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Pierre-Francois Loos 2019-04-12 10:28:00 +02:00
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Manuscript/G2-srDFT.tex
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@ -87,8 +87,7 @@
% basis sets % basis sets
\newcommand{\Bas}{\mathcal{B}} \newcommand{\Bas}{\mathcal{B}}
\newcommand{\Basval}{\mathcal{B}_\text{val}} \newcommand{\Val}{\text{val}}
\newcommand{\Val}{\mathcal{V}}
\newcommand{\Cor}{\mathcal{C}} \newcommand{\Cor}{\mathcal{C}}
% operators % operators
@ -281,6 +280,10 @@ for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $
%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. %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. %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}
%=================================================================
\alert{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.} \alert{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.}
Although this choice is not unique, we choose here the range-separation function Although this choice is not unique, we choose here the range-separation function
\begin{equation} \begin{equation}
@ -295,8 +298,11 @@ such that the long-range interaction
What about $\rsmu{\Bas}{}(\br{1},\br{2}) = \sqrt{\rsmu{\Bas}{}(\br{1}) \rsmu{\Bas}{}(\br{2})}$ and a proper spherical average to get $\rsmu{\Bas}{}(r_{12})$?} What about $\rsmu{\Bas}{}(\br{1},\br{2}) = \sqrt{\rsmu{\Bas}{}(\br{1}) \rsmu{\Bas}{}(\br{2})}$ and a proper spherical average to get $\rsmu{\Bas}{}(r_{12})$?}
coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{})$. coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{})$.
%=================================================================
%\subsection{Short-range correlation functionals}
%=================================================================
Once defined, $\rsmu{\Bas}{}(\br{})$ can be used in RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$. Once defined, $\rsmu{\Bas}{}(\br{})$ can be used in RS-DFT functionals 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} 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}
\begin{multline} \begin{multline}
\label{eq:ec_md_mu} \label{eq:ec_md_mu}
\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
@ -319,19 +325,20 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following l
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:large_mu_ecmd} \label{eq:large_mu_ecmd}
\lim_{\mu \to \infty} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] & = 0, \lim_{\mu \to \infty} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = 0,
\\ \\
\label{eq:small_mu_ecmd} \label{eq:small_mu_ecmd}
\lim_{\mu \to 0} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] & = \Ec[\n{}{}(\br{})], \lim_{\mu \to 0} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = \Ec[\n{}{}(\br{})],
\end{align} \end{align}
\end{subequations} \end{subequations}
where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT. where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
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}]. 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 that of the ECMD functionals [Eq.~\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 $\wf{}{\rsmu{\Bas}{}}$ 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. %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. %This makes them particularly well adapted to the present context where one aims at correcting a general WFT method.
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 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} \begin{equation}
\label{eq:def_lda_tot} \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{}, \bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
@ -358,10 +365,13 @@ Therefore, the PBE complementary functional reads
\end{equation} \end{equation}
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}. 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}.
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 $\Cor \bigcap \Val = \O$. %\subsection{Valence approximation}
%=================================================================
We therefore define the valence-only 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 $\Cor \bigcap \Val = \O$.
Therefore, we define the valence-only effective interaction
\begin{equation} \begin{equation}
\W{\Bas}{\Val}(\br{1},\br{2}) = \W{\Bas}{\Val}(\br{1},\br{2}) =
\begin{cases} \begin{cases}
@ -381,7 +391,7 @@ with
= \sum_{pqrs \in \Val} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, = \sum_{pqrs \in \Val} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{gather} \end{gather}
\end{subequations} \end{subequations}
and the corresponding valence range separation function and the corresponding valence range-separation function
\begin{equation} \begin{equation}
\label{eq:muval} \label{eq:muval}
\rsmu{\Bas}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\Val}(\br{},\br{}). \rsmu{\Bas}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\Val}(\br{},\br{}).
@ -390,6 +400,9 @@ It is worth noting that, within the present definition, $\W{\Bas}{\Val}(\br{1},\
Defining $\n{\modZ}{\Val}$ as the valence one-electron density obtained with the model $\modZ$, the valence part of the complementary functional $\bE{}{\Val}[\n{\modZ}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\modZ}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modZ}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$. Defining $\n{\modZ}{\Val}$ as the valence one-electron density obtained with the model $\modZ$, the valence part of the complementary functional $\bE{}{\Val}[\n{\modZ}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\modZ}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modZ}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$.
%=================================================================
%\subsection{Computational considerations}
%=================================================================
Regarding now the main computational source of the present approach, it consists in the evaluation Regarding now the main computational source of the present approach, it consists in the evaluation
of $\W{\Bas}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. of $\W{\Bas}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point.
All through this paper, we use pair density matrix of a single Slater determinant (typically HF) All through this paper, we use pair density matrix of a single Slater determinant (typically HF)