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59
Manuscript/SI/srDFT_SC-SI.tex
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@ -163,7 +163,9 @@
\newcommand{\twodmrdiagpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsitot}[0]{ \ntwo_{\wf{}{A+B}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsiaa}[0]{ \ntwo_{\wf{}{AA}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsiaad}[0]{ \ntwo_{\wf{}{AA}}(\rr{}{})}
\newcommand{\twodmrdiagpsibb}[0]{ \ntwo_{\wf{}{BB}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsibbd}[0]{ \ntwo_{\wf{}{BB}}(\rr{}{})}
\newcommand{\twodmrdiagpsiab}[0]{ \ntwo_{\wf{}{AB}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsival}[0]{ \ntwo_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
\newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]}
@ -296,7 +298,7 @@ Such an integral can be rewritten as the sum of the contribution on $A$ and $B$
& \efuncdenpbe{\argebasis} = \\ & \int_{ \br{} \in A} \text{d}\br{} \,\denr \ecmd(\argrebasis) \\ & + \int_{ \br{} \in B} \text{d}\br{} \,\denr \ecmd(\argrebasis),
\end{aligned}
\end{equation}
Therefore, a sufficient condition to obtain size extensivity in the limit of dissociated fragments is that all arguments entering in the function $\ecmd(\argrebasis)$ are \textit{intensive}, which means that they are \textit{locally} the same in the system $A$ and in the sub system $A$ of the super system $A+B$.
Therefore, a sufficient condition to obtain size extensivity in the limit of dissociated fragments is that all arguments entering in the function $\ecmd(\argrebasis)$ are \textit{intensive}, which means that they \textit{locally} coincide in the system $A$ and in the sub system $A$ of the super system $A+B$.
Regarding the density and its gradients, these are necessary intensive quantities. The remaining questions are therefore the local range-separation parameter $\murpsi$ and the on-top pair density.
@ -305,21 +307,21 @@ A crucial ingredient in the type of functionals used in the present paper togeth
\begin{equation}
\ntwo_{\wf{}{}}(\br{}) = \sum_{pqrs} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
\end{equation}
with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$.
Assume now that the wave function $\wf{}{\Bas}$ can be written as a product of two wave functions defined on two non-overlapping and non-interacting fragments $A$ and $B$
with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$.
Assume now that the wave function $\wf{A+B}{}$ of the super system $A+B$ can be written as a product of two wave functions defined on two non-overlapping and non-interacting fragments $A$ and $B$
\begin{equation}
\ket{\wf{A+B}{}} = \ket{\wf{A}{}} \times \ket{\wf{B}{}}.
\end{equation}
Labelling the orbitals of fragment $A$ as $p_A,q_A,r_A,s_A$ and of fragment $B$ as $p_B,q_B,r_B,s_B$ and assuming that they don't overlap, one can split the two-body operator as
Labelling the orbitals of fragment $A$ as $p_A,q_A,r_A,s_A$ and of fragment $B$ as $p_B,q_B,r_B,s_B$ and assuming that they don't overlap, one can split the two-body density operator as
\begin{equation}
\begin{aligned}
\hat{\Gamma} = \hat{\Gamma}_{AA}{} + \hat{\Gamma}_{BB}{} + \hat{\Gamma}_{AB}{}
\hat{\Gamma}(\br{1},\br{2}) = \hat{\Gamma}_{AA}{}(\br{1},\br{2}) + \hat{\Gamma}_{BB}{}(\br{1},\br{2}) + \hat{\Gamma}_{AB}{}(\br{1},\br{2})
\end{aligned}
\end{equation}
with
\begin{equation}
\begin{aligned}
\hat{\Gamma}_{AA} = \sum_{p_A,q_A,r_A,s_A} \aic{r_{A,\downarrow}}\aic{s_{A,\uparrow}}\ai{q_{A,\uparrow}}\ai{p_{A,\downarrow}},
\hat{\Gamma}_{AA}(\br{1},\br{2}) = \sum_{p_A,q_A,r_A,s_A}& \SO{r_A}{1} \SO{s_A}{2} \SO{p_A}{1} \SO{q_A}{2} \\ & \aic{r_{A,\downarrow}}\aic{s_{A,\uparrow}}\ai{q_{A,\uparrow}}\ai{p_{A,\downarrow}} ,
\end{aligned}
\end{equation}
(and equivalently for $B$),
@ -331,7 +333,7 @@ with
and
\begin{equation}
\begin{aligned}
\hat{\Gamma}_{AB} = \sum_{p_A,q_B,r_A,s_B} \left( \aic{r_{A,\downarrow}}\ai{p_{A,\downarrow}} \aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}} + \aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}} \aic{r_{A,\downarrow}}\ai{p_{A,\downarrow}} \right) .
\hat{\Gamma}_{AB} = \sum_{p_A,q_B,r_A,s_B} & \SO{r_A}{1} \SO{s_B}{2} \SO{p_A}{1} \SO{q_B}{2} \\ & \left( \aic{r_{A,\downarrow}}\ai{p_{A,\downarrow}} \aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}} + \aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}} \aic{r_{A,\downarrow}}\ai{p_{A,\downarrow}} \right) .
\end{aligned}
\end{equation}
Therefore, one can express the two-body density as
@ -341,15 +343,10 @@ Therefore, one can express the two-body density as
where $\twodmrdiagpsiaa$ and $\twodmrdiagpsibb$ are the two-body densities of the isolated fragments
\begin{equation}
\begin{aligned}
& \twodmrdiagpsiaa = \\ & \sum_{p_A q_A r_A s_A} \SO{p_A}{1} \SO{q_A}{2} \bra{\wf{A}{}}\hat{\Gamma}_{AA}\ket{\wf{A}{}} \SO{r_A}{1} \SO{s_A}{2},
& \twodmrdiagpsiaa = \bra{\wf{A}{}} \hat{\Gamma}_{AA}(\br{1},\br{2}) \ket{\wf{A}{}}
\end{aligned}
\end{equation}
(and equivalently for $B$),
%\begin{equation}
% \begin{aligned}
% & \twodmrdiagpsibb = \\ & \sum_{p_B q_B r_B s_B} \SO{p_B}{1} \SO{q_B}{2} \bra{\wf{B}{}}\hat{\Gamma}_{BB}\ket{\wf{B}{}} \SO{r_B}{1} \SO{s_B}{2},
% \end{aligned}
%\end{equation}
and $\twodmrdiagpsiab$ is simply the product of the one body densities of the sub systems
\begin{equation}
\begin{aligned}
@ -358,33 +355,17 @@ and $\twodmrdiagpsiab$ is simply the product of the one body densities of the su
\end{equation}
\begin{equation}
\begin{aligned}
& n_{B}(\br{}) = \sum_{p_B r_B} \SO{p_B}{} \bra{\wf{B}{}}\aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}}\ket{\wf{B}{}} \SO{r_B}{} ,
& n_{A}(\br{}) = \sum_{p_A r_A} \SO{p_A}{} \bra{\wf{A}{}}\aic{s_{A,\uparrow}}\ai{q_{A,\uparrow}}\ket{\wf{A}{}} \SO{r_A}{} ,
\end{aligned}
\end{equation}
(and equivalently for $B$),
%\begin{equation}
% \begin{aligned}
% & n_{A}(\br{}) = \sum_{p_A r_A} \SO{p_A}{} \bra{\wf{A}{}}\aic{s_{A,\uparrow}}\ai{q_{A,\uparrow}}\ket{\wf{A}{}} \SO{r_A}{}.
% \end{aligned}
%\end{equation}
Based on these considerations, one can express the on-top pair density as the sum of the on-top pair densities of the isolated systems
(and equivalently for $B$).
As the densities of $A$ and $B$ are by definition non overlapping, one can express the on-top pair density as the sum of the on-top pair densities of the isolated systems
\begin{equation}
\begin{aligned}
\ntwo_{\wf{A+B}{}}(\br{}) = \ntwo_{\wf{A/A}{}}(\br{}) + \ntwo_{\wf{B/B}{}}(\br{})
\ntwo_{\wf{A+B}{}}(\br{}) = \twodmrdiagpsiaad + \twodmrdiagpsibbd
\end{aligned}
\end{equation}
with
\begin{equation}
\begin{aligned}
& \ntwo_{\wf{}{A/A}}(\br{}) = \sum_{p_A q_A r_A s_A} \SO{p_A}{} \SO{q_A}{} \Gam{p_A q_A}{r_A s_A} \SO{r_A}{} \SO{s_A}{}, \\
& \Gam{p_A q_A}{r_A s_A} = \bra{\Psi_A} \aic{r_{A,\downarrow}}\aic{s_{A,\uparrow}}\ai{q_{A,\uparrow}}\ai{p_{A,\downarrow}} \ket{\Psi_A}.
\end{aligned}
\end{equation}
(and equivalently for $B$).
%\begin{equation}
% \ntwo_{\wf{}{B/B}}(\br{}) = \sum_{p_B q_B r_B s_B} \SO{p_B}{} \SO{q_B}{} \Gam{p_B q_B}{r_B s_B} \SO{r_B}{} \SO{s_B}{}
%\end{equation}
As $\ntwo_{\wf{}{A/A}}(\br{}) = 0 \text{ if }\br{} \in B$ (and equivalently for $\ntwo_{\wf{}{B/B}}(\br{}) $ on $B$), one can conclude that provided that the wave function is multiplicative, the on-top pair density is a local intensive quantity.
As $\ntwo_{\wf{}{A/A}}(\br{}) = 0 \text{ if }\br{} \in B$ (and equivalently for $\ntwo_{\wf{}{B/B}}(\br{}) $ on $A$), one can conclude that provided that the wave function is multiplicative, the on-top pair density is a local intensive quantity.
\subsection{Property of the local-range separation parameter}
The local range separation parameter depends on the on-top pair density at a given point $\br{}$ and on the numerator
\begin{equation}
@ -412,17 +393,19 @@ with
% & f_{\wf{BB}{}}(\bfr{},\bfr{}) = \\ &\sum_{p_B q_B r_B s_B t_B u_B} \SO{p_B }{ } \SO{q_B}{ } \V{p_B q_B}{r_B s_B} \Gam{r_B s_B}{t_B u_B} \SO{t_B}{ } \SO{u_B}{ }.
% \end{aligned}
%\end{equation}
As a consequence, the local range-separation parameter in the super system $A+B$ with a multiplicative wave function is simply
As a consequence, the local range-separation parameter in the super system $A+B$
\begin{equation}
\label{eq:def_mur}
\murpsi = \frac{\sqrt{\pi}}{2} \frac{f_{\wf{A+B}{}}(\bfr{},\bfr{})}{\ntwo_{\wf{A+B}{}}(\br{})}
\end{equation}
which is nothing but
which, in the case of a multiplicative wave function is nothing but
\begin{equation}
\label{eq:def_mur}
\murpsi = \murpsia + \murpsib
\murpsi = \murpsia + \murpsib.
\end{equation}
and therefore is an intensive quantity. The conclusion of this paragraph is that, provided that the wave function for the system $A+B$ is multiplicative in the limit of the dissociated fragments, all quantities used for the basis set correction are intensive and therefore the basis set correction is size consistent.
As $\murpsia = 0 \text{ if }\br{} \in B$ (and equivalently for $\murpsib $ on $B$), $\murpsi$ is an intensive quantity. The conclusion of this paragraph is that, provided that the wave function for the system $A+B$ is multiplicative in the limit of the dissociated fragments, all quantities used for the basis set correction are intensive and therefore the basis set correction is size consistent.
\section{Computational considerations}
\bibliography{../srDFT_SC}

2
Manuscript/srDFT_SC.tex
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@ -572,7 +572,7 @@ An alternative way to eliminate the $S_z$ dependency is to simply set $\zeta=0$,
\subsubsection{Size consistency}
Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments \ce{A\bond{...}B}, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem \ce{A\bond{...}B}. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\Psi_{\ce{A\bond{...}B}}^{\basis} = \Psi_{\ce{A}}^{\basis} \Psi_{\ce{B}}^{\basis}$. In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, a properly chosen CAS wave function is sufficient to recover this property. \titou{The underlying active space must however be chosen in such a way that it leads to size-consistent energies in the limit of dissociated fragments.}
Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments \ce{A\bond{...}B}, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem \ce{A\bond{...}B}. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\Psi_{\ce{A\bond{...}B}}^{\basis} = \Psi_{\ce{A}}^{\basis} \Psi_{\ce{B}}^{\basis}$ \manu{(see SI for more detailed demonstration of that statement)}. In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, a properly chosen CAS wave function is sufficient to recover this property. \titou{The underlying active space must however be chosen in such a way that it leads to size-consistent energies in the limit of dissociated fragments.}
\subsection{\titou{Complementary density functional approximations}}