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Merge branch 'master' of https://github.com/pfloos/srDFT_G2

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Manuscript/G2-srDFT.tex
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@ -172,6 +172,10 @@
\newcommand{\n}[2]{n_{#1}^{#2}}
\newcommand{\E}[2]{E_{#1}^{#2}}
\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
\newcommand{\bEc}[1]{\Bar{E}_\text{c}^{#1}}
\newcommand{\e}[2]{\varepsilon_{#1}^{#2}}
\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
\newcommand{\bec}[1]{\Bar{e}^{#1}}
\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
\newcommand{\W}[2]{W_{#1}^{#2}}
\newcommand{\w}[2]{w_{#1}^{#2}}
@ -390,29 +394,30 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only so
% \EexFCIinfty \approx \EexFCIbasis + \efuncden{\dencipsi}
%\end{equation}
However, in addition of being unknown, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
One of the consequences of the incompleteness of $\Bas$ is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points.
As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent Coulomb interaction at $r_{12} = 0$.
Therefore, it feels natural to evaluate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals.
Contrary to conventional RS-DFT schemes which require a range-separated parameter $\rsmu{}{}$, we must know the value of $\rsmu{}{}$ at any point in space due to the spatial inhomogeneity of $\Bas$, hence defining a range-separated \textit{function} $\rsmu{}{}(\br{})$.
%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}).
%=================================================================
%\subsection{General scheme for the approximation of the unknown complementary functional $\efuncbasis$}
%=================================================================
However, in addition of being unknown, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
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 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}).
Then, 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}) .
%=================================================================
%\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\Bas$}
%\label{sec:weff}
%=================================================================
One of the consequences of the incompleteness of $\Bas$ is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points.
As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could also originate from a Hamiltonian with a non-divergent Coulomb interaction.
Therefore, the impact of the incompleteness of $\Bas$ can be viewed as a removal of the divergence of the Coulomb interaction at $r_{12} = 0$.
The present paragraph briefly describes how to obtain an effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
%=================================================================
\subsection{Effective Coulomb operator}
%=================================================================
%The present section briefly describes how to obtain an effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ defined such that
\begin{equation}
\label{eq:int_eq_wee}
@ -456,7 +461,7 @@ Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-1
%=================================================================
\subsubsection{Range-separation function}
\subsection{Range-separation function}
%=================================================================
To be able to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ thanks to functionals developed in the field of RS-DFT, we fit the effective interaction with a long-range interaction having a range-separation parameter \textit{varying in space}.
More precisely, if we define the value of the interaction at coalescence as
@ -476,7 +481,7 @@ where the long-range-like interaction is defined as
Equation \eqref{eq:def_wcoal} is equivalent to the following condition
\begin{equation}
\label{eq:mu_of_r}
\rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{})
\rsmu{\wf{}{\Bas}}{}(\br{}) = \W{\wf{}{\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}
@ -486,7 +491,7 @@ Equation \eqref{eq:def_wcoal} is equivalent to the following condition
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}\wbasis = r_{12}^{-1} \quad \forall (\bx{1},\bx{2})
% &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, ,
\end{equation}
one has $\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = \infty$
@ -499,15 +504,16 @@ and therefore
\end{equation}
%=================================================================
\subsection{Complementary functional}
%=================================================================
\label{sec:ecmd}
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}
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we propose here to approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of SR-DFT energy functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
\begin{multline}
\label{eq:ec_md_mu}
\ecmubis = \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{}{}}
\\
- \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
\end{multline}
@ -516,30 +522,68 @@ where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimizati
\label{eq:argmin}
\wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
\end{equation}
and
with
\begin{equation}
\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{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})
\end{equation}
is the long-range Coulomb operator with
and
\begin{equation}
\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}
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 $\rsmu{}{}$
is the long-range part of the Coulomb operator.
The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following limiting forms:
\begin{subequations}
\begin{align}
\label{eq:large_mu_ecmd}
\lim_{\mu \rightarrow \infty} \ecmubis & = 0 \quad & \forall \n{}{}(\br{})
\lim_{\mu \to \infty} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = 0,
\\
\label{eq:small_mu_ecmd}
\lim_{\mu \to 0} \ecmubis & = \Ec[\denr] \quad & \forall \n{}{}(\br{})
\lim_{\mu \to 0} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = \Ec[\n{}{}(\br{})],
\end{align}
\end{subequations}
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.
%--------------------------------------------
%\subsubsection{Local density approximation}
%--------------------------------------------
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we define the LDA version of ECMD as
\begin{equation}
\label{eq:def_lda_tot}
\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
\end{equation}
where $\be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}]$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) 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 recently proposed functional of some of the present authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{\PBE}{}(\n{}{},\nabla \n{}{})$ for small $\rsmu{}{}$ and the exact large-$\rsmu{}{}$ behaviour, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGori-PRB-06} yielding
\begin{subequations}
\begin{gather}
\label{eq:epsilon_cmdpbe}
\be{\PBE}{\sr}(\n{}{},\nabla \n{}{},\rsmu{}{}) = \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{1 + \beta(n,\nabla n, \rsmu{}{})\rsmu{}{3} }
\\
\label{eq:epsilon_cmdpbe}
\beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}.
\end{gather}
\end{subequations}
The difference between the ECMD PBE functional defined in Ref.~\cite{FerGinTou-JCP-18} and the present expression is that we approximate the \textit{exact} ground-state on-top pair density of the system $\n{}{(2)}(\br{})$ by its UEG version, i.e.
\begin{equation}
\label{eq:ueg_ontop}
\n{}{(2)}(\br{}) \approx \n{\UEG}{(2)}(\br{}) = \n{}{}(\br{})^2 g_0[\n{}{}(\br{})]
\end{equation}
where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
Therefore, the PBE complementary function reads
\begin{equation}
\label{eq:def_lda_tot}
\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\PBE}{\sr}[\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
\end{equation}
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 $\denmodel$
@ -554,62 +598,6 @@ It is important to notice that in the limit of a complete basis set, according t
\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.
%--------------------------------------------
\subsubsection{Local density approximation}
%--------------------------------------------
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we define the LDA version of ECMD as
\begin{equation}
\label{eq:def_lda_tot}
\bE{\LDA}{\Bas,\wf{}{\Bas}}[\n{}{}(\br{})] = \int \n{}{}(\br{}) \emuldamodel \dbr{}
\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-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 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 $\mu \rightarrow \infty$.
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
\begin{equation}
\label{eq:ecmd_large_mu}
\ecmubis = \frac{2\sqrt{\pi}\left(1 - \sqrt{2}\right)}{3\,\mu^3} \int \text{d}{\bf r} \,\, n^{(2)}({\bf} r)
\end{equation}
where $ n^{(2)}({\bf r}) $ is the \textit{exact} on-top pair density for the ground state of the system.
As the exact ground state on-top pair density $n^{(2)}({\bf} r)$ is not known, we propose here to approximate it by that of the UEG at the density of the system:
\begin{equation}
\label{eq:ueg_ontop}
n^{(2)}({\bf} r) \approx n^{(2)}_{\text{UEG}}(n_{\uparrow}({\bf} r) , \, n_{\downarrow}({\bf} r))
\end{equation}
where $n_{\uparrow}({\bf} r)$ and $ n_{\downarrow}({\bf} r)$ are, respectively, the up and down spin densities of the physical system at ${\bf} r$, $n^{(2)}_{\text{UEG}}(n_{\uparrow} , n_{\downarrow})$ is the UEG on-top pair density
\begin{equation}
\label{eq:ueg_ontop}
n^{(2)}_{\text{UEG}}(n_{\uparrow} , n_{\downarrow}) = 4\, n_{\uparrow} \, n_{\downarrow} \, g_0(n_{\uparrow},\, n_{\downarrow})
\end{equation}
and $g_0(n_{\uparrow} ,\, n_{\downarrow})$ is the correlation factor of the UEG whose parametrization can be found in equation (46) of \onlinecite{GorSav-PRA-06}.
As the form in \eqref{eq:ecmd_large_mu} diverges for small values of $\mu$ as $1/\mu^3$, we follow the work proposed in \cite{FerGinTou-JCP-18} and interpolate between the large-$\mu$ limit and the $\mu = 0$ limit where the $\ecmubis$ reduces to the Kohn-Sham correlation functional (see equation \eqref{eq:small_mu_ecmd}), for which we take the PBE approximation as in \cite{FerGinTou-JCP-18}.
More precisely, we propose the following expression for the
\begin{equation}
\label{eq:ecmd_large_mu}
\ecmubis = \int \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(n({\bf} r),\nabla n({\bf} r);\,\mu)
\end{equation}
with
\begin{equation}
\label{eq:epsilon_cmdpbe}
\bar{e}_{\text{c,md}}^\text{PBE}(n,\nabla n;\,\mu) = \frac{e_c^{PBE}(n,\nabla n)}{1 + \beta_{\text{c,md}\,\text{PBE}}(n,\nabla n;\,\mu)\mu^3 }
\end{equation}
\begin{equation}
\label{eq:epsilon_cmdpbe}
\beta(n,\nabla n;\,\mu) = \frac{3 e_c^{PBE}(n,\nabla n)}{2\sqrt{\pi}\left(1 - \sqrt{2}\right)n^{(2)}_{\text{UEG}}(n_{\uparrow} , n_{\downarrow})}.
\end{equation}
Therefore, we propose this approximation for the complementary functional $\ecompmodel$:
\begin{equation}
\label{eq:def_lda_tot}
\ecompmodelpbe = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(n({\bf r}),\nabla n({\bf r});\,\mur)
\end{equation}
%=================================================================
\subsection{Valence effective interaction}