loos/srDFT_G2
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Bread on the board

Browse Source
This commit is contained in:
Pierre-Francois Loos 2019-04-05 12:21:37 +02:00
parent d99dac8be8
commit 412fab2084
1 changed files with 70 additions and 69 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

133
Manuscript/G2-srDFT.tex
View File

@ -159,6 +159,9 @@
% methods % methods
\newcommand{\UEG}{\text{UEG}}
\newcommand{\LDA}{\text{LDA}}
\newcommand{\PBE}{\text{PBE}}
\newcommand{\FCI}{\text{FCI}} \newcommand{\FCI}{\text{FCI}}
\newcommand{\CCSDT}{\text{CCSD(T)}} \newcommand{\CCSDT}{\text{CCSD(T)}}
\newcommand{\lr}{\text{lr}} \newcommand{\lr}{\text{lr}}
@ -177,7 +180,7 @@
\newcommand{\SO}[2]{\phi_{#1}(\bx{#2})} \newcommand{\SO}[2]{\phi_{#1}(\bx{#2})}
\newcommand{\modX}{\text{X}} \newcommand{\modX}{\text{X}}
\newcommand{\modY}{Y} \newcommand{\modY}{\text{Y}}
% basis sets % basis sets
\newcommand{\Bas}{\mathcal{B}} \newcommand{\Bas}{\mathcal{B}}
@ -267,13 +270,13 @@ We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} fo
%================================================================= %=================================================================
%\subsection{Correcting the basis set error of a general WFT model} %\subsection{Correcting the basis set error of a general WFT model}
%================================================================= %=================================================================
Let us assume we have both the density $\n{\modX}{\Bas}$ and energy $\E{\modX}{\Bas}$ of a $\Nel$-electron system described by a method $\modX$ in an incomplete basis set $\Bas$. Let us assume we have both the energy $\E{\modX}{\Bas}$ and density $\n{\modY}{\Bas}$ of a $\Nel$-electron system described by two methods $\modX$ and $\modY$ (potentially identical) in an incomplete basis set $\Bas$.
According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\n{\modX}{\Bas}$ is a resonable approximation of the \textit{exact} ground state density $\n{}{}$, one may approximate the \textit{exact} ground state energy as According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and $\n{\modY}{\Bas}$ are reasonable approximations of the \textit{exact} ground state energy $\E{}{}$ and density $\n{}{}$, one may write
\begin{equation} \begin{equation}
\label{eq:e0basis} \label{eq:e0basis}
\E{}{} \E{}{}
\approx \E{\modX}{\Bas} \approx \E{\modX}{\Bas}
+ \bE{}{\Bas}[\n{\wf{\modX}{\Bas}}{}], + \bE{}{\Bas}[\n{\wf{\modY}{\Bas}}{}],
\end{equation} \end{equation}
where where
\begin{equation} \begin{equation}
@ -291,11 +294,11 @@ Both wave functions yield the same target density $\n{}{}$.
An important aspect of such theory is that, in the limit of a complete basis set $\Bas$ (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that An important aspect of such theory is that, in the limit of a complete basis set $\Bas$ (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that
\begin{equation} \begin{equation}
\label{eq:limitfunc} \label{eq:limitfunc}
\lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\wf{\modX}{\Bas}}{}] ) = \E{\modX}{} \approx E, \lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\wf{\modY}{\Bas}}{}] ) = \E{\modX}{} \approx E,
\end{equation} \end{equation}
where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set. where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set.
In the case of $\modX = \FCI$, we have as strict equality as $\E{\FCI}{} = E$. In the case of $\modX = \FCI$, we have as 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 method $\modX$. 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 $\modY$.
%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 $\emodel$. %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 %As any wave function model is necessary an approximation to the FCI model, one can write
@ -415,7 +418,7 @@ In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ de
\label{eq:int_eq_wee} \label{eq:int_eq_wee}
\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2}, \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
\end{equation} \end{equation}
(where $\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})$ is the two-body density associated with $\wf{}{\Bas}$, $\bx{} = \qty(\br{},\sigma)$ collects space and spin variables, and $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$), it is key to realise that (where $\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})$ is the two-body density associated with $\wf{}{\Bas}$, $\bx{} = \qty(\br{},\sigma)$ collects space and spin variables, and $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$), one must realise that
\begin{equation} \begin{equation}
\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}}, \mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}},
\end{equation} \end{equation}
@ -424,7 +427,7 @@ which states that the expectation value of $\hWee{}$ over $\wf{}{\Bas}$ is equal
\label{eq:WeeB} \label{eq:WeeB}
\hWee{\Bas} = \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i} \hWee{\Bas} = \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i}
\end{equation} \end{equation}
over the same wave function $\wf{}{\Bas}$, where the indices run over all \alert{occupied} spinorbitals $\SO{i}{}$ in $\Bas$ and $\vijkl$ are the usual two-electron Coulomb integrals. over the same wave function $\wf{}{\Bas}$, where the indices run over all spinorbitals $\SO{i}{}$ in $\Bas$ and $\vijkl$ are the usual two-electron Coulomb integrals.
Because one can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that Because one can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
@ -440,7 +443,7 @@ where
\label{eq:fbasis} \label{eq:fbasis}
\f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \f{\wf{}{\Bas}}{}(\bx{1},\bx{2})
\\ \\
= \sum_{ijklmn \in \Bas} \SO{i}{1} \SO{j}{2} \vijkl \Gam{mn}{pq}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1}, = \sum_{ijklmn \in \Bas} \SO{i}{1} \SO{j}{2} \vijkl \Gam{mn}{kl}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1},
\end{multline} \end{multline}
and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\wf{}{\Bas}}$ is the two-body density tensor of $\wf{}{\Bas}$, it comes naturally that and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\wf{}{\Bas}}$ is the two-body density tensor of $\wf{}{\Bas}$, it comes naturally that
\begin{equation} \begin{equation}
@ -448,7 +451,7 @@ and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m}
\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})}{\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})}. \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})}{\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})}.
\end{equation} \end{equation}
As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries and is necessarily \textit{finite} at $r_{12} = 0$ for an incomplete basis set $\Bas$. As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries and is necessarily \textit{finite} at $r_{12} = 0$ for an incomplete basis set.
Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = r_{12}^{-1}$. Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = r_{12}^{-1}$.
@ -495,81 +498,47 @@ and therefore
%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, . %\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
\end{equation} \end{equation}
%=================================================================
\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 split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively, and $\Cor \bigcap \Val = \O$.
According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ satisfying
\begin{equation}
\label{eq:expectweebval}
\mel*{\wf{}{\Bas}}{\hWee{\Val}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
\end{equation}
where $\hWee{\Val}$, the valence part of the Coulomb operator, has a similar expression as $\hWee{\Bas}$ in Eq.~\eqref{eq:WeeB}.
%\begin{equation}
% \hWee{\Val} = \frac{1}{2} \sum_{ijkl \in \Val} \vijkl \aic{k}\aic{l}\ai{j}\ai{i},
%\end{equation}
Following the spirit of Eq.~\eqref{eq:fbasis}, the function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ can be defined as
\begin{multline}
\label{eq:fbasisval}
\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})
\\
= \sum_{ij \in \Bas} \sum_{klmn \in \Val} \SO{i}{1} \SO{j}{2} \vijkl \gammaklmn{\wf{}{\Bas}} \SO{n}{2} \SO{m}{1}.
\end{multline}
and, the valence part of the effective interaction is
\begin{equation}
\label{eq:def_weebasis}
\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) }{\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})},
\end{equation}
where $\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})$ is the two body density associated to the valence electrons.
%\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, $\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ and $\murpsival$ fulfils Eqs.~\eqref{eq:lim_W} and \eqref{eq:lim_mur}.
%================================================================= %=================================================================
\subsection{Complementary functional} \subsection{Complementary functional}
%================================================================= %=================================================================
\label{sec:ecmd} \label{sec:ecmd}
In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} the authors have proposed to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ using a specific class of SR-DFT energy functionals, namely the ECMD whose general definition is \cite{TouGorSav-TCA-05} In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} the authors proposed to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ using a specific class of SR-DFT energy functionals, namely the ECMD whose general definition is \cite{TouGorSav-TCA-05}
\begin{multline} \begin{multline}
\label{eq:ec_md_mu} \label{eq:ec_md_mu}
\ecmubis = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}} \ecmubis = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
\\ \\
- \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}, - \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
\end{multline} \end{multline}
where the wave function $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
\begin{equation} \begin{equation}
\label{eq:argmin} \label{eq:argmin}
\wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}}, \wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
\end{equation} \end{equation}
where and
\begin{equation} \begin{equation}
\label{eq:weemu} \label{eq:weemu}
\hWee{\lr,\rsmu{}{}} = \frac{1}{2} \iint \w{}{\lr,\rsmu{}{}}(r_{12}) \hn{}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2}, \hWee{\lr,\rsmu{}{}} = \frac{1}{2} \iint \w{}{\lr,\rsmu{}{}}(r_{12}) \hn{}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2},
\end{equation} \end{equation}
is the long-range electron-electron interaction operator with is the long-range Coulomb operator with
\begin{equation} \begin{equation}
\label{eq:erf} \label{eq:erf}
\w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}, \w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}},
\end{equation} \end{equation}
and the pair-density operator $\hn{}{(2)}(\br{1},\br{2}) =\hn{}{}(\br{1}) \hn{}{}(\br{2}) - \delta (\br{1}-\br{2}) \hn{}{}(\br{1})$. and $\hn{}{(2)}(\br{1},\br{2}) =\hn{}{}(\br{1}) \hn{}{}(\br{2}) - \delta (\br{1}-\br{2}) \hn{}{}(\br{1})$ is the pair-density operator.
The ECMD functionals admit two limits as function of $\mu$ The ECMD functionals admit two limits as function of $\rsmu{}{}$
\begin{equation} \begin{subequations}
\begin{align}
\label{eq:large_mu_ecmd} \label{eq:large_mu_ecmd}
\lim_{\mu \rightarrow \infty} \ecmubis = 0 \quad \forall\,\,\denr \lim_{\mu \rightarrow \infty} \ecmubis & = 0 \quad & \forall \n{}{}(\br{})
\end{equation} \\
\begin{equation}
\label{eq:small_mu_ecmd} \label{eq:small_mu_ecmd}
\lim_{\mu \rightarrow 0} \ecmubis = E_{\text{c}}[\denr]\quad \forall\,\,\denr \lim_{\mu \to 0} \ecmubis & = \Ec[\denr] \quad & \forall \n{}{}(\br{})
\end{equation} \end{align}
where $E_{\text{c}}[\denr]$ is the usual universal correlation functional defined in the Kohn-Sham DFT. \end{subequations}
These functionals differ from the standard RSDFT correlation functional by the fact that the reference is not the Kohn-Sham Slater determinant but a multi determinant wave function, which makes them much more adapted in the present context where one aims at correcting the general multi-determinant WFT model. where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
These 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, which makes them much more adapted in the present context where one aims at correcting a general multi-determinant WFT model.
The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$ labelled by ECMD-$\mathcal{X}$. The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$ labelled by ECMD-$\mathcal{X}$.
Such a functional of the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$. Such a functional of the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$.
@ -588,17 +557,18 @@ for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to
%-------------------------------------------- %--------------------------------------------
\subsubsection{Local density approximation} \subsubsection{Local density approximation}
%-------------------------------------------- %--------------------------------------------
As done in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, one can define an LDA-like approximation for $\ecompmodel$ as Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we define the LDA version of ECMD as
\begin{equation} \begin{equation}
\label{eq:def_lda_tot} \label{eq:def_lda_tot}
\ecompmodellda = \int \, \text{d}{\bf r} \,\, \denmodelr \,\, \emuldamodel\,, \bE{\LDA}{\Bas,\wf{}{\Bas}}[\n{}{}(\br{})] = \int \n{}{}(\br{}) \emuldamodel \dbr{}
\end{equation} \end{equation}
where $\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}(n,\mu)$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}. In practice, for open-shell systems, we use the spin-polarized version of this functional (i.e., depending on the spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case. where $\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}(n,\mu)$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas 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} \subsubsection{New PBE functional}
%-------------------------------------------- %--------------------------------------------
The LDA-like functional defined in \eqref{eq:def_lda_tot} relies only on the transferability of the physics of UEG which is certainly valid for large values of $\mu$ but which is known to over correlate for small values of $\mu$. The LDA-like functional defined in \eqref{eq:def_lda_tot} relies only on the transferability of the physics of the uniform electron gas (UEG) which is certainly valid for large values of $\mu$ but which is known to over correlate for small values of $\mu$.
In order to correct such a defect, we propose here a new ECMD functional inspired by the recently proposed functional of some of the present authors\cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional when $\mu \rightarrow 0$ and the exact behaviour which is known when $\mu \rightarrow \infty$. In order to correct such a defect, we propose here a new ECMD functional inspired by the recently proposed functional of some of the present authors\cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional when $\mu \rightarrow 0$ and the exact behaviour which is known when $\mu \rightarrow \infty$.
Thanks to the study of the behaviour in the large $\mu$ limit of the various quantities appearing in the ECMD\cite{TouColSav-PRA-04,GoriSav-PRA-06,PazMorGori-PRB-06}, one can have an analytical expression of $\ecmubis$ in that regime Thanks to the study of the behaviour in the large $\mu$ limit of the various quantities appearing in the ECMD\cite{TouColSav-PRA-04,GoriSav-PRA-06,PazMorGori-PRB-06}, one can have an analytical expression of $\ecmubis$ in that regime
@ -641,9 +611,40 @@ Therefore, we propose this approximation for the complementary functional $\ecom
\ecompmodelpbe = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(n({\bf r}),\nabla n({\bf r});\,\mur) \ecompmodelpbe = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(n({\bf r}),\nabla n({\bf r});\,\mur)
\end{equation} \end{equation}
%-------------------------------------------- %=================================================================
\subsection{Valence-only approximation for the complementary functional} \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 split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively, and $\Cor \bigcap \Val = \O$.
According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ satisfying
\begin{equation}
\label{eq:expectweebval}
\mel*{\wf{}{\Bas}}{\hWee{\Val}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
\end{equation}
where $\hWee{\Val}$, the valence part of the Coulomb operator, has a similar expression as $\hWee{\Bas}$ in Eq.~\eqref{eq:WeeB}.
%\begin{equation}
% \hWee{\Val} = \frac{1}{2} \sum_{ijkl \in \Val} \vijkl \aic{k}\aic{l}\ai{j}\ai{i},
%\end{equation}
Following the spirit of Eq.~\eqref{eq:fbasis}, the function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ can be defined as
\begin{multline}
\label{eq:fbasisval}
\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})
\\
= \sum_{ij \in \Bas} \sum_{klmn \in \Val} \SO{i}{1} \SO{j}{2} \vijkl \gammaklmn{\wf{}{\Bas}} \SO{n}{2} \SO{m}{1}.
\end{multline}
and, the valence part of the effective interaction is
\begin{equation}
\label{eq:def_weebasis}
\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) }{\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})},
\end{equation}
where $\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})$ is the two body density associated to the valence electrons.
%\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, $\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ and $\murpsival$ fulfils Eqs.~\eqref{eq:lim_W} and \eqref{eq:lim_mur}.
We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models. We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models.
Defining the valence one-body spin density matrix as Defining the valence one-body spin density matrix as
\begin{equation} \begin{equation}