Bread on the board

Manuscript/G2-srDFT.tex

@@ -161,6 +161,8 @@
 % methods
 \newcommand{\FCI}{\text{FCI}}
 \newcommand{\CCSDT}{\text{CCSD(T)}}
+\newcommand{\lr}{\text{lr}}
+\newcommand{\sr}{\text{sr}}
 
 \newcommand{\Nel}{N}
 
@@ -169,6 +171,9 @@
 \newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
 \newcommand{\wf}[2]{\Psi_{#1}^{#2}}
 \newcommand{\W}[2]{W_{#1}^{#2}}
+\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{\modX}{\text{X}}
@@ -389,8 +394,8 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only so
 %=================================================================
 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 parameter $\mu(\br{})$ varying in space. %(see Sec.~\ref{sec:weff}). 
+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 $\n{\modX}{\Bas}$ with $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) .   
+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}) .   
 
 
 %=================================================================
@@ -402,9 +407,9 @@ As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12
 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}
-%----------------------------------------------------------------
+%=================================================================
 In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ defined such that 
 \begin{equation}
 	\label{eq:int_eq_wee}
@@ -446,42 +451,11 @@ and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m}
 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 $\Bas$. 
 Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = r_{12}^{-1}$. 
 
-%----------------------------------------------------------------
-\subsubsection{Definition of a 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}
+\subsubsection{Range-separation function}
-   \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},
+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}. 
-\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 $\Basval$. 
-
-
-\subsubsection{Definition of a range-separation parameter varying in space}
-To be able to approximate the complementary functional $\efuncbasis$ thanks to functionals developed in the field of RSDFT, 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 
 \begin{equation}
  \label{eq:def_wcoal}
@@ -490,71 +464,101 @@ More precisely, if we define the value of the interaction at coalescence as
 where $(\bfr{},\bar{{\bf x}}_{})$ means a couple of anti-parallel spins at the same point in $\bfrb{}$, 
 we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction:
 \begin{equation}
-  \wbasiscoal{} = w^{\text{lr},\murpsi}(\bfrb{},\bfrb{})
+  \wbasiscoal{} = \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\bfrb{},\bfrb{})
 \end{equation}
-where the long-range-like interaction is defined as: 
+where the long-range-like interaction is defined as
 \begin{equation}
- w^{\text{lr},\mur}(\bfrb{1},\bfrb{2}) = \frac{ 1 }{2} \bigg( \frac{\text{erf}\big( \murr{1} \, r_{12}\big)}{r_{12}} + \frac{\text{erf}\big( \murr{2} \, r_{12}\big)}{ r_{12}}\bigg).  
+	\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}} }.  
 \end{equation}
-The equation \eqref{eq:def_wcoal} is equivalent to the following condition for $\murpsi$:
+Equation \eqref{eq:def_wcoal} is equivalent to the following condition
 \begin{equation}
  \label{eq:mu_of_r}
- \murpsi = \frac{\sqrt{\pi}}{2} \, \wbasiscoal{} \, .
+ 	\rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \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}
- \label{eq:mu_of_r_val}
- \murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, .
 \end{equation}
+%As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as
+%\begin{equation}
+% \label{eq:mu_of_r_val}
+% \murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, .
+%\end{equation}
 An important point to notice is that, in the limit of a complete basis set $\Bas$, as 
 \begin{equation}
- \begin{aligned}
+\label{eq:lim_W}
- &\lim_{\Bas \rightarrow \infty}\wbasis = 1/r_{12}  \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\\
+	\lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1}  \quad \forall  (\br{1},\br{2})
- &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, , 
+% &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, , 
- \end{aligned}
-\end{equation}
-one has
-\begin{equation}
- \begin{aligned}
- &\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = +\infty\,\,, \\
- &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,, 
- \end{aligned}
 \end{equation}
+one has $\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = \infty$
+% &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,, 
 and therefore 
 \begin{equation}
- \label{eq:lim_mur}
+\label{eq:lim_mur}
- \begin{aligned}
+	\lim_{\Bas \rightarrow \infty} \rsmu{\wf{}{\Bas}}{}(\br{}) = \infty
- &\lim_{\Bas \rightarrow \infty} \murpsi = +\infty \,\, \\
+%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
- &\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
- \end{aligned}
 \end{equation}
 
-\subsection{Approximations for the complementary functional $\ecompmodel$}
+%=================================================================
-\subsubsection{General scheme}
+\subsection{Valence effective interaction}
-\label{sec:ecmd}
+%=================================================================
-In \onlinecite{GinPraFerAssSavTou-JCP-18} the authors have proposed to approximate the complementary functional $\efuncbasis$ by using a specific class of SRDFT energy functionals, namely the ECMD whose general definition is\cite{TouGorSav-TCA-05}: 
+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}
- \begin{aligned}
+	\label{eq:expectweebval}
-  \label{eq:ec_md_mu}
+	\mel*{\wf{}{\Bas}}{\hWee{\Val}}{\wf{}{\Bas}} =  \frac{1}{2} \iint \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
-  \ecmubis = & \min_{\Psi \rightarrow \denr}\elemm{\Psi}{\kinop +\weeop}{\Psi}\\-\;&\elemm{\psimu[\denr]}{\kinop+\weeop}{\psimu[\denr]},
- \end{aligned}
 \end{equation}
-where the wave function $\psimu[\denr]$ is defined by the constrained minimization
+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}
+%=================================================================
+\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}
+\begin{multline}
+  \label{eq:ec_md_mu}
+  \ecmubis = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
+  \\
+  - \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
+\end{multline}
+where the wave function $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
 \begin{equation}
 \label{eq:argmin}
-\psimu[\denr] = \arg \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop + \weeopmu}{\Psi},
+	\wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
 \end{equation}
-where $\weeopmu$ is the long-range electron-electron interaction operator
+where
 \begin{equation}
-  \label{eq:weemu}
+\label{eq:weemu}
-  \weeopmu = \frac{1}{2} \iint \text{d}{\bf r}_1  \text{d}{\bf r}_2 \; w^{\text{lr},\mu}(|{\bf r}_1 - {\bf r}_2|) \hat{n}^{(2)}({\bf r}_1,{\bf r}_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}
-with
+is the long-range electron-electron interaction operator with
 \begin{equation}
 \label{eq:erf}
-w^{\text{lr},\mu}(r_{12}) = \frac{\text{erf}(\mu r_{12})}{r_{12}},
+	\w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}},
 \end{equation}
-and the pair-density operator $\hat{n}^{(2)}({\bf r}_1,{\bf r}_2) =\hat{n}({\bf r}_1) \hat{n}({\bf r}_2) - \delta ({\bf r}_1-{\bf r}_2) \hat{n}({\bf r}_1)$.
+and the pair-density operator $\hn{}{(2)}(\br{1},\br{2}) =\hn{}{}(\br{1}) \hn{}{}(\br{2}) - \delta (\br{1}-\br{2}) \hn{}{}(\br{1})$.
 The ECMD functionals admit two limits as function of $\mu$
 \begin{equation}
  \label{eq:large_mu_ecmd}
@@ -581,7 +585,9 @@ 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{LDA approximation for the complementary functional}
+%--------------------------------------------
+\subsubsection{Local density approximation}
+%--------------------------------------------
 As done in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, one can define an LDA-like approximation for $\ecompmodel$ as 
 \begin{equation}
  \label{eq:def_lda_tot}
@@ -589,7 +595,9 @@ As done in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, one can define an LDA-li
 \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.
 
-\subsubsection{New PBE interpolated ECMD 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 $\mu$. 
 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 $\mu \rightarrow \infty$. 
 
@@ -633,7 +641,9 @@ 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)
 \end{equation}
 
+%--------------------------------------------
 \subsection{Valence-only approximation for the complementary functional}
+%--------------------------------------------
 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}