notations alsmost OK
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@ -94,14 +94,13 @@
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% operators
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\newcommand{\hT}{\Hat{T}}
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\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
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\newcommand{\updw}{\uparrow\downarrow}
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\newcommand{\f}[2]{f_{#1}^{#2}}
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\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
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% coordinates
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\newcommand{\br}[1]{\mathbf{r}_{#1}}
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%\newcommand{\br}[1]{\mathbf{x}_{#1}}
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\newcommand{\dbr}[1]{d\br{#1}}
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%\newcommand{\dbr}[1]{d\br{#1}}
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\newcommand{\ra}{\rightarrow}
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\newcommand{\De}{D_\text{e}}
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@ -215,99 +214,9 @@ Importantly, in the limit of a complete basis set (which we refer to as $\Bas \t
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\lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\modY}{\Bas}] ) = \E{\modX}{} \approx E,
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\end{equation}
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where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set.
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In the case $\modX = \FCI$, we have as strict equality as $\E{\FCI}{} = \E{}{}$.
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In the case $\modX = \FCI$, 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 $\modX$ and $\modY$ for the \titou{FCI} energy and density within $\Bas$, respectively.
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%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 $\emodel$.
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%As any wave function model is necessary an approximation to the FCI model, one can write
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%\begin{equation}
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% \efci \approx \emodel
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%\end{equation}
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%and
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%\begin{equation}
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% \denfci \approx \denmodel
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%\end{equation}
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%and by defining the energy provided by the model $\model$ in the complete basis set
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%\begin{equation}
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% \emodelcomplete = \lim_{\Bas \rightarrow \infty} \emodel\,\, ,
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%\end{equation}
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%we can then write
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%\begin{equation}
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% \emodelcomplete \approx \emodel + \ecompmodel
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%\end{equation}
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%which verifies the correct limit since
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%\begin{equation}
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% \lim_{\Bas \rightarrow \infty} \ecompmodel = 0\,\, .
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%\end{equation}
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%=================================================================
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%\subsection{Basis set correction for the CIPSI algorithm and the CCSD(T) ansatz}
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%=================================================================
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%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.
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%=================================================================
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%\subsubsection{Basis set correction for the CCSD(T) energy}
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%=================================================================
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%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.
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%Defining $\ecc$ as the CCSD(T) energy obtained in $\Bas$, in the present notations we have
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%\begin{equation}
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% \emodel = \ecc \,\, .
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%\end{equation}
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%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
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%\begin{equation}
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% \denmodel = \denhf \,\, .
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%\end{equation}
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%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.
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%Therefore, we approximate the complete basis set CCSD(T) energy $\ecccomplete$ by
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%\begin{equation}
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% \ecccomplete \approx \ecc + \efuncden{\denhf} \,\, .
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%\end{equation}
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%=================================================================
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%\subsubsection{Correction of the CIPSI algorithm}
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%=================================================================
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%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.
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%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}
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%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}
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%The CIPSI algorithm used in this work uses iteratively enlarged selected CI spaces and
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%Epstein--Nesbet\cite{epstein,nesbet} multi-reference perturbation theory. Within a basis set $\Bas$, the CIPSI energy is
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%\begin{align}
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% E_\mathrm{CIPSI}^{\Bas} &= E_\text{v} + E^{(2)} \,\,,
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%\end{align}
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%where $E_\text{v}$ is the variational energy
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%\begin{align}
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% E_\text{v} &= \min_{\{ c_{\rm I}\}} \frac{\elemm{\Psi^{(0)}}{\hat{H}}{\Psi^{(0)}} }{\ovrlp{\Psi^{(0)}}{\Psi^{(0)}}}\,\,,
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%\end{align}
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%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
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%\begin{align}
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% E^{(2)} &= \sum_{\kappa} \frac{|\elemm{\Psi^{(0)}}{\hat{H}}{\kappa}|^2}{E_\text{v} - \elemm{\kappa}{H}{\kappa}} = \sum_{\kappa} \,\, e_{\kappa}^{(2)} \,\, ,
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%\end{align}
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%where $\kappa$ denotes a determinant outside $\mathcal{R}$.
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%To reduce the cost of the evaluation of the second-order energy correction, the semi-stochastic multi-reference approach
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%of Garniron \textit{et al.} \cite{stochastic_pt_yan} was used, adopting the technical specifications recommended in that work.
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%The CIPSI energy is systematically refined by doubling the size of the CI reference space at each iteration, selecting
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%the determinants $\kappa$ with the largest $\vert e_{\kappa}^{(2)} \vert$.
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%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 $\EexFCIbasis$.
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%
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%In the context of the basis set correction, we use the following conventions
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%\begin{equation}
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% \emodel = \EexFCIbasis
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%\end{equation}
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%\begin{equation}
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% \denmodelr = \dencipsir
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%\end{equation}
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%where the density $\dencipsir$ is defined as
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%\begin{equation}
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% \dencipsi = \sum_{ij \in \Bas} \elemm{\Psi^{(0)}}{\aic{i}\ai{j}}{\Psi^{(0)}} \phi_i(\bfrb{} ) \phi_j(\bfrb{} ) \,\, ,
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%\end{equation}
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%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 $\denfci$.
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%
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%Finally, we approximate complete basis set exFCI energy $\EexFCIinfty$ as
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%\begin{equation}
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% \EexFCIinfty \approx \EexFCIbasis + \efuncden{\dencipsi}
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%\end{equation}
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Rigorously speaking, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
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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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@ -318,32 +227,16 @@ Contrary to the conventional RS-DFT scheme which requires a range-separated \tex
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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 $\Bas$.
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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} alongside $\rsmu{\Bas}{}(\br{})$ as range separation.
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%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}].
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%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}) .
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%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}).
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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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%=================================================================
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%\subsection{General scheme for the approximation of the unknown complementary functional $\efuncbasis$}
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%=================================================================
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%=================================================================
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%\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis set $\Bas$}
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%\label{sec:weff}
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%=================================================================
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%=================================================================
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%\subsection{Effective Coulomb operator}
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%=================================================================
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We define the effective operator $\W{\Bas}{}(\br{1},\br{2})$ as
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We define the effective operator as
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\begin{equation}
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\label{eq:def_weebasis}
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\W{\Bas}{}(\br{1},\br{2}) =
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\begin{cases}
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\f{\Bas}{}(\br{1},\br{2})/\n{2}{}(\br{1},\br{2}), & \text{if $\n{2}{}(\br{1},\br{2}) \ne 0$,}
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\\
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\infty, & \text{otherwise.}
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\infty, & \text{otherwise,}
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\end{cases}
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\end{equation}
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where
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@ -352,108 +245,56 @@ where
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\n{2}{}(\br{1},\br{2})
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= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}
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\end{equation}
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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
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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,
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\begin{equation}
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\label{eq:fbasis}
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\f{\Bas}{}(\br{1},\br{2})
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= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
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\end{equation}
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and $\V{pq}{rs}$ are the usual Coulomb two-electron integrals.
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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.
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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 do not have any basis set correction.
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\PFL{I don't agree with this. There must be a correction for one-electron system.
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However, it does not come from the e-e cusp but from the e-n cusp.}
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With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ verifies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
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With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
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\begin{equation}
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\label{eq:int_eq_wee}
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\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \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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where here $\hWee{}$ contains only the opposite-spins component of the two-electron interaction, and \eqref{eq:int_eq_wee} can be rewritten as
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where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$.
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Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as
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\begin{equation}
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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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which intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
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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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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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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
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An important quantity to define in the present context 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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\label{eq:wcoal}
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\W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}),
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\end{equation}
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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.
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and which is necessarily \textit{finite} for an incomplete basis set as long as the on-top two-body density is non vanishing.
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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
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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})
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\begin{equation}
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\label{eq:lim_W}
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\lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = r_{12}^{-1}\
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\end{equation}
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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.
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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.
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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.
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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.
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%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.
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%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.
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%=================================================================
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%\subsection{Range-separation function}
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%=================================================================
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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.
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To do so, we choose a range-separation \textit{function} $\rsmu{\Bas}{}(\br{})$
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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.
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To do so, we choose a range-separation \textit{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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\end{equation}
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such that the long-range interaction $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2})$
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such that the long-range interaction
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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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\PFL{This expression looks like a cheap spherical average.}
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coincides with the effective interaction $\W{\Bas}{}(\br{})$ for all points in $\mathbb{R}^3$
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\begin{equation}
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\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{}).
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\end{equation}
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%More precisely, if we define the value of the interaction at coalescence as
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%\begin{equation}
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% \label{eq:def_wcoal}
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% \wbasiscoal{} = W_{\wf{}{\Bas}}(\bfr{},\bar{{\bf x}}_{}).
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%\end{equation}
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%where $(\br{},\Bar{\br{}})$ means a couple of anti-parallel spins at the same position $\br{}$,
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%we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction:
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%\begin{equation}
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% \wbasiscoal{} = \w{}{\lr,\rsmu{\Bas}{}}(\bfrb{},\bfrb{})
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%\end{equation}
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%where the long-range-like interaction is defined as
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%\begin{equation}
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% \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}} }.
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%\end{equation}
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%Equation \eqref{eq:def_wcoal} is equivalent to the following condition
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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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%\end{equation}
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%As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as
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%\begin{equation}
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% \label{eq:mu_of_r_val}
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% \murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, .
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%\end{equation}
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%An important point to notice is that, in the limit of a complete basis set $\Bas$, as
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%\begin{equation}
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%\label{eq:lim_W}
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% \lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1} \quad \forall (\br{1},\br{2})
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%% &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, ,
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%\end{equation}
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%one has $\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = \infty$
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%% &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,,
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%and therefore
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%\begin{equation}
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%\label{eq:lim_mur}
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% \lim_{\Bas \rightarrow \infty} \rsmu{\Bas}{}(\br{}) = \infty
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%%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
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%\end{equation}
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%=================================================================
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%\subsection{Complementary functional}
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%=================================================================
|
||||
%\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}
|
||||
|
|
|
|||
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Reference in New Issue
Block a user