Bread on the board

Manuscript/G2-srDFT.tex

@@ -159,6 +159,9 @@
 
 
 % methods
+\newcommand{\UEG}{\text{UEG}}
+\newcommand{\LDA}{\text{LDA}}
+\newcommand{\PBE}{\text{PBE}}
 \newcommand{\FCI}{\text{FCI}}
 \newcommand{\CCSDT}{\text{CCSD(T)}}
 \newcommand{\lr}{\text{lr}}
@@ -177,7 +180,7 @@
 \newcommand{\SO}[2]{\phi_{#1}(\bx{#2})}
 
 \newcommand{\modX}{\text{X}}
-\newcommand{\modY}{Y}
+\newcommand{\modY}{\text{Y}}
 
 % basis sets 
 \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}
 %=================================================================
-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}
 	\label{eq:e0basis}
 	\E{}{} 
 	\approx \E{\modX}{\Bas} 
-	+ \bE{}{\Bas}[\n{\wf{\modX}{\Bas}}{}],
+	+ \bE{}{\Bas}[\n{\wf{\modY}{\Bas}}{}],
 \end{equation}
 where
 \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
 \begin{equation}
   \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}
 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 $\E{\FCI}{} = E$.
-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 $\emodel$. 
 %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}
 	\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}
-(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}
 	\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}},
 \end{equation}
@@ -424,7 +427,7 @@ which states that the expectation value of $\hWee{}$ over $\wf{}{\Bas}$ is equal
 \label{eq:WeeB}
 	\hWee{\Bas} =  \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i}
 \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  
 \begin{subequations}
 \begin{align}
@@ -440,7 +443,7 @@ where
 	\label{eq:fbasis}
 	\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}
 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}
@@ -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})}.
 \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}, $\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = r_{12}^{-1}$. 
 
 
@@ -495,81 +498,47 @@ and therefore
 %\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
 \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}
 %=================================================================
 \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}
   \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
+where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
 \begin{equation}
 \label{eq:argmin}
 	\wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
 \end{equation}
-where
+and
 \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},
 \end{equation}
-is the long-range electron-electron interaction operator with
+is the long-range Coulomb operator with
 \begin{equation}
 \label{eq:erf}
 	\w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}},
 \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}
- \label{eq:large_mu_ecmd}
+\begin{align}
- \lim_{\mu \rightarrow \infty} \ecmubis = 0 \quad \forall\,\,\denr
+	\label{eq:large_mu_ecmd}
-\end{equation}
+	\lim_{\mu \rightarrow \infty} \ecmubis & = 0 \quad & \forall \n{}{}(\br{})
-\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-$\mathcal{X}$. 
 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}
 %--------------------------------------------
-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}
- \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}
-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}
 %--------------------------------------------
-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 $\mu \rightarrow \infty$. 
 
 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)
 \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. 
 Defining the valence one-body spin density matrix as
 \begin{equation}