%\end{equation}
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%is the long-range part of the Coulomb operator.
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The ECMD functionals admit, for any $\n{}{}(\br{})$, the following two limiting forms
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The ECMD functionals admit, for any $\n{}{}$, the following two limiting forms
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\begin{subequations}
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\begin{align}
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\label{eq:large_mu_ecmd}
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\lim_{\mu \to \infty} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = 0,
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\lim_{\mu \to \infty} \bE{}{\sr}[\n{}{},\rsmu{}{}] & = 0,
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\\
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\label{eq:small_mu_ecmd}
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\lim_{\mu \to 0} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = \Ec[\n{}{}(\br{})],
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\lim_{\mu \to 0} \bE{}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
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\end{align}
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\end{subequations}
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where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
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where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KS-DFT.
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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}].
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Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) = \W{\Bas}{}(\br{1},\br{2})$, then $\wf{}{\rsmu{\Bas}{}}$ and $\wf{}{\Bas}$ coincide.
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%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.
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%This makes them particularly well adapted to the present context where one aims at correcting a general WFT method.
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Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by ECMD functionals evaluated with the range-separation function $\rsmu{\Bas}{}(\br{})$.
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The LDA version of the ECMD complementary functional is defined as
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The local-density approximation (LDA) of the ECMD complementary functional is defined as
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\begin{equation}
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\label{eq:def_lda_tot}
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\bE{\LDA}{\Bas}[\n{}{}(\br{}),\rsmu{\Bas}{}(\br{})] = \int \be{\LDA}{\sr}\qty(\n{}{}(\br{}),\rsmu{\Bas}{}(\br{})) \n{}{}(\br{}) \dbr{},
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\bE{\LDA}{\Bas}[\n{}{},\rsmu{\Bas}{}] = \int \! \n{}{}(\br{}) \; \be{\text{c,md}}{\sr,\LDA}\qty(\{\n{\sigma}{}(\br{})\},\rsmu{\Bas}{}(\br{})) \dbr{},
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\end{equation}
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where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the reduced (i.e.~per electron) ECMD functional of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
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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 $\mu$.
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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 $\mu$.
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In order to correct such a defect, we propose here a new PBE ECMD functional
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\begin{equation}
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In order to correct such a defect, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional
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\begin{eqnarray}
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\label{eq:def_pbe_tot}
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\bE{\PBE}{\Bas}[\n{}{}(\br{}),\rsmu{\Bas}{}(\br{})] = \int \be{\PBE}{\sr}\qty(\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{\Bas}{}(\br{})) \n{}{}(\br{}) \dbr{}
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\end{equation}
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inspired by the recent functional proposed by some of the authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{\PBE}{}(\n{}{},\nabla \n{}{})$ for $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
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\bE{\PBE}{\Bas}[\n{}{},\rsmu{\Bas}{}] = \phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxx}
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\nonumber\\
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\int \! \n{}{}(\br{}) \; \be{\text{c,md}}{\sr,\PBE}\qty(\{\n{\sigma}{}(\br{})\},\{\nabla \n{\sigma}{}(\br{})\},\rsmu{\Bas}{}(\br{})) \dbr{},
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\end{eqnarray}
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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{\PBE}{\sr}(\n{}{},\nabla \n{}{},\rsmu{}{}) = \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{1 + \beta(n,\nabla n, \rsmu{}{})\rsmu{}{3} },
|
||||
\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,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{2}{\UEG}(\n{}{})}.
|
||||
\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}(\{\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} is that we approximate here the \textit{exact} ground-state on-top pair density by its UEG version, i.e.~$\n{2}{}(\br{}) \approx \n{2}{\UEG}(\n{}{}(\br{})) = \n{}{}(\br{})^2 g_0(\n{}{}(\br{}))$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
|
||||
This represents a major computational saving without loss of performance as we eschew the computation of $\n{2}{}(\br{})$.
|
||||
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 \textit{exact} on-top pair density by its UEG version, i.e.~$\n{2}{}(\br{},\br{}) \approx \n{2}{\UEG}(0,\n{}{}(\br{}))$, with $\n{2}{\UEG}(0,n) = 4 \; n_{\uparrow} \; n_{\downarrow} \; g(0,n)$ and the UEG on-top pair-distribution function $g(0,n)$ 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 $\n{2}{}(\br{})$.
|
||||
|
||||
Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{}]$ is then equal to $\bE{\LDA}{\Bas}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
|
||||
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
|
||||
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}
|
||||
@ -406,12 +413,12 @@ and the corresponding FC range-separation function
|
||||
\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 model $\modZ$, the FC contribution of the complementary functional is then evaluated as $\bE{\LDA}{\Bas}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$.
|
||||
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 $\W{\Bas}{}(\br{})$ [see Eq.~\eqref{eq:wcoal}] at each quadrature grid point.
|
||||
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.
|
||||
@ -419,10 +426,10 @@ In our current implementation, the computational bottleneck is the four-index tr
|
||||
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 model that provides an energy and a density,
|
||||
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 limit of a complete basis set, hence guaranteeing an unaltered CBS limit for a given WFT model.
|
||||
iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a given WFT method.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\section{Results}
|
||||
@ -484,53 +491,53 @@ iii) vanishes in the limit of a complete basis set, hence guaranteeing an unalte
|
||||
\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).
|
||||
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 exhausively 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.
|
||||
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 energy.
|
||||
For exFCI, the one-electron density is computed from a very large CIPSI expansion containing several million determinants.
|
||||
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 MOs $\BasFC$ involved in the definition of the effective interaction refers to the non-frozen MOs.
|
||||
The FC density-based correction is set consistently when the FC approximation was applied in WFT methods.
|
||||
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 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}, we can label these atomization energies as near-FCI.
|
||||
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 diatomics 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).
|
||||
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 SR-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better.
|
||||
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}
|
||||
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.
|
||||
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.
|
||||
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.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
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