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Manuscript/G2-srDFT.tex
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@ -235,7 +235,7 @@ We report a universal density-based basis set incompleteness correction that can
\maketitle \maketitle
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction} %\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} 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 advantages and shortcomings. Although both spring from the same Schr\"odinger equation, each of these philosophies has its own advantages and shortcomings.
@ -263,13 +263,11 @@ Using accurate and rigorous WFT methods, some of us have developed radical gener
In that respect range-separated DFT (RS-DFT) is particularly promising as it allows to perform multi-configurational DFT calculations within a rigorous mathematical framework. In that respect range-separated DFT (RS-DFT) is particularly promising as it allows to perform multi-configurational DFT calculations within a rigorous mathematical framework.
Range-separated hybrids, i.e.~single-determinant approximations of RS-DFT, correct for the wrong long-range behavior of the usual hybrid approximations thanks to the inclusion of the long-range part of the Hartree-Fock (HF) exchange. Range-separated hybrids, i.e.~single-determinant approximations of RS-DFT, correct for the wrong long-range behavior of the usual hybrid approximations thanks to the inclusion of the long-range part of the Hartree-Fock (HF) exchange.
%The present manuscript is organised as follows.
The present manuscript is organised as follows.
Unless otherwise stated, atomic used are used. Unless otherwise stated, atomic used are used.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory} %\section{Theory}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
The basis set correction employed here relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the non-completeness of the one-electron basis set. The basis set correction employed here relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the non-completeness of the one-electron basis set.
Here, we only provide the main working equations. Here, we only provide the main working equations.
@ -294,7 +292,7 @@ where
= \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}} = \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}}
- \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}} - \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
\end{equation} \end{equation}
is the basis-dependent complementary density functional defined in Eq.~(8) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, and $\hT$ and $\hWee{}$ are the kinetic and interelectronic repulsion operators, respectively. is the basis-dependent complementary density functional defined in Eq.~(8) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, and $\hT$ and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ are the kinetic and interelectronic repulsion operators, respectively.
In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Nel$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively. In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Nel$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively.
Both wave functions yield the same target density $\n{}{}$. Both wave functions yield the same target density $\n{}{}$.
@ -423,7 +421,7 @@ The final step employs $\rsmu{}{}(\br{})$ within short-range density functionals
%\label{sec:weff} %\label{sec:weff}
%================================================================= %=================================================================
%================================================================= %=================================================================
\subsection{Effective Coulomb operator} %\subsection{Effective Coulomb operator}
%================================================================= %=================================================================
In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ defined such that In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ defined such that
\begin{equation} \begin{equation}
@ -460,67 +458,81 @@ where
and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\wf{}{\Bas}}$ is the two-body density tensor of $\wf{}{\Bas}$, it comes naturally that and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\wf{}{\Bas}}$ is the two-body density tensor of $\wf{}{\Bas}$, it comes naturally that
\begin{equation} \begin{equation}
\label{eq:def_weebasis} \label{eq:def_weebasis}
\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})}{\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})}. \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \f{\wf{}{\Bas}}{}(\bx{1},\bx{2})/\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}).
\end{equation} \end{equation}
As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries and is necessarily \textit{finite} at $r_{12} = 0$ for an incomplete basis set. As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries and is necessarily \textit{finite} at $r_{12} = 0$ for an incomplete basis set.
Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = r_{12}^{-1}$. Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}
\begin{equation}
\label{eq:lim_W}
\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = r_{12}^{-1}.
\end{equation}
%================================================================= %=================================================================
\subsection{Range-separation function} %\subsection{Range-separation function}
%================================================================= %=================================================================
To be able to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ thanks to functionals developed in the field of RS-DFT, we fit the effective interaction to a long-range interaction characterised by a range-separation function $\rsmu{}{}(\br{})$, i.e~varying in space. To be able to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ thanks to functionals developed in the field of RS-DFT, we associate the effective interaction to a long-range interaction characterised by a range-separation function $\rsmu{}{}(\br{})$.
More precisely, if we define the value of the interaction at coalescence as Although this choice is not unique, the long-range interaction we have chosen is
\begin{equation}
\label{eq:def_wcoal}
\wbasiscoal{} = W_{\wf{}{\Bas}}(\bfr{},\bar{{\bf x}}_{}).
\end{equation}
where $(\bx{},\Bar{\bx{}})$ means a couple of anti-parallel spins at the same position $\br{}$,
we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction:
\begin{equation}
\wbasiscoal{} = \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\bfrb{},\bfrb{})
\end{equation}
where the long-range-like interaction is defined as
\begin{equation} \begin{equation}
\w{}{\lr,\rsmu{}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \murr{1} \, r_{12}]}{r_{12}} + \frac{\erf[ \murr{2} r_{12}]}{ r_{12}} }. \w{}{\lr,\rsmu{}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \murr{1} \, r_{12}]}{r_{12}} + \frac{\erf[ \murr{2} r_{12}]}{ r_{12}} }.
\end{equation} \end{equation}
Equation \eqref{eq:def_wcoal} is equivalent to the following condition and ensuring that $\w{}{\lr,\rsmu{}{}}(\br{1},\br{2})$ and $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ have the same value at coalescence of same-spin electron pairs yields
\begin{equation} \begin{equation}
\label{eq:mu_of_r} \label{eq:mu_of_r}
\rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{2}{\sqrt{\pi}} \W{\wf{}{\Bas}}{}(\br{}) \rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{}).
\end{equation} \end{equation}
%More precisely, if we define the value of the interaction at coalescence as
%\begin{equation}
% \label{eq:def_wcoal}
% \wbasiscoal{} = W_{\wf{}{\Bas}}(\bfr{},\bar{{\bf x}}_{}).
%\end{equation}
%where $(\bx{},\Bar{\bx{}})$ means a couple of anti-parallel spins at the same position $\br{}$,
%we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction:
%\begin{equation}
% \wbasiscoal{} = \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\bfrb{},\bfrb{})
%\end{equation}
%where the long-range-like interaction is defined as
%\begin{equation}
% \w{}{\lr,\rsmu{}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \murr{1} \, r_{12}]}{r_{12}} + \frac{\erf[ \murr{2} r_{12}]}{ r_{12}} }.
%\end{equation}
%Equation \eqref{eq:def_wcoal} is equivalent to the following condition
%\begin{equation}
% \label{eq:mu_of_r}
% \rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{})
%\end{equation}
%As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as %As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as
%\begin{equation} %\begin{equation}
% \label{eq:mu_of_r_val} % \label{eq:mu_of_r_val}
% \murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, . % \murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, .
%\end{equation} %\end{equation}
An important point to notice is that, in the limit of a complete basis set $\Bas$, as %An important point to notice is that, in the limit of a complete basis set $\Bas$, as
\begin{equation} %\begin{equation}
\label{eq:lim_W} %\label{eq:lim_W}
\lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1} \quad \forall (\bx{1},\bx{2}) % \lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1} \quad \forall (\bx{1},\bx{2})
% &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, , %% &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, ,
\end{equation} %\end{equation}
one has $\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = \infty$ %one has $\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = \infty$
% &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,, %% &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,,
and therefore %and therefore
\begin{equation} %\begin{equation}
\label{eq:lim_mur} %\label{eq:lim_mur}
\lim_{\Bas \rightarrow \infty} \rsmu{\wf{}{\Bas}}{}(\br{}) = \infty % \lim_{\Bas \rightarrow \infty} \rsmu{\wf{}{\Bas}}{}(\br{}) = \infty
%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, . %%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
\end{equation} %\end{equation}
%================================================================= %=================================================================
\subsection{Complementary functional} %\subsection{Complementary functional}
%================================================================= %=================================================================
\label{sec:ecmd} %\label{sec:ecmd}
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we propose here to approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of SR-DFT energy functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05} Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we propose here to approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
\begin{multline} \begin{multline}
\label{eq:ec_md_mu} \label{eq:ec_md_mu}
\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
\\ \\
- \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}, - \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
\end{multline} \end{multline}
@ -529,25 +541,21 @@ where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimizati
\label{eq:argmin} \label{eq:argmin}
\wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}}, \wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
\end{equation} \end{equation}
with with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
\begin{equation} %and
\label{eq:weemu} %\begin{equation}
\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij}) %\label{eq:erf}
\end{equation} % \w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}.
and %\end{equation}
\begin{equation} %is the long-range part of the Coulomb operator.
\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 density $\n{}{}(\br{})$, the two following limiting forms: The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following limiting forms:
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:large_mu_ecmd} \label{eq:large_mu_ecmd}
\lim_{\mu \to \infty} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = 0, \lim_{\mu \to \infty} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] & = 0,
\\ \\
\label{eq:small_mu_ecmd} \label{eq:small_mu_ecmd}
\lim_{\mu \to 0} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = \Ec[\n{}{}(\br{})], \lim_{\mu \to 0} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] & = \Ec[\n{}{}(\br{})],
\end{align} \end{align}
\end{subequations} \end{subequations}
where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT. where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
@ -559,9 +567,9 @@ These functionals differ from the standard RS-DFT correlation functional by the
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we define the LDA version of ECMD as Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we define the LDA version of ECMD as
\begin{equation} \begin{equation}
\label{eq:def_lda_tot} \label{eq:def_lda_tot}
\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{} \bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] \n{}{}(\br{}) \dbr{},
\end{equation} \end{equation}
where $\be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}]$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}. where $\be{\UEG}{\sr}(\n{}{},\rsmu{}{})$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
%In practice, for open-shell systems, we use the spin-resolved version of this functional (i.e., depending on both spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case. %In practice, for open-shell systems, we use the spin-resolved version of this functional (i.e., depending on both spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
%-------------------------------------------- %--------------------------------------------
@ -578,41 +586,37 @@ In order to correct such a defect, we propose here a new ECMD functional inspire
\beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}. \beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}.
\end{gather} \end{gather}
\end{subequations} \end{subequations}
The difference between the ECMD PBE functional defined in Ref.~\cite{FerGinTou-JCP-18} and the present expression is that we approximate the \textit{exact} ground-state on-top pair density of the system $\n{}{(2)}(\br{})$ by its UEG version, i.e. The difference between the ECMD PBE functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression is that we approximate the \textit{exact} ground-state on-top pair density of the system $\n{}{(2)}(\br{})$ by its UEG version, i.e.~$\n{}{(2)} \approx \n{\UEG}{(2)} = \n{}{2} g_0(\n{}{})$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
\begin{equation}
\label{eq:ueg_ontop}
\n{}{(2)}(\br{}) \approx \n{\UEG}{(2)}(\br{}) = \n{}{}(\br{})^2 g_0[\n{}{}(\br{})]
\end{equation}
where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
Therefore, the PBE complementary function reads Therefore, the PBE complementary function reads
\begin{equation} \begin{equation}
\label{eq:def_lda_tot} \label{eq:def_lda_tot}
\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\PBE}{\sr}[\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{} \bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}[\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})] \n{}{}(\br{}) \dbr{}
\end{equation} \end{equation}
The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$ labelled by ECMD-$\mathcal{X}$. %The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$ labelled by ECMD-$\mathcal{X}$.
Such a functional of the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$. %Such a functional of the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$.
A general scheme to approximate $\ecompmodel$ is to use $\ecmuapprox$ with the $\mur$ defined in \eqref{eq:def_weebasis} and to evaluate it at the density defined by the model $\denmodel$ %A general scheme to approximate $\ecompmodel$ is to use $\ecmuapprox$ with the $\mur$ defined in \eqref{eq:def_weebasis} and to evaluate it at the density defined by the model $\denmodel$
\begin{equation} %\begin{equation}
\label{eq:approx_ecfuncbasis} % \label{eq:approx_ecfuncbasis}
\ecompmodel \approx \ecmuapproxmurmodel % \ecompmodel \approx \ecmuapproxmurmodel
\end{equation} %\end{equation}
Therefore, any approximated ECMD can be used to estimate $\ecompmodel$. %Therefore, any approximated ECMD can be used to estimate $\ecompmodel$.
It is important to notice that in the limit of a complete basis set, according to equations \eqref{eq:lim_mur} and \eqref{eq:large_mu_ecmd} one has %It is important to notice that in the limit of a complete basis set, according to equations \eqref{eq:lim_mur} and \eqref{eq:large_mu_ecmd} one has
\begin{equation} %\begin{equation}
\lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad , % \lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
\end{equation} %\end{equation}
for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD. %for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
%================================================================= %=================================================================
\subsection{Valence effective interaction} %\subsection{Valence effective interaction}
%================================================================= %=================================================================
As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a general subset of molecular orbitals. As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a general subset of molecular orbitals.
We then split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively, and $\Cor \bigcap \Val = \O$. We then split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively, and $\Cor \bigcap \Val = \O$.
According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ satisfying %According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$.
Accounting solely for the valence electrons, Eq.~\eqref{eq:expectweeb} becomes
\begin{equation} \begin{equation}
\label{eq:expectweebval} \label{eq:expectweebval}
\mel*{\wf{}{\Bas}}{\hWee{\Val}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) \dbx{1} \dbx{2}, \mel*{\wf{}{\Bas}}{\hWee{\Val}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
@ -631,14 +635,14 @@ Following the spirit of Eq.~\eqref{eq:fbasis}, the function $\f{\wf{}{\Bas}}{\Va
and, the valence part of the effective interaction is and, the valence part of the effective interaction is
\begin{equation} \begin{equation}
\label{eq:def_weebasis} \label{eq:def_weebasis}
\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) }{\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})}, \W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})/\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2}),
\end{equation} \end{equation}
where $\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})$ is the two body density associated to the valence electrons. where $\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})$ is the two body density associated to the valence electrons.
%\begin{equation} %\begin{equation}
% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} . % \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} .
%\end{equation} %\end{equation}
%It is worth noting that, in Eq.~\eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\Bas$, and the $(k,l,m,n)$, which span only the valence space $\Basval$. Only with such a definition, one can show (see annex) that $\fbasisval$ fulfills \eqref{eq:expectweebval} and tends to the exact interaction $1/r_{12}$ in the limit of a complete basis set $\Bas$, whatever the choice of subset $\Basval$. %It is worth noting that, in Eq.~\eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\Bas$, and the $(k,l,m,n)$, which span only the valence space $\Basval$. Only with such a definition, one can show (see annex) that $\fbasisval$ fulfills \eqref{eq:expectweebval} and tends to the exact interaction $1/r_{12}$ in the limit of a complete basis set $\Bas$, whatever the choice of subset $\Basval$.
It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ and $\murpsival$ fulfils Eqs.~\eqref{eq:lim_W} and \eqref{eq:lim_mur}. It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ satisfies Eq.~\eqref{eq:lim_W}.
We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models. We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models.
Defining the valence one-body spin density matrix as Defining the valence one-body spin density matrix as
@ -663,59 +667,10 @@ Therefore, we propose the following valence-only approximations for the compleme
\end{equation} \end{equation}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section{Results} %\section{Results}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\begin{table} %%% TABLE I %%%
\caption{
Statistical analysis (in \kcal) of the G2 atomization energies depicted in Fig.~\ref{fig:G2_AE}.
Mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS reference data.
\label{tab:stats}}
\begin{ruledtabular}
\begin{tabular}{lddd}
Method & \tabc{MAD} & \tabc{RMSD} & \tabc{MAX} \\
\hline
CCSD(T)/cc-pVDZ & 22.81 & 25.82 & 72.08 \\
CCSD(T)/cc-pVTZ & 7.95 & 8.99 & 25.99 \\
CCSD(T)/cc-pVQZ & 3.24 & 3.67 & 11.66 \\
CCSD(T)/cc-pV5Z & 1.39 & 1.54 & 3.46 \\
\\
CCSD(T)+LDA/cc-pVDZ & 11.75 & 13.99 & 54.88 \\
CCSD(T)+LDA/cc-pVTZ & 3.11 & 3.94 & 16.77 \\
CCSD(T)+LDA/cc-pVQZ & 0.87 & 1.36 & 6.22 \\
\\
CCSD(T)+PBE/cc-pVDZ & 8.68 & 10.92 & 45.81 \\
CCSD(T)+PBE/cc-pVTZ & 2.66 & 3.52 & 14.73 \\
CCSD(T)+PBE/cc-pVQZ & 0.88 & 1.36 & 5.96 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure*}
\includegraphics[width=\linewidth]{VDZ}
\includegraphics[width=\linewidth]{VTZ}
\includegraphics[width=\linewidth]{VQZ}
\caption{
Deviation (in \kcal) from CCSD(T)/CBS reference atomization energies obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
\label{fig:G2_AE}}
\end{figure*}
\subsection{Comparison between the CIPSI and CCSD(T) models in the case of C$_2$, N$_2$, O$_2$, F$_2$}
We begin the investigation of the behavior of the basis-set correction by the study of the atomization energies of the C$_2$, N$_2$, O$_2$ and F$_2$ homo-nuclear diatomic molecules in the Dunning cc-pVXZ and cc-pCVXZ in the case of N$_2$ (X=D,T,Q,5) using both the CIPSI algorithm and the CCSD(T). N$_2$, O$_2$ and F$_2$ belong to the G2 set and can be considered as weakly correlated, whereas C$_2$ contains already a non negligible non dynamic correlation component.
All through this work, we follow the frozen core (FC) convention of Klopper \textit{et. al}\cite{HauKlo-JCP-12} which consists in all-electron calculations for Li-Be, a He core for B-Na atoms and a Ne core for the Al-Cl series. In the context of the DFT correction for the basis-set, this implies that, for a given system in a given basis set $\Bas$, the set of valence orbitals $\Basval$ involved in the definition of the valence interaction $\wbasisval$ and density $\onedmval$ refers to all MOs except the core.
In order to estimate the CBS limit of each model we use the two-point extrapolation of Ref. \onlinecite{HalHelJorKloKocOls-CPL-98} for the correlation energies and report the corresponding atomization energy which are referred as $D_e^{Q5Z}$ and $D_e^{C(Q5)Z}$ for the cc-pVXZ and cc-pCVXZ basis sets, respectively. All through this work, the valence interaction and density was used when the frozen core approximation was done on the WFT model.
\subsection{Convergence of the atomization energies with the WFT models }
As the exFCI calculations were converged with a precision of about 0.2 mH, we can consider the atomization energies computed at this level as near FCI values which we will consider as the reference for a given system in a given basis set. The results for these molecules are shown in Table \ref{tab:diatomics}.
As one can notice from the data, the convergence of the exFCI atomization energies is slow with respect to the basis set, and the chemical accuracy is barely reached for C$_2$, O$_2$ and F$_2$ even at the cc-pv5Z basis set. Also, the atomization energies are always too small, reflecting the fact that, in a given basis set, a molecule is always more poorly described than the atoms due to the larger number of interacting pairs of electrons in the molecule.
The same behaviours hold for the CCSD(T) model, and one can notice that the atomization energies of the CCSD(T) are always slightly underestimated with respect to the CIPSI ones, showing that the CCSD(T) ansatz is better suited for the atoms than for the molecule.
\subsection{The effect of the basis set correction within the LDA and PBE approximation}
Regarding the effect of the basis set correction, both for the CIPSI and CCSD(T) models, several observations can be done.
First, in a given basis set, the addition of the basis set correction, both at the LDA and PBE level, improves the result even if it can overestimates the estimated CBS atomization energies by a few tens of kcal/mol (the largest deviation being 0.6 kcal/mol for N$_2$ at the (FC)CCSD(T)+PBE-val level in the cc-pv5z basis). Nevertheless, the deviations observed in the largest basis sets are typically in the range of the accuracy of the atomization energies computed with the CBS extrapolation technique.
Also, the values obtained with the largest basis sets tends to converge toward a value close to the estimated CBS values. Also, one can observe that the sensitivity to the functional is quite large for the double- and triple-zeta basis sets, where clearly the PBE functional performs better. Nevertheless, from the quadruple-zeta basis set, the LDA and PBE functional agrees within a few tens of kcal/mol.
\begin{table*} \begin{table*}
\caption{ \caption{
\label{tab:diatomics} \label{tab:diatomics}
@ -766,68 +721,57 @@ Also, the values obtained with the largest basis sets tends to converge toward a
\fnt[2]{``Full'' calculation, i.e., all electrons are correlated. All spinorbitals 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*} \end{table*}
%%% TABLE II %%%
\begin{table}
\caption{
Statistical analysis (in \kcal) of the G2 atomization energies depicted in Fig.~\ref{fig:G2_AE}.
Mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS reference data.
\label{tab:stats}}
\begin{ruledtabular}
\begin{tabular}{lddd}
Method & \tabc{MAD} & \tabc{RMSD} & \tabc{MAX} \\
\hline
CCSD(T)/cc-pVDZ & 22.81 & 25.82 & 72.08 \\
CCSD(T)/cc-pVTZ & 7.95 & 8.99 & 25.99 \\
CCSD(T)/cc-pVQZ & 3.24 & 3.67 & 11.66 \\
CCSD(T)/cc-pV5Z & 1.39 & 1.54 & 3.46 \\
\\
CCSD(T)+LDA/cc-pVDZ & 11.75 & 13.99 & 54.88 \\
CCSD(T)+LDA/cc-pVTZ & 3.11 & 3.94 & 16.77 \\
CCSD(T)+LDA/cc-pVQZ & 0.87 & 1.36 & 6.22 \\
\\
CCSD(T)+PBE/cc-pVDZ & 8.68 & 10.92 & 45.81 \\
CCSD(T)+PBE/cc-pVTZ & 2.66 & 3.52 & 14.73 \\
CCSD(T)+PBE/cc-pVQZ & 0.88 & 1.36 & 5.96 \\
\end{tabular}
\end{ruledtabular}
\end{table}
%%% FIGURE 1 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{VDZ}
\includegraphics[width=\linewidth]{VTZ}
\includegraphics[width=\linewidth]{VQZ}
\caption{
Deviation (in \kcal) from CCSD(T)/CBS reference atomization energies obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
\label{fig:G2_AE}}
\end{figure*}
[ %\subsection{Comparison between the CIPSI and CCSD(T) models in the case of C$_2$, N$_2$, O$_2$, F$_2$}
[ We begin the investigation of the behavior of the basis-set correction by the study of the atomization energies of the C$_2$, N$_2$, O$_2$ and F$_2$ homo-nuclear diatomic molecules in the Dunning cc-pVXZ and cc-pCVXZ in the case of N$_2$ (X=D,T,Q,5) using both the CIPSI algorithm and the CCSD(T). N$_2$, O$_2$ and F$_2$ belong to the G2 set and can be considered as weakly correlated, whereas C$_2$ contains already a non negligible non dynamic correlation component.
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All through this work, we follow the frozen core (FC) convention of Klopper \textit{et. al}\cite{HauKlo-JCP-12} which consists in all-electron calculations for Li-Be, a He core for B-Na atoms and a Ne core for the Al-Cl series. In the context of the DFT correction for the basis-set, this implies that, for a given system in a given basis set $\Bas$, the set of valence orbitals $\Basval$ involved in the definition of the valence interaction $\wbasisval$ and density $\onedmval$ refers to all MOs except the core.
In order to estimate the CBS limit of each model we use the two-point extrapolation of Ref. \onlinecite{HalHelJorKloKocOls-CPL-98} for the correlation energies and report the corresponding atomization energy which are referred as $D_e^{Q5Z}$ and $D_e^{C(Q5)Z}$ for the cc-pVXZ and cc-pCVXZ basis sets, respectively. All through this work, the valence interaction and density was used when the frozen core approximation was done on the WFT model.
%\subsection{Convergence of the atomization energies with the WFT models }
As the exFCI calculations were converged with a precision of about 0.2 mH, we can consider the atomization energies computed at this level as near FCI values which we will consider as the reference for a given system in a given basis set. The results for these molecules are shown in Table \ref{tab:diatomics}.
As one can notice from the data, the convergence of the exFCI atomization energies is slow with respect to the basis set, and the chemical accuracy is barely reached for C$_2$, O$_2$ and F$_2$ even at the cc-pv5Z basis set. Also, the atomization energies are always too small, reflecting the fact that, in a given basis set, a molecule is always more poorly described than the atoms due to the larger number of interacting pairs of electrons in the molecule.
The same behaviours hold for the CCSD(T) model, and one can notice that the atomization energies of the CCSD(T) are always slightly underestimated with respect to the CIPSI ones, showing that the CCSD(T) ansatz is better suited for the atoms than for the molecule.
%\subsection{The effect of the basis set correction within the LDA and PBE approximation}
Regarding the effect of the basis set correction, both for the CIPSI and CCSD(T) models, several observations can be done.
First, in a given basis set, the addition of the basis set correction, both at the LDA and PBE level, improves the result even if it can overestimates the estimated CBS atomization energies by a few tens of kcal/mol (the largest deviation being 0.6 kcal/mol for N$_2$ at the (FC)CCSD(T)+PBE-val level in the cc-pv5z basis). Nevertheless, the deviations observed in the largest basis sets are typically in the range of the accuracy of the atomization energies computed with the CBS extrapolation technique.
Also, the values obtained with the largest basis sets tends to converge toward a value close to the estimated CBS values. Also, one can observe that the sensitivity to the functional is quite large for the double- and triple-zeta basis sets, where clearly the PBE functional performs better. Nevertheless, from the quadruple-zeta basis set, the LDA and PBE functional agrees within a few tens of kcal/mol.
\bibliography{G2-srDFT} \bibliography{G2-srDFT}