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Emmanuel Giner
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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: Application to Coupled Cluster}
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\author{Bath\'elemy Pradines}
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\affiliation{\LCT}
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@ -223,7 +223,7 @@ Because the e-e cusp originates from the divergence of the Coulomb operator at $
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Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals which deal with a smooth long-range electron interaction.
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Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, here we use a range-separated \textit{function} $\rsmu{\Bas}{}(\br{})$ which 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 \manu{effect of the basis set incompleteness on the }Coulomb operator \trashMG{in a finite 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 \manu{effect of the basis set incompleteness of $\Bas$ on the }Coulomb operator \trashMG{in a finite basis $\Bas$}.
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%The present definition ensures that $\W{\Bas}{}(\br{1},\br{2})$ is finite at the e-e coalescence point as long as an incomplete basis set is used, and tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb interaction as $\Bas \to \infty$.
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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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@ -265,8 +265,8 @@ Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as
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\iint r_{12}^{-1} \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{equation}
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it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
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Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems are free of correction as the present approach must only correct the basis set incompleteness error originates from the e-e cusp.
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A similar correction for the electron-nucleus cusp is currently under active development.
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Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} \manu{does not affect \eqref{eq:int_eq_wee} and} ensures that one-electron systems are free of correction as the present approach must only correct the basis set incompleteness error originates from the e-e cusp.
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\trashMG{A similar correction for the electron-nucleus cusp is currently under active development.}
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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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@ -306,7 +306,7 @@ coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\B
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%=================================================================
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%\subsection{Short-range correlation functionals}
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%=================================================================
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\manu{Once defined $\rsmu{\Bas}{}(\br{})$, we can use RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$, and
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\manu{Once defined a range separation function $\rsmu{\Bas}{}(\br{})$, we can use RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$, and
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}
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as in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
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\begin{multline}
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@ -321,7 +321,7 @@ where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization
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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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\end{equation}
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with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
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with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \manu{\hat{\w{}{\lr,\rsmu{}{}}}(\hat{r}_{ij})}$.
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%\begin{multline}
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% \label{eq:ec_md_mu}
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% \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
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@ -351,7 +351,7 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the following two l
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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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The choice of ECMD in the present scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [Eq.~\eqref{eq:E_funcbasis}] and the ECMD functionals [Eq.~\eqref{eq:ec_md_mu}].
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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 functionals [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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@ -359,7 +359,7 @@ Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluate
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The LDA version 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}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
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\bE{\LDA}{\manu{\Bas}}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
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\end{equation}
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where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the \trashMG{short-range} reduced (i.e.~per electron) ECMD of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
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@ -367,7 +367,7 @@ The short-range LDA correlation functional relies on the transferability of the
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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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\label{eq:def_pbe_tot}
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\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}\big(\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{}
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\bE{\PBE}{\manu{\Bas}}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}\big(\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})\big) \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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\begin{subequations}
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@ -379,17 +379,18 @@ inspired by the recent functional proposed by some of the authors \cite{FerGinTo
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\beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{2}{\UEG}(\n{}{})}.
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\end{gather}
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\end{subequations}
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The difference between the ECMD PBE 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}.
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The difference between the ECMD \trashMG{PBE} 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}.
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This represents a major computational saving without loss of performance as we eschew the computation of $\n{2}{}(\br{})$.
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Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
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Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{}]$ is then equal to $\bE{\LDA}{\manu{\Bas}}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\manu{\Bas}}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
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%=================================================================
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%\subsection{Valence approximation}
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%=================================================================
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As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a subset of spinorbitals.
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We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ is the set of core spinorbitals, and define the FC version of the effective interaction as
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\manu{I like the idea of defining the $\BasFC$ as the complementary of $\Cor$, but the line over $\Bas$ is barely visible ... :(}
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We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ is the set of core \trashMG{spinorbitals} \manu{spatial orbitals}, and define the FC version of the effective interaction as
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\begin{equation}
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\W{\Bas}{\FC}(\br{1},\br{2}) =
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\begin{cases}
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@ -416,13 +417,13 @@ and the corresponding FC range-separation function
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\end{equation}
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It is worth not\manu{ic}ing that, within the present definition, $\W{\Bas}{\FC}(\br{1},\br{2})$ still \trashMG{satisfies Eq.~\eqref{eq:lim_W}} \manu{tends to the regular Coulomb interaction when $\Bas \to \infty$}.
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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 $\bE{\LDA}{\sr}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$.
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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 $\bE{\LDA}{\manu{\Bas}}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$ or $\bE{\PBE}{\manu{\Bas}}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$.
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%=================================================================
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%\subsection{Computational considerations}
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%=================================================================
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One of the most computationally intensive task of the present approach is the evaluation of $\W{\Bas}{}(\br{})$ [see Eqs.~\eqref{eq:wcoal}] at each quadrature grid point.
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Yet embarrassingly parallel, this step scales, in the general (multi-determinantal) case, as \manu{$\Ng \Nb^6$} (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ in the case of a single Slater determinant.
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Yet embarrassingly parallel, this step scales, in the general (multi-determinantal) case, as \manu{$\Ng \Nb^6$} (where $\Nb$ is the number of basis functions in $\Bas$) but is \manu{strongly} reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ in the case of a single Slater determinant, \manu{which is the case used all through this work}.
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%\begin{equation}
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% \label{eq:fcoal}
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% \f{\Bas}{\HF}(\br{}) = \sum_{pq \in \Bas} \sum_{ij}^{\occ} \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{},
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@ -432,7 +433,7 @@ Nevertheless, this step usually has to be performed for most correlated WFT calc
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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.
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%When the four-index transformation become prohibitive, by performing successive matrix multiplications, one could rewrite the equations directly in the AO basis where it scales formally as $\order{\Ng \Nb^4}$ but where one can take advantage of the sparsity atomic-orbital-based algorithms to significantly speed up the calculations.
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To conclude this section, we point out that, independently of the DFT functional, the present basis set correction
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To conclude this section, we point out that \manu{because of the definitions \eqref{eq:def_weebasis},\eqref{eq:mu_of_r} and the properties \eqref{eq:lim_W} and \eqref{eq:large_mu_ecmd}}, independently of the DFT functional, the present basis set correction
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i) can be applied to any WFT model that provides an energy and a density,
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ii) does not correct one-electron systems, and
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iii) vanishes in the limit of a complete basis set, hence guaranteeing an unaltered CBS limit for a given WFT model.
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