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Manuscript/srDFT_SC.bbl
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@ -568,15 +568,6 @@
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\
}\textbf {\bibinfo {volume} {77}},\ \bibinfo {pages} {3865} (\bibinfo {year}
{1996})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Gori-Giorgi}\ and\ \citenamefont
{Savin}(2006{\natexlab{b}})}]{GoriSav-PRA-06}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
{Gori-Giorgi}}\ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Savin}},\ }\href {\doibase 10.1103/PhysRevA.73.032506} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {73}},\
\bibinfo {pages} {032506} (\bibinfo {year} {2006}{\natexlab{b}})}\BibitemShut
{NoStop}%
\bibitem [{\citenamefont {Paziani}\ \emph {et~al.}(2006)\citenamefont
{Paziani}, \citenamefont {Moroni}, \citenamefont {Gori-Giorgi},\ and\
\citenamefont {Bachelet}}]{PazMorGorBac-PRB-06}%
@ -588,6 +579,15 @@
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. B}\
}\textbf {\bibinfo {volume} {73}},\ \bibinfo {pages} {155111} (\bibinfo
{year} {2006})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Gori-Giorgi}\ and\ \citenamefont
{Savin}(2006{\natexlab{b}})}]{GoriSav-PRA-06}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
{Gori-Giorgi}}\ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Savin}},\ }\href {\doibase 10.1103/PhysRevA.73.032506} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {73}},\
\bibinfo {pages} {032506} (\bibinfo {year} {2006}{\natexlab{b}})}\BibitemShut
{NoStop}%
\bibitem [{\citenamefont {Gritsenko}, \citenamefont {van Meer},\ and\
\citenamefont {Pernal}(2018)}]{GritMeePer-PRA-18}%
\BibitemOpen

69
Manuscript/srDFT_SC.tex
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@ -77,7 +77,7 @@
\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\wf{}{\Bas})\right)}
\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\wf{}{\Bas})\right)}
\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)}
\newcommand{\ecmd}[0]{\varepsilon^{\text{c,md}}_{\text{PBE}}}
\newcommand{\ecmd}[0]{\bar{\varepsilon}_{\text{c,md}}^{\text{sr},\text{PBE}}}
\newcommand{\psibasis}[0]{\Psi^{\basis}}
\newcommand{\BasFC}{\mathcal{A}}
@ -109,12 +109,12 @@
%%%%%% arguments
\newcommand{\argepbe}[0]{\den,\zeta,s}
\newcommand{\argebasis}[0]{\den,\zeta,s,\ntwo,\mu_{\Psi^{\basis}}}
\newcommand{\argebasis}[0]{\den,\zeta,\ntwo,\mu}
\newcommand{\argecmd}[0]{\den,\zeta,s,\ntwo,\mu}
\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s(\br{}),\ntwo(\br{}),\mu(\br{})}
\newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
@ -244,6 +244,7 @@
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
\newcommand{\f}[2]{f_{#1}^{#2}}
\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
\newcommand{\isEquivTo}[1]{\underset{#1}{\sim}}
% coordinates
\newcommand{\br}[1]{{\mathbf{r}_{#1}}}
@ -434,9 +435,7 @@ Because of the very definition of $\wbasis$, one has the following property in t
which is again fundamental to guarantee the correct behavior of the theory in the CBS limit.
\subsubsection{Frozen-core approximation}
As all WFT calculations for the purpose of that work are performed within the frozen core approximation, we use the valence-only versions of the various quantities needed for the complementary basis set functional introduced in Ref. \cite{LooPraSceTouGin-JCPL-19}.
We split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ and $\BasFC$ are the sets of core and active MOs, respectively)
and define the valence only range separation parameter
As all WFT calculations in this work are performed within the frozen-core approximation, we use the valence-only version of the various quantities needed for the complementary basis functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and active orbitals, respectively, and define the valence-only local range-separation parameter as
\begin{equation}
\label{eq:def_mur_val}
\murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{},
@ -446,69 +445,67 @@ where $\wbasisval$ is the valence-only effective interaction defined as
\label{eq:wbasis_val}
\wbasisval =
\begin{cases}
\fbasisval /\twodmrdiagpsi, & \text{if $\twodmrdiagpsival \ne 0$,}
\fbasisval /\twodmrdiagpsival, & \text{if $\twodmrdiagpsival \ne 0$,}
\\
\infty, & \text{otherwise,}
\end{cases}
\end{equation}
where $\fbasisval$ is defined as
where $\fbasisval$ and $\twodmrdiagpsival$ are defined as
\begin{equation}
\label{eq:fbasis_val}
\fbasisval
= \sum_{pq\in \Bas} \sum_{rstu \in \BasFC} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation}
and $\twodmrdiagpsival$
and
\begin{equation}
\label{eq:twordm_val}
\twodmrdiagpsival = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\twodmrdiagpsival = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}.
\end{equation}
Notice the summations on the active set of orbitals in equations \eqref{eq:fbasis_val} and \eqref{eq:twordm_val}.
It is noteworthy that, within the present definition, $\wbasisval$ still tends to the regular Coulomb interaction as $\Bas \to \CBS$.
Notice the restrictions of the sums to the set of active orbitals in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}.
It is noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$.
\subsection{Generic form and properties of the approximations for $\efuncden{\denr}$ }
\subsection{Generic form and properties of the approximations for $\efuncden{\den}$ }
\label{sec:functional}
\subsubsection{Generic form of the approximated functionals}
\subsubsection{Generic form of the approximate functionals}
\label{sec:functional_form}
As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis set functional $\efuncden{\denr}$ by using the so-called multi-determinant correlation functional (ECMD) introduced by Toulouse and co-workers\cite{TouGorSav-TCA-05}.
Following the recent work of some of the present authors\cite{LooPraSceTouGin-JCPL-19}, we propose to use a PBE-like functional which uses the total density $\denr$, spin polarisation $\zeta(\br{})$, reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$ and the on-top pair density $\ntwo(\br{})$. In the present work, all the density-related quantities are computed with the same wave function $\psibasis$ used to define $\murpsi$.
Therefore, a given approximation X of $\efuncden{\denr}$ have the following generic form
As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis functional $\efuncden{\den}$ by using the so-called correlation energy functional with multideterminant reference (ECMD) introduced by Toulouse \textit{et al.}~\cite{TouGorSav-TCA-05}. Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to use a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the total density $\denr$, the spin polarisation $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$, the reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$, and the on-top pair density $\ntwo(\br{})\equiv \ntwo(\br{},\br{})$. In the present work, all these quantities are computed with the same wave function $\psibasis$ used to define $\mur \equiv\murpsi$. Therefore, a given approximation X of $\efuncden{\den}$ will have the following generic local form
\begin{equation}
\begin{aligned}
\label{eq:def_ecmdpbebasis}
\efuncdenpbe{\argebasis} = &\int d\br{} \,\denr \\ & \ecmd(\argrebasis)
&\efuncdenpbe{\argebasis} = \;\;\;\;\;\;\;\; \\ &\int \d\br{} \,\denr \ecmd(\argrebasis),
\end{aligned}
\end{equation}
where $\ecmd(\argecmd)$ is the ECMD correlation energy density defined as
where $\ecmd(\argecmd)$ is the correlation energy per particle taken as
\begin{equation}
\label{eq:def_ecmdpbe}
\ecmd(\argecmd) = \frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{1+ \mu^3 \beta(\argepbe)}
\ecmd(\argecmd) = \frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{1+ \beta(\argepbe) \; \mu^3},
\end{equation}
with
\begin{equation}
\label{eq:def_beta}
\beta(\argebasis) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{\ntwo/\den},
\beta(\argepbe) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{\ntwo/\den},
\end{equation}
and where $\varepsilon_{\text{c,PBE}}(\argepbe)$ is the usual PBE correlation energy density\cite{PerBurErn-PRL-96}. Before introducing the different flavour of approximated functionals that we will use here (see \ref{sec:def_func}), we would like to give some motivations for the such a choice of functional form.
where $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ is the usual PBE correlation energy per particle~\cite{PerBurErn-PRL-96}. Before introducing the different flavors of approximate functionals that we will use here (see Section~\ref{sec:def_func}), we would like to give some motivations for this choice of functional form.
The actual functional form of $\ecmd(\argecmd)$ have been originally proposed by some of the present authors in the context of RSDFT~\cite{FerGinTou-JCP-18} in order to fulfill the two following limits
The functional form of $\ecmd(\argecmd)$ in Eq.~\ref{eq:def_ecmdpbe} has been originally proposed in Ref.~\onlinecite{FerGinTou-JCP-18} in the context of RSDFT. In the $\mu\to 0$ limit, it reduces to the usual PBE correlation functional
\begin{equation}
\lim_{\mu \rightarrow 0} \ecmd(\argecmd) = \varepsilon_{\text{c,PBE}}(\argepbe),
\lim_{\mu \rightarrow 0} \ecmd(\argecmd) = \varepsilon_{\text{c}}^{\text{PBE}}(\argepbe),
\end{equation}
which can be qualified as the weak correlation regime, and the large $\mu$ limit
which is relevant in the weak-correlation (or high-density) limit. In the large-$\mu$ limit, it behaves as
\begin{equation}
\label{eq:lim_mularge}
\ecmd(\argecmd) = \frac{1}{\mu^3} \ntwo + o(\frac{1}{\mu^5}),
\ecmd(\argecmd) \isEquivTo{\mu\to\infty} \frac{2\sqrt{\pi}(1 - \sqrt{2})}{3 \mu^3} \frac{\ntwo}{n},
\end{equation}
which, as it was previously shown\cite{TouColSav-PRA-04, GoriSav-PRA-06,PazMorGorBac-PRB-06} by various authors, is the exact expression for the ECMD in the limit of large $\mu$, provided that $\ntwo$ is the \textit{exact} on-top pair density of the system.
In the context of RSDFT, some of the present authors have illustrated in Ref.~\onlinecite{FerGinTou-JCP-18} that the on-top pair density involved in eq. \eqref{eq:def_ecmdpbe} plays a crucial role when reaching the strong correlation regime. The importance of the on-top pair density in the strong correlation regime have been also acknowledged by Pernal and co-workers\cite{GritMeePer-PRA-18} and Gagliardi and co-workers\cite{CarTruGag-JPCA-17}.
Also, $\ecmd(\argecmd) $ vanishes when $\ntwo$ vanishes
which is the exact large-$\mu$ behavior of the exact ECMD correlation energy~\cite{PazMorGorBac-PRB-06,FerGinTou-JCP-18}. Of course, for a specific system, the large-$\mu$ behavior will be exact only if one uses for $n_2$ the \textit{exact} on-top pair density of this system. This large-$\mu$ limit in Eq.~\eqref{eq:lim_mularge} is relevant in the strong-correlation (or low-density) limit. In the context of RSDFT, some of the present authors have illustrated in Ref.~\onlinecite{FerGinTou-JCP-18} that the on-top pair density involved in Eq. \eqref{eq:def_ecmdpbe} plays indeed a crucial role when reaching the strong-correlation regime. The importance of the on-top pair density in the strong-correlation regime have been also recently acknowledged by Gagliardi and coworkers~\cite{CarTruGag-JPCA-17} and Pernal and coworkers\cite{GritMeePer-PRA-18}.
Note also that $\ecmd(\argecmd)$ vanishes when $\ntwo$ vanishes
\begin{equation}
\label{eq:lim_n2}
\lim_{\ntwo \rightarrow 0} \ecmd(\argecmd) = 0
\lim_{\ntwo \rightarrow 0} \ecmd(\argecmd) = 0,
\end{equation}
which is exact for systems with a vanishing on-top pair density, such as the totally dissociated H$_2$ which is the archetype of strongly correlated systems.
Also, the function $\ecmd(\argecmd)$ vanishes when $\mu \rightarrow \infty$ as all RSDFT functionals
\begin{equation}
which is expected for systems with a vanishing on-top pair density, such as the totally dissociated H$_2$ molecule which is the archetype of strongly correlated systems. Finally, the function $\ecmd(\argecmd)$ vanishes when $\mu \rightarrow \infty$ like all RSDFT short-range functionals \begin{equation}
\label{eq:lim_muinf}
\lim_{\mu \rightarrow \infty} \ecmd(\argecmd) = 0.
\end{equation}
@ -535,7 +532,7 @@ Another important requirement is the independence of the energy with respect to
Such a property is also important in the context of covalent bond breaking where the ground state of the super system $A+B$ is in general of low spin while the ground states of the fragments $A$ and $B$ are in high spin which can have multiple $S_z$ components.
\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain $S_z$ invariance}
A sufficient condition to achieve $S_z$ invariance is to eliminate all dependency to $S_z$, which in the case of $\ecmd(\argecmd)$ is the spin polarisation $\zeta(\br{})$ involved in the correlation energy density $\varepsilon_{\text{c,PBE}}(\argepbe)$ (see equation \eqref{eq:def_ecmdpbe}).
A sufficient condition to achieve $S_z$ invariance is to eliminate all dependency to $S_z$, which in the case of $\ecmd(\argecmd)$ is the spin polarisation $\zeta(\br{})$ involved in the correlation energy density $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ (see equation \eqref{eq:def_ecmdpbe}).
As originally shown by Perdew and co-workers\cite{PerSavBur-PRA-95}, the dependence on the spin polarisation in the KS-DFT framework can be removed by the rewriting the spin polarisation of a single Slater determinant with only the on-top pair density and the total density. In other terms, the spin density dependence usually introduced in the correlation functionals of KS-DFT tries to mimic the effect of the on-top pair density.
Based on this reasoning, a similar approach has been used in the context of multi configurational DFT in order to remove the $S_z$ dependency.
In practice, these approaches introduce the effective spin polarisation
@ -553,7 +550,7 @@ The advantages of this approach are at least two folds: i) the effective spin p
Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $n^2 - 4 \ntwo^{\psibasis}<0$ and also
the formula of equation \eqref{eq:def_effspin} is exact only when the density $n$ and on-top pair density $\ntwo^{\psibasis}$ are obtained from a single determinant\cite{PerSavBur-PRA-95}, but it is applied to multi configurational wave functions.
An alternative to eliminate the $S_z$ dependency would be to simply set $\zeta(\br{})=0$, but this would lower the accuracy of the usual correlation functional, such as the PBE correlation functional used here $\varepsilon_{\text{c,PBE}}(\argepbe)$. Nevertheless, as the spin polarisation usually tries to mimic the on-top pair density and the function $\ecmd(\argecmd)$ explicitly depends on the on-top pair density (see equations \eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}), we propose here to use the $\ecmd$ functional with \textit{a zero spin polarisation}. This ensures a $S_z$ invariance and, as will be numerically shown, very weakly affect the accuracy of the functional.
An alternative to eliminate the $S_z$ dependency would be to simply set $\zeta(\br{})=0$, but this would lower the accuracy of the usual correlation functional, such as the PBE correlation functional used here $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, as the spin polarisation usually tries to mimic the on-top pair density and the function $\ecmd(\argecmd)$ explicitly depends on the on-top pair density (see equations \eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}), we propose here to use the $\ecmd$ functional with \textit{a zero spin polarisation}. This ensures a $S_z$ invariance and, as will be numerically shown, very weakly affect the accuracy of the functional.
\subsubsection{Conditions on $\psibasis$ for the extensivity}
In the case of the present basis set correction, as $\efuncdenpbe{\argebasis}$ is an integral over $\mathbb{R}^3$ of local quantities, in the case of non overlapping fragments $A\ldots B$ it can be written as the sum of two local contributions: one coming from the integration over the region of the sub-system $A$ and the other one from the region of the sub-system $B$.
@ -569,7 +566,7 @@ The condition for the active space involved in the CASSCF wave function is that
As the present work focusses on the strong correlation regime, we propose here to investigate only approximated functionals which are $S_z$ invariant and size extensive in the case of covalent bond breaking. Therefore, the wave function $\psibasis$ used throughout this paper are of CASSCF type in order to ensure extensivity of all density related quantities.
The difference between the different flavours of functionals are only on i) the type of on-top pair density used, and ii) the type of spin polarisation used.
Regarding the spin polarisation that enters into $\varepsilon_{\text{c,PBE}}(\argepbe)$, two different types of $S_z$ invariant formulations are used: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in equation \eqref{eq:def_effspin}, and iii) a \textit{zero} spin polarization.
Regarding the spin polarisation that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$ invariant formulations are used: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in equation \eqref{eq:def_effspin}, and iii) a \textit{zero} spin polarization.
Regarding the approximation to the \textit{exact} on-top pair density entering in equation \eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads
\begin{equation}