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working on the theory section

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Manuscript/srDFT_SC.aux
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@ -30,29 +30,37 @@
\citation{G2,excited}
\citation{kato}
\citation{GinPraFerAssSavTou-JCP-18}
\bibdata{srDFT_SCNotes,srDFT_SC}
\bibstyle{aipnum4-1}
\citation{REVTEX41Control}
\citation{aip41Control}
\citation{GinPraFerAssSavTou-JCP-18}
\@writefile{toc}{\contentsline {subsection}{\numberline {A}Basic formal equations}{2}{section*.5}}
\newlabel{eq:levy}{{1}{2}{}{equation.2.1}{}}
\newlabel{eq:levy_func}{{2}{2}{}{equation.2.2}{}}
\newlabel{eq:e0approx}{{5}{2}{}{equation.2.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B}Definition of an effective interaction within $\mathcal {B}$}{2}{section*.6}}
\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{2}{section*.7}}
\newlabel{LastBibItem}{{0}{2}{}{figure.7}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{3}{figure.1}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {C}Definition of an range-separation parameter varying in real space}{2}{section*.7}}
\citation{GinPraFerAssSavTou-JCP-18}
\bibdata{srDFT_SCNotes,srDFT_SC}
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\citation{REVTEX41Control}
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\newlabel{eq:weelr}{{11}{3}{}{equation.2.11}{}}
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\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{3}{section*.8}}
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Manuscript/srDFT_SC.out
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@ -4,4 +4,5 @@
\BOOKMARK [1][-]{section*.4}{Theory}{section*.2}% 4
\BOOKMARK [2][-]{section*.5}{Basic formal equations}{section*.4}% 5
\BOOKMARK [2][-]{section*.6}{Definition of an effective interaction within B}{section*.4}% 6
\BOOKMARK [1][-]{section*.7}{Results}{section*.2}% 7
\BOOKMARK [2][-]{section*.7}{Definition of an range-separation parameter varying in real space}{section*.4}% 7
\BOOKMARK [1][-]{section*.8}{Results}{section*.2}% 8

63
Manuscript/srDFT_SC.tex
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@ -105,6 +105,7 @@
\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
\newcommand{\wbasiscoal}[0]{W_{\wf{}{\Bas}}(\bfr{},\bfr{})}
\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
@ -115,7 +116,7 @@
\newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]}
\newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}}
\newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]}
\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})}
%\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})}
\newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
\newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})}
\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
@ -170,6 +171,8 @@
\newcommand{\sr}{\text{sr}}
\newcommand{\Nel}{N}
\newcommand{\V}[2]{V_{#1}^{#2}}
\newcommand{\n}[2]{n_{#1}^{#2}}
\newcommand{\E}[2]{E_{#1}^{#2}}
@ -183,7 +186,7 @@
\newcommand{\w}[2]{w_{#1}^{#2}}
\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
\newcommand{\SO}[2]{\phi_{#1}(\bx{#2})}
\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
\newcommand{\modX}{\text{X}}
\newcommand{\modY}{\text{Y}}
@ -280,7 +283,7 @@ The exact ground state energy $E_0$ of a $N-$electron system can be obtained by
\label{eq:levy}
E_0 = \min_{\denr} \bigg\{ F[\denr] + (v_{\text{ne}} (\br{}) |\denr) \bigg\},
\end{equation}
where $(v_{ne}(\br)|\denr)$ is the nuclei-electron interaction for a given density $\denr$ and $F[\denr]$ is the so-called Levy-Liev universal density functional
where $(v_{ne}(\br{})|\denr)$ is the nuclei-electron interaction for a given density $\denr$ and $F[\denr]$ is the so-called Levy-Liev universal density functional
\begin{equation}
\label{eq:levy_func}
F[\denr] = \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi}.
@ -292,17 +295,17 @@ Following equation (7) of \cite{GinPraFerAssSavTou-JCP-18}, we split $F[\denr]$
\begin{equation}
F[\denr] = \min_{\wf{}{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\denr}
\end{equation}
where $\wf{}{\Bas}$ refer to $N-$electron wave functions expanded in $\Bas$, and
where $\efuncden{\denr}$ is the density functional complementary to the basis set $\Bas$ defined as
\begin{equation}
\begin{aligned}
\efuncden{\denr} =& \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi} \\ 
&- \min_{\Psi^{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}},
&- \min_{\Psi^{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}.
\end{aligned}
\end{equation}
and $\wf{}{\Bas}$ refer to $N-$electron wave functions expanded in $\Bas$.
The functional $\efuncden{\denr}$ must therefore recover all physical effects not included in the basis set $\Bas$.
Assuming that the FCI density $\denFCI$ in $\Bas$ is a good approximation of the exact density (see equations 12-15 of \cite{GinPraFerAssSavTou-JCP-18}), one obtains the following approximation for the exact ground state density
Assuming that the FCI density $\denFCI$ in $\Bas$ is a good approximation of the exact density, one obtains the following approximation for the exact ground state density (see equations 12-15 of \cite{GinPraFerAssSavTou-JCP-18})
\begin{equation}
\label{eq:e0approx}
E_0 = \efci + \efuncbasisFCI
@ -317,17 +320,55 @@ As it was originally derived in \cite{GinPraFerAssSavTou-JCP-18} (see section D
More specifically, we define the effective interaction associated to a given wave function $\wf{}{\Bas}$ as
\begin{equation}
\wbasis = \fbasis/\twodmrdiagpsi
\label{eq:wbasis}
\wbasis =
\begin{cases}
\fbasis /\twodmrdiagpsi, & \text{if $\twodmrdiagpsi \ne 0$,}
\\
\infty, & \text{otherwise,}
\end{cases}
\end{equation}
where $\twodmrdiagpsi$ is the opposite spin two-body density associated to $\wf{}{\Bas}$
\begin{equation}
\twodmrdiagpsi = \sum_{pqrs} \phi_{p}(\br) \phi_{q}(\br) \Gam{pq}{rs} \phi_{r}(\br) \phi_{s}(\br),
\twodmrdiagpsi = \sum_{pqrs} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{equation}
$\Gam{pq}{rs}$
$\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated two-body tensor, $\SO{p}{}$ are the spatial orthonormal orbitals,
\begin{equation}
\int \int \dr{1} \dr{2} \wbasis \twodmrdiagpsi = \elemm{\wf{}{\Bas}}{\weeop}{\wf{}{\Bas}},
\label{eq:fbasis}
\fbasis
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation}
where $\twodmrdiagpsi$ is the two-body density of
and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals.
With such a definition, one can show that $\wbasis$ satisfies
\begin{equation}
\int \int \dr{1} \dr{2} \wbasis \twodmrdiagpsi = \int \int \dr{1} \dr{2} \frac{\twodmrdiagpsi}{|\br{1}-\br{2}|}.
\end{equation}
As it was shown in \cite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessary finite at coalescence for an incomplete basis set, and tends to the regular coulomb interaction in the limit of a complete basis set, that is
\begin{equation}
\label{eq:cbs_wbasis}
\lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|}.
\end{equation}
The condition of equation \eqref{eq:cbs_wbasis} is fundamental as it guarantees the good behaviour of all the theory in the limit of a complete basis set.
\subsection{Definition of an range-separation parameter varying in real space}
As the effective interaction within a basis set $\wbasis$ is non divergent, one can fit such a function with a long-range interaction defined in the framework of RSDFT which depends on the range-separation parameter $\mu$
\begin{equation}
\label{eq:weelr}
w_{ee}^{\lr}(\mu;\br{1},\br{2}) = \frac{\text{erf}\big(\mu \,|\br{1}-\br{2}| \big)}{|\br{1}-\br{2}|}.
\end{equation}
As originally proposed in \cite{GinPraFerAssSavTou-JCP-18}, we introduce a range-separation parameter $\murpsi$ varying in real space
\begin{equation}
\murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal
\end{equation}
such that
\begin{equation}
w_{ee}^{\lr}(\murpsi;\br{ },\br{ }) = \wbasiscoal.
\end{equation}
Because of the very definition of $\wbasis$, one has the following properties at the CBS limit (see \eqref{eq:cbs_wbasis})
\begin{equation}
\label{eq:cbs_mu}
\lim_{\Bas \rightarrow \text{CBS}} \murpsi = \infty,
\end{equation}
which is fundamental to guarantee the good behaviour of the theory at the CBS limit.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
%%%%%%%%%%%%%%%%%%%%%%%%