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Manuscript/srDFT_SC.tex
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@ -828,57 +828,53 @@ Also, it is shown that the basis-set correction gives substantial differential c
\subsection{Sufficient condition for size consistency}
The basis-set correction is expressed as an integral in real space
\begin{equation}
\begin{multline}
\label{eq:def_ecmdpbebasisAnnex}
\begin{aligned}
& \efuncdenpbe{\argebasis} = \\ & \int \text{d}\br{} \,\denr \ecmd(\argrebasis),
\end{aligned}
\end{equation}
\efuncdenpbe{\argebasis} = \\ \int \text{d}\br{} \,\denr \ecmd(\argrebasis),
\end{multline}
where all the local quantities $\argrebasis$ are obtained from the same wave function $\Psi$. In the limit of two non-overlapping and non-interacting dissociated fragments $\text{A}+\text{B}$, this integral can be rewritten as the sum of the integral over the region $\Omega_\text{A}$ and the integral over the region $\Omega_\text{B}$
\begin{equation}
\begin{multline}
\label{eq:def_ecmdpbebasisAB}
\begin{aligned}
& \efuncdenpbeAB{\argebasis} = \\ & \int_{\Omega_\text{A}} \text{d}\br{} \,\denr \ecmd(\argrebasis) \\ & + \int_{\Omega_\text{B}} \text{d}\br{} \,\denr \ecmd(\argrebasis).
\end{aligned}
\end{equation}
Therefore, a sufficient condition to obtain size consistency is that all the local quantities $\argrebasis$ are \textit{intensive}, i.e. that they \textit{locally} coincide in the supersystem $\text{A}+\text{B}$ and in each isolated fragment $\text{X}=\text{A}$ or $\text{B}$. Hence, for $\br{} \in \Omega_\text{X}$, we should have
\efuncdenpbeAB{\argebasis} =
\\
\int_{\Omega_\text{A}} \text{d}\br{} \,\denr \ecmd(\argrebasis)
\\
+ \int_{\Omega_\text{B}} \text{d}\br{} \,\denr \ecmd(\argrebasis).
\end{multline}
Therefore, a sufficient condition to obtain size consistency is that all the local quantities $\argrebasis$ are \textit{intensive}, \ie, they \textit{locally} coincide in the supersystem $\text{A}+\text{B}$ and in each isolated fragment $\text{X}=\text{A}$ or $\text{B}$. Hence, for $\br{} \in \Omega_\text{X}$, we must have
\begin{subequations}
\begin{equation}
\begin{gather}
n_\text{A+B}(\br{}) = n_\text{X}(\br{}),
\label{nAB}
\end{equation}
\begin{equation}
\\
\zeta_\text{A+B}(\br{}) = \zeta_\text{X}(\br{}),
\label{zAB}
\end{equation}
\begin{equation}
\\
s_\text{A+B}(\br{}) = s_\text{X}(\br{}),
\label{sAB}
\end{equation}
\begin{equation}
\\
n_{2,\text{A+B}}(\br{}) = n_{2,\text{X}}(\br{}),
\label{n2AB}
\end{equation}
\begin{equation}
\\
\mu_{\text{A+B}}(\br{}) = \mu_{\text{X}}(\br{}),
\label{muAB}
\end{equation}
\end{gather}
\end{subequations}
where the left-hand-side quantities are for the supersystem and the right-hand-side quantities for an isolated fragment. Such conditions can be difficult to fulfill in the presence of degeneracies since the system X can be in a different mixed state (i.e. ensemble) in the supersystem $\text{A}+\text{B}$ and in the isolated fragment~\cite{Sav-CP-09}. Here, we will consider the simple case where the wave function of the supersystem is multiplicatively separable, i.e.
where the left-hand-side quantities are for the supersystem and the right-hand-side quantities for an isolated fragment. Such conditions can be difficult to fulfill in the presence of degeneracies since the system X can be in a different mixed state (\ie, ensemble) in the supersystem $\text{A}+\text{B}$ and in the isolated fragment. \cite{Sav-CP-09} Here, we will consider the simple case where the wave function of the supersystem is multiplicatively separable, \ie,
\begin{equation}
\ket{\wf{\text{A}+\text{B}}{}} = \ket{\wf{\text{A}}{}} \otimes \ket{\wf{\text{B}}{}},
\ket*{\wf{\text{A}+\text{B}}{}} = \ket*{\wf{\text{A}}{}} \otimes \ket*{\wf{\text{B}}{}},
\label{PsiAB}
\end{equation}
where $\otimes$ is the antisymmetric tensor product. In this case, it is easy to shown that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid, as well known, and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid. Before showing this, we note that even though we do not explicitly consider the case of degeneracies, the lack of size consistency which could arise from spin-multiplet degeneracies can be avoided by the same strategy used for imposing the energy independence on $S_z$, i.e. by using the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ or a zero spin polarization $\zeta(\br{}) = 0$. Moreover, the lack of size consistency which could arise from spatial degeneracies (e.g., coming from atomic p states) can also be avoided by selecting the same member of the ensemble in the supersystem and in the isolated fragment. This applies to the systems treated in this work.
where $\otimes$ is the antisymmetric tensor product. In this case, it is easy to shown that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid, as well known, and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid. Before showing this, we note that even though we do not explicitly consider the case of degeneracies, the lack of size consistency (which could arise from spin-multiplet degeneracies) can be avoided by the same strategy used for imposing the energy independence on $S_z$, \ie, by using the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ or a zero spin polarization $\zeta(\br{}) = 0$. Moreover, the lack of size consistency, which could arise from spatial degeneracies (\eg, coming from atomic $p$ states), can also be avoided by selecting the same member of the ensemble in the supersystem and in the isolated fragment. This applies to the systems treated in this work.
\subsection{Intensivity of the on-top pair density and of the local range-separation function}
\subsection{Intensivity of the on-top pair density and the local range-separation function}
The on-top pair density can be written in an orthonormal spatial orbital basis $\{\SO{p}{}\}$ as
\begin{equation}
\label{eq:def_n2}
n_{2{}}(\br{}) = \sum_{pqrs \in \Bas} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
\end{equation}
with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$. As the summations run over all orbitals in the basis set $\Bas$, $n_{2{}}(\br{})$ is invariant to orbital rotations and can thus be expressed in terms of localized orbitals. For two non-overlapping fragments $\text{A}+\text{B}$, the basis set can then partitioned into orbitals localized on the fragment A and orbitals localized on B, i.e. $\Bas=\Bas_\text{A}\cup \Bas_\text{B}$. Therefore, we see that the on-top pair density of the supersystem $\text{A}+\text{B}$ is additively separable
with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$. As the summations run over all orbitals in the basis set $\Bas$, $n_{2{}}(\br{})$ is invariant to orbital rotations and can thus be expressed in terms of localized orbitals. For two non-overlapping fragments $\text{A}+\text{B}$, the basis set can then be partitioned into orbitals localized on the fragment A and orbitals localized on B, \ie, $\Bas=\Bas_\text{A}\cup \Bas_\text{B}$. Therefore, we see that the on-top pair density of the supersystem $\text{A}+\text{B}$ is additively separable
\begin{equation}
\label{eq:def_n2}
n_{2,\text{A}+\text{B}}(\br{}) = n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{}),
@ -888,9 +884,9 @@ where $n_{2,\text{X}}(\br{})$ is the on-top pair density of the fragment X
\label{eq:def_n2}
n_{2,\text{X}}(\br{}) = \sum_{pqrs \in \Bas_\text{X}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
\end{equation}
in which the elements $\Gam{pq}{rs}$ with orbital indices restricted to the fragment X are $\Gam{pq}{rs} = 2 \mel*{\wf{\text{A}+\text{B}}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{\text{A}+\text{B}}{}} = 2 \mel*{\wf{\text{X}}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{\text{X}}{}}$, owing to the multiplicative structure of the wave function [Eq.~\eqref{PsiAB}]. This shows that the on-top pair density is a local intensive quantity.
in which the elements $\Gam{pq}{rs}$ with orbital indices restricted to the fragment X are $\Gam{pq}{rs} = 2 \mel*{\wf{\text{A}+\text{B}}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{\text{A}+\text{B}}{}} = 2 \mel*{\wf{\text{X}}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{\text{X}}{}}$, owing to the multiplicative structure of the wave function [see Eq.~\eqref{PsiAB}]. This shows that the on-top pair density is a local intensive quantity.
The local range-separation function is defined by
The local range-separation function is defined as
\begin{equation}
\label{eq:def_murAnnex}
\mur = \frac{\sqrt{\pi}}{2} \frac{f(\bfr{},\bfr{})}{n_{2}(\br{})},
@ -923,7 +919,7 @@ which gives
\end{equation}
with $\mu_{\text{X}}(\bfr{}) = (\sqrt{\pi}/2) f_{\text{X}}(\bfr{},\bfr{})/n_{2,\text{X}}(\br{})$. The local range-separation function is thus a local intensive quantity.
In conclusion, if the wave function of the supersystem $\text{A}+\text{B}$ is multiplicative separable, all local quantities used in the basis-set correction functional are intensive and therefore the basis-set correction is size consistent.
We can therefore conclude that, if the wave function of the supersystem $\text{A}+\text{B}$ is multiplicative separable, all local quantities used in the basis-set correction functional are intensive and therefore the basis-set correction is size consistent.
\section{Computation cost of the basis-set correction for a CASSCF wave function}
\label{computational}