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18
Manuscript/srDFT_SC.bbl
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@ -6,7 +6,7 @@
%Control: page (0) single
%Control: year (1) truncated
%Control: production of eprint (0) enabled
\begin{thebibliography}{68}%
\begin{thebibliography}{70}%
\makeatletter
\providecommand \@ifxundefined [1]{%
\@ifx{#1\undefined}
@ -596,6 +596,14 @@
{\doibase 10.1103/PhysRevA.98.062510} {\bibfield {journal} {\bibinfo
{journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {98}},\ \bibinfo
{pages} {062510} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Moscard\'o}\ and\ \citenamefont
{San-Fabi\'an}(1991)}]{MosSan-PRA-91}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont
{Moscard\'o}}\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{San-Fabi\'an}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. A}\ }\textbf {\bibinfo {volume} {44}},\ \bibinfo {pages} {1549}
(\bibinfo {year} {1991})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Becke}, \citenamefont {Savin},\ and\ \citenamefont
{Stoll}(1995)}]{BecSavSto-TCA-95}%
\BibitemOpen
@ -605,6 +613,14 @@
{\bibfield {journal} {\bibinfo {journal} {Theoret. Chim. Acta}\ }\textbf
{\bibinfo {volume} {{91}}},\ \bibinfo {pages} {147} (\bibinfo {year}
{1995})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Savin}(1996)}]{Sav-INC-96a}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Savin}},\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {Recent Advances
in Density Functional Theory}}},\ \bibinfo {editor} {edited by\ \bibinfo
{editor} {\bibfnamefont {D.~P.}\ \bibnamefont {Chong}}}\ (\bibinfo
{publisher} {World Scientific},\ \bibinfo {year} {1996})\ pp.\ \bibinfo
{pages} {129--148}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Perdew}, \citenamefont {Savin},\ and\ \citenamefont
{Burke}(1995)}]{PerSavBur-PRA-95}%
\BibitemOpen

15
Manuscript/srDFT_SC.tex
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@ -535,22 +535,21 @@ Another important requirement is spin-multiplet degeneracy, i.e. the independenc
\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain spin-multiplet degeneracy}
A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$, which in the case of the functional $\ecmd(\argecmd)$ means removing the dependence on the spin polarization $\zeta(\br{})$ used in the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ (see Eq. \eqref{eq:def_ecmdpbe}).
As originally shown by Perdew and co-workers\cite{BecSavSto-TCA-95,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
A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependence on $S_z$, which in the case of the functional $\ecmd(\argecmd)$ means removing the dependency on the spin polarization $\zeta(\br{})$ used in the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ (see Eq. \eqref{eq:def_ecmdpbe}). It has been proposed to replace in functionals the dependence on the spin polarization by the dependence on the on-top pair density~\cite{MosSan-PRA-91,BecSavSto-TCA-95,Sav-INC-96a,Sav-INC-96a} (see, also, Refs.~\onlinecite{PerSavBur-PRA-95,StaDav-CPL-01}). Most often, the on-top pair density is used through an effective spin polarisation
%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
\begin{equation}
\label{eq:def_effspin}
\tilde{\zeta}(n,\ntwo^{\psibasis}) =
\tilde{\zeta}(n_{2}) =
% \begin{cases}
\sqrt{ n^2 - 4 \ntwo^{\psibasis} }
\sqrt{ 1 - 2 n_{2}/n^2 }
% 0 & \text{otherwise.}
% \end{cases}
\end{equation}
which uses the on-top pair density $\ntwo^{\psibasis}$ of a given wave function $\psibasis$.
which uses the on-top pair density $n_{2,\psibasis}$ of a given wave function $\psibasis$.
The advantages of this approach are at least two folds: i) the effective spin polarisation $\tilde{\zeta}$ is $S_z$ invariant, ii) it introduces an indirect dependency on the on-top pair density of the wave function $\psibasis$ which usually improves the treatment of strong correlation.
Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $n^2 - 4 \ntwo^{\psibasis}<0$ and also
Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $n^2 - 4 n_{2,\psibasis}<0$ 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 $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. 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.