diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index 46a5ee3..e5ac1a5 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -94,14 +94,13 @@ % operators \newcommand{\hT}{\Hat{T}} \newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}} +\newcommand{\updw}{\uparrow\downarrow} \newcommand{\f}[2]{f_{#1}^{#2}} \newcommand{\Gam}[2]{\Gamma_{#1}^{#2}} % coordinates \newcommand{\br}[1]{\mathbf{r}_{#1}} -%\newcommand{\br}[1]{\mathbf{x}_{#1}} \newcommand{\dbr}[1]{d\br{#1}} -%\newcommand{\dbr}[1]{d\br{#1}} \newcommand{\ra}{\rightarrow} \newcommand{\De}{D_\text{e}} @@ -215,99 +214,9 @@ Importantly, in the limit of a complete basis set (which we refer to as $\Bas \t \lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\modY}{\Bas}] ) = \E{\modX}{} \approx E, \end{equation} where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set. -In the case $\modX = \FCI$, we have as strict equality as $\E{\FCI}{} = \E{}{}$. +In the case $\modX = \FCI$, we have a strict equality as $\E{\FCI}{} = \E{}{}$. Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modX$ and $\modY$ for the \titou{FCI} energy and density within $\Bas$, respectively. -%Here we propose to generalize such approach to a general WFT model, referred here as $\model$, projected in a basis set $\Bas$ which must provides a density $\denmodel$ and an energy $\emodel$. -%As any wave function model is necessary an approximation to the FCI model, one can write -%\begin{equation} -% \efci \approx \emodel -%\end{equation} -%and -%\begin{equation} -% \denfci \approx \denmodel -%\end{equation} -%and by defining the energy provided by the model $\model$ in the complete basis set -%\begin{equation} -% \emodelcomplete = \lim_{\Bas \rightarrow \infty} \emodel\,\, , -%\end{equation} -%we can then write -%\begin{equation} -% \emodelcomplete \approx \emodel + \ecompmodel -%\end{equation} -%which verifies the correct limit since -%\begin{equation} -% \lim_{\Bas \rightarrow \infty} \ecompmodel = 0\,\, . -%\end{equation} - - -%================================================================= -%\subsection{Basis set correction for the CIPSI algorithm and the CCSD(T) ansatz} -%================================================================= -%In this work we propose to apply the basis set correction to a selected CI algorithm, namely the CIPSI algorithm, and to the CCSD(T) ansatz in %order to speed-up the basis set convergence of these models. - -%================================================================= -%\subsubsection{Basis set correction for the CCSD(T) energy} -%================================================================= -%The CCSD(T) method is a very popular WFT approach which is known to provide very good estimation of the correlation energies for weakly correlated systems, whose wave function are dominated by the HF Slater determinant. -%Defining $\ecc$ as the CCSD(T) energy obtained in $\Bas$, in the present notations we have -%\begin{equation} -% \emodel = \ecc \,\, . -%\end{equation} -%In the context of the basis set correction, one needs to choose a density as the density of the model $\denmodel$, and we chose here the HF density -%\begin{equation} -% \denmodel = \denhf \,\, . -%\end{equation} -%Such a choice can be motivated by the fact that the correction to the HF density brought by the excited Slater determinants are at least of second-order in perturbation theory. -%Therefore, we approximate the complete basis set CCSD(T) energy $\ecccomplete$ by -%\begin{equation} -% \ecccomplete \approx \ecc + \efuncden{\denhf} \,\, . -%\end{equation} - -%================================================================= -%\subsubsection{Correction of the CIPSI algorithm} -%================================================================= -%The CIPSI algorithm approximates the FCI wave function through an iterative selected CI procedure, and the FCI energy through a second-order multi-reference perturbation theory. -%The CIPSI algorithm belongs to the general class of methods build upon selected CI\cite{bender,malrieu,buenker1,buenker-book,three_class_CIPSI,harrison,hbci} -%which have been successfully used to converge to FCI correlation energies, one-body properties, and nodal surfaces.\cite{three_class_CIPSI,Rubio198698,cimiraglia_cipsi,cele_cipsi_zeroth_order,Angeli2000472,canadian,atoms_3d,f2_dmc,atoms_dmc_julien,GinTewGarAla-JCTC-18,SceBenJacCafLoo-JCP-18,LooSceBloGarCafJac-JCTC-18,LooBogSceCafJAc-JCTC-19} -%The CIPSI algorithm used in this work uses iteratively enlarged selected CI spaces and -%Epstein--Nesbet\cite{epstein,nesbet} multi-reference perturbation theory. Within a basis set $\Bas$, the CIPSI energy is -%\begin{align} -% E_\mathrm{CIPSI}^{\Bas} &= E_\text{v} + E^{(2)} \,\,, -%\end{align} -%where $E_\text{v}$ is the variational energy -%\begin{align} -% E_\text{v} &= \min_{\{ c_{\rm I}\}} \frac{\elemm{\Psi^{(0)}}{\hat{H}}{\Psi^{(0)}} }{\ovrlp{\Psi^{(0)}}{\Psi^{(0)}}}\,\,, -%\end{align} -%where the reference wave function $\ket{\Psi^{(0)}} = \sum_{{\rm I}\,\in\,\mathcal{R}} \,\,c_{\rm I} \,\,\ket{\rm I}$ is expanded in Slater determinants I within the CI reference space $\mathcal{R}$, and $E^{(2)}$ is the second-order energy correction -%\begin{align} -% E^{(2)} &= \sum_{\kappa} \frac{|\elemm{\Psi^{(0)}}{\hat{H}}{\kappa}|^2}{E_\text{v} - \elemm{\kappa}{H}{\kappa}} = \sum_{\kappa} \,\, e_{\kappa}^{(2)} \,\, , -%\end{align} -%where $\kappa$ denotes a determinant outside $\mathcal{R}$. -%To reduce the cost of the evaluation of the second-order energy correction, the semi-stochastic multi-reference approach -%of Garniron \textit{et al.} \cite{stochastic_pt_yan} was used, adopting the technical specifications recommended in that work. -%The CIPSI energy is systematically refined by doubling the size of the CI reference space at each iteration, selecting -%the determinants $\kappa$ with the largest $\vert e_{\kappa}^{(2)} \vert$. -%In order to reach a faster convergence of the estimation of the FCI energy, we use the extrapolated FCI energy (exFCI) proposed by Holmes \textit{et al}\cite{HolUmrSha-JCP-17} which we refer here as $\EexFCIbasis$. -% -%In the context of the basis set correction, we use the following conventions -%\begin{equation} -% \emodel = \EexFCIbasis -%\end{equation} -%\begin{equation} -% \denmodelr = \dencipsir -%\end{equation} -%where the density $\dencipsir$ is defined as -%\begin{equation} -% \dencipsi = \sum_{ij \in \Bas} \elemm{\Psi^{(0)}}{\aic{i}\ai{j}}{\Psi^{(0)}} \phi_i(\bfrb{} ) \phi_j(\bfrb{} ) \,\, , -%\end{equation} -%and $\phi_i(\bfrb{} )$ are the spin orbitals in the MO basis evaluated at $\bfrb{}$. As it was shown in \onlinecite{GinPraFerAssSavTou-JCP-18} that the CIPSI density converges rapidly with the size of $\Psi^{(0)}$ for weakly correlated systems, $\dencipsir$ can be thought as a reasonable approximation of the FCI density $\denfci$. -% -%Finally, we approximate complete basis set exFCI energy $\EexFCIinfty$ as -%\begin{equation} -% \EexFCIinfty \approx \EexFCIbasis + \efuncden{\dencipsi} -%\end{equation} - Rigorously speaking, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct for the lack of cusp in $\wf{}{\Bas}$ (i.e.~a discontinuous derivative) at the e-e coalescence points, a universal condition of exact wave functions. @@ -318,32 +227,16 @@ Contrary to the conventional RS-DFT scheme which requires a range-separated \tex The first step of the basis set correction consists in obtaining an effective two-electron interaction $\W{\Bas}{}(\br{1},\br{2})$ which represents the effect of the projection of the Coulomb operator in an incomplete basis set $\Bas$. 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 $\Bas \to \infty$. In a second step, we shall link $\W{\Bas}{}(\br{1},\br{2})$ to $\rsmu{\Bas}{}(\br{})$. -In the final step, we employ short-range density functionals\cite{TouGorSav-TCA-05} alongside $\rsmu{\Bas}{}(\br{})$ as range separation. -%Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}]. -%First, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05} that we evaluate at $\n{\modX}{\Bas}$ alongside $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) . -%Second, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation \textit{function} $\mu(\br{})$ defined in real space. %(see Sec.~\ref{sec:weff}). +In the final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{\Bas}{}(\br{})$ as range separation. - -%================================================================= -%\subsection{General scheme for the approximation of the unknown complementary functional $\efuncbasis$} -%================================================================= - - -%================================================================= -%\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis set $\Bas$} -%\label{sec:weff} -%================================================================= -%================================================================= -%\subsection{Effective Coulomb operator} -%================================================================= -We define the effective operator $\W{\Bas}{}(\br{1},\br{2})$ as +We define the effective operator as \begin{equation} \label{eq:def_weebasis} \W{\Bas}{}(\br{1},\br{2}) = \begin{cases} \f{\Bas}{}(\br{1},\br{2})/\n{2}{}(\br{1},\br{2}), & \text{if $\n{2}{}(\br{1},\br{2}) \ne 0$,} \\ - \infty, & \text{otherwise.} + \infty, & \text{otherwise,} \end{cases} \end{equation} where @@ -352,108 +245,56 @@ where \n{2}{}(\br{1},\br{2}) = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2} \end{equation} -is the opposite-spin two-body density associated with $\wf{}{\Bas}$, $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ is the opposite-spin two-body density tensor of $\wf{}{\Bas}$, $\SO{p}{}$ is a spinorbital, $\f{\Bas}{}(\br{1},\br{2})$ is defined as +and $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ are the opposite-spin two-body density and density tensor (respectively) associated with $\wf{}{\Bas}$, $\SO{p}{}$ is a spinorbital, \begin{equation} \label{eq:fbasis} \f{\Bas}{}(\br{1},\br{2}) = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, \end{equation} and $\V{pq}{rs}$ are the usual Coulomb two-electron integrals. -The definition of equation \eqref{eq:def_weebasis} is the same of equation (27) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, but here we add the extra condition that $\W{\Bas}{}(\br{1},\br{2})$ diverges when the two-body density vanishes, which ensures that one-electron systems do not have any basis set correction. +Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems do not have any basis set correction. \PFL{I don't agree with this. There must be a correction for one-electron system. However, it does not come from the e-e cusp but from the e-n cusp.} -With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ verifies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) +With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) \begin{equation} \label{eq:int_eq_wee} - \mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2}, + \mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2}, \end{equation} -where here $\hWee{}$ contains only the opposite-spins component of the two-electron interaction, and \eqref{eq:int_eq_wee} can be rewritten as +where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$. +Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as \begin{equation} \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}, \end{equation} -which intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction. +it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction. 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. -An important quantity to define in the present context is $\W{\Bas}{}(\br{})$ which is the value of the effective interaction at $\br{}$ for opposite spins at coalescence +An important quantity to define in the present context is the value of the effective interaction at coalescence of opposite-spin electrons \begin{equation} - \label{eq:wcoal} - \W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}) + \label{eq:wcoal} + \W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}), \end{equation} -and which is necessarily \textit{finite} at for an \textit{incomplete} basis set as long as the on-top two-body density is non vanishing. +and which is necessarily \textit{finite} for an incomplete basis set as long as the on-top two-body density is non vanishing. -Of course, there exists \textit{a priori} an infinite set of functions in $\mathbb{R}^6$ satisfying \eqref{eq:int_eq_wee}, but thanks to its very definition one can show (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that +Of course, there exists \textit{a priori} an infinite set of functions in $\mathbb{R}^6$ satisfying \eqref{eq:int_eq_wee}, but thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) \begin{equation} \label{eq:lim_W} \lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = r_{12}^{-1}\ \end{equation} -for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $\wf{}{\Bas}$, which therefore guarantees a physically satisfying limit. -An important point here is that, with the present definition of $\W{\Bas}{}(\br{1},\br{2})$, one can quantify the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence. -As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\Bas}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of $\Bas$ for weakly correlated systems. +for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $\wf{}{\Bas}$, which guarantees a physically satisfying limit. +%An important point here is that, with the present definition of $\W{\Bas}{}(\br{1},\br{2})$, one can quantify the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence. +%As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\Bas}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of $\Bas$ for weakly correlated systems. -%================================================================= -%\subsection{Range-separation function} -%================================================================= -As we can map the Coulomb operator within a basis set $\Bas$ with a non divergent two-electron interaction, we can link the present theory with the RS-DFT which uses the so-called long-range interaction which are smooth bounded two-electron operators. -To do so, we choose a range-separation \textit{function} $\rsmu{\Bas}{}(\br{})$ +Because the Coulomb operator within a basis set $\Bas$ is a non divergent two-electron interaction, we can straightforwardly link the present theory with the RS-DFT which uses the so-called long-range interaction which are smooth bounded two-electron operators. +To do so, we choose a range-separation \textit{function} \begin{equation} \label{eq:mu_of_r} \rsmu{\Bas}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{}) \end{equation} -such that the long-range interaction $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2})$ +such that the long-range interaction \begin{equation} \w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{\Bas}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{\Bas}{}(\br{2}) r_{12}]}{ r_{12}} } \end{equation} \PFL{This expression looks like a cheap spherical average.} -coincides with the effective interaction $\W{\Bas}{}(\br{})$ for all points in $\mathbb{R}^3$ -\begin{equation} - \w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{}). -\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 $(\br{},\Bar{\br{}})$ 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{\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{\Bas}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{}) -%\end{equation} -%As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as -%\begin{equation} -% \label{eq:mu_of_r_val} -% \murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, . -%\end{equation} -%An important point to notice is that, in the limit of a complete basis set $\Bas$, as -%\begin{equation} -%\label{eq:lim_W} -% \lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1} \quad \forall (\br{1},\br{2}) -%% &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, , -%\end{equation} -%one has $\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = \infty$ -%% &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,, -%and therefore -%\begin{equation} -%\label{eq:lim_mur} -% \lim_{\Bas \rightarrow \infty} \rsmu{\Bas}{}(\br{}) = \infty -%%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, . -%\end{equation} - - - -%================================================================= -%\subsection{Complementary functional} -%================================================================= -%\label{sec:ecmd} +coincides with the effective interaction $\W{\Bas}{}(\br{})$ at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{})$. Once defined the range-separation function $\rsmu{\Bas}{}(\br{})$, we can use the functionals defined in the field of RS-DFT to approximate $\bE{}{\Bas}[\n{}{}]$. As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we 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} @@ -486,24 +327,17 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following l \end{subequations} where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT. The choice of the ECMD as the functionals to be used in this scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [see equation \eqref{eq:E_funcbasis}] and that of the ECMD functionals [see equation \eqref{eq:ec_md_mu}]. -Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) \approx \W{\Bas}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\Bas}{}(\br{})}$ coincides with $\wf{}{\Bas}$. +Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) \approx \W{\Bas}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\Bas}{}}$ coincides with $\wf{}{\Bas}$. %The ECMD functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function. %This makes them particularly well adapted to the present context where one aims at correcting a general WFT method. -%-------------------------------------------- -%\subsubsection{Local density approximation} -%-------------------------------------------- Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{\Bas}{}(\br{})$. Therefore, we define the LDA version of $\bE{}{\Bas}[\n{}{}]$ as \begin{equation} \label{eq:def_lda_tot} \bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{}, \end{equation} where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the short-range ECMD per particle of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} 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. -%-------------------------------------------- -%\subsubsection{New PBE functional} -%-------------------------------------------- The short-range LDA correlation functional defined in Eq.~\eqref{eq:def_lda_tot} relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$. In order to correct such a defect, we propose here a new ECMD functional inspired by the recent functional proposed by some of the present 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 \begin{subequations} @@ -515,8 +349,8 @@ 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{2}{\UEG}(\n{}{})}. \end{gather} \end{subequations} -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 of the system $\n{2}{}(\br{})$ by its UEG version, i.e.~$\n{2}{}(\br{}) \approx \n{2}{\UEG}(\br{}) = \left(\n{}{}(\br{})\right)^2 g_0(\n{}{}(\br{}) )$, 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 $\n{2}{}$. +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 of the system $\n{2}{}(\br{})$ by its UEG version, i.e.~$\n{2}{}(\br{}) \approx \n{2}{\UEG}(\br{}) = \n{}{}(\br{})^2 g_0(\n{}{}(\br{}) )$, 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 $\n{2}{}(\br{})$. Therefore, the PBE complementary functional reads \begin{equation} \label{eq:def_pbe_tot} @@ -524,27 +358,9 @@ Therefore, the PBE complementary functional reads \end{equation} Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modY}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\modY}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modY}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}. -%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 $\mu$. -%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} -% \label{eq:approx_ecfuncbasis} -% \ecompmodel \approx \ecmuapproxmurmodel -%\end{equation} -%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 -%\begin{equation} -% \lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad , -%\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. - -%================================================================= -%\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. We then naturally 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 $\wf{}{\Bas}$. We therefore define the valence-only effective interaction \begin{equation} \W{\Bas}{\Val}(\br{1},\br{2}) = @@ -559,44 +375,19 @@ with \begin{gather} \label{eq:fbasisval} \f{\Bas}{\Val}(\br{1},\br{2}) - = \sum_{pq \in \Bas} \sum_{rstu \in \Val} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[{\wf{}{\Bas}}] \SO{t}{1} \SO{u}{2}, + = \sum_{pq \in \Bas} \sum_{rstu \in \Val} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, \\ \n{2}{\Val}(\br{1},\br{2}) = \sum_{pqrs \in \Val} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, \end{gather} \end{subequations} -and the corresponding valence range separation function $\rsmu{\Bas}{\Val}(\br{})$ +and the corresponding valence range separation function \begin{equation} \label{eq:muval} \rsmu{\Bas}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\Val}(\br{},\br{}). \end{equation} -%\begin{equation} -% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} . -%\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, within the present definition, $\W{\Bas}{\Val}(\br{1},\br{2})$ still 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. -%Defining the valence one-body spin density matrix as -%\begin{equation} -% \begin{aligned} -% \onedmval[\wf{}{\Bas}] & = \elemm{\wf{}{\Bas}}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\wf{}{\Bas}} \qquad \text{if }(i,j)\in \Basval \\ -% & = 0 \qquad \text{in other cases} -% \end{aligned} -%\end{equation} -%then one can define the valence density as: -%\begin{equation} -% \denval_{\sigma}({\bf r}) = \sum_{i,j} \onedmval[\wf{}{\Bas}] \phi_i({\bf r}) \phi_j({\bf r}) -%\end{equation} -%Therefore, we propose the following valence-only approximations for the complementary functional -%\begin{equation} -% \label{eq:def_lda_tot} -% \ecompmodelldaval = \int \, \text{d}{\bf r} \,\, \denval({\bf r}) \,\, \emuldaval\,, -%\end{equation} -%\begin{equation} -% \label{eq:def_lda_tot} -% \ecompmodelpbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval) -%\end{equation} Defining $\n{\modY}{\Val}$ as the valence one-electron density obtained with the model $\modY$, the valence part of the complementary functional $\bE{}{\Val}[\n{\modY}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\modY}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modY}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$. Regarding now the main computational source of the present approach, it consists in the evaluation @@ -606,15 +397,13 @@ for $\Gam{rs}{tu}$ and therefore the computational bottleneck reduces to the eva at each quadrature grid point of \begin{equation} \label{eq:fcoal} - \f{\Bas}{\HF}(\br{}) = \sum_{p,q\in\Bas} \sum_{i\in \nocca} \sum_{j\in \noccb} \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{} + \f{\Bas}{\HF}(\br{}) = \sum_{pq \in \Bas} \sum_{i \in \nocca} \sum_{j\in \noccb} \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{} \end{equation} which scales as $\Nb^2\times N_{elec}^2 \times \Ng$ and is embarassingly parallel. Within the present formulation, the bottleneck is the four-index transformation to obtain the two-electron integrals on the MO basis which appear in \eqref{eq:fcoal}. Nevertheless, this step has in general to be performed before a correlated WFT calculations and therefore it represent a minor limitation. 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. - To conclude this theory session, it is important to notice that the basis set correction proposed here has the folowing properties whatever the approximations made in the DFT part: i) it can be applied to any WFT model that provides an energy and a density, ii) it vanishes for one-electron systems, iii) it vanishes in the limit of a complete basis set and thus garentees the correct CBS limit of the WFT model. -%, because of the behaviour of the effective interaction [see \eqref{eq:lim_W}], the range separation function $\rsmu{\Bas}{}(\br{})$ tends to infinity. Therefore, in the limit of a complete basis set, according to equation \eqref{eq:large_mu_ecmd}, the complementary functional tends to zero \textit{whatever the approximated functional} used for the DFT part. %%%%%%%%%%%%%%%%%%%%%%%% \section{Results}