From 08bc5043e642d2d34137d0bb23bc8708599fac21 Mon Sep 17 00:00:00 2001 From: Julien Toulouse Date: Mon, 9 Dec 2019 17:55:58 +0100 Subject: [PATCH] changes --- Manuscript/srDFT_SC.bbl | 18 +++++++++++++++++- Manuscript/srDFT_SC.tex | 15 +++++++-------- 2 files changed, 24 insertions(+), 9 deletions(-) diff --git a/Manuscript/srDFT_SC.bbl b/Manuscript/srDFT_SC.bbl index d351fe4..20d7cb3 100644 --- a/Manuscript/srDFT_SC.bbl +++ b/Manuscript/srDFT_SC.bbl @@ -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 diff --git a/Manuscript/srDFT_SC.tex b/Manuscript/srDFT_SC.tex index 1947e50..21a8379 100644 --- a/Manuscript/srDFT_SC.tex +++ b/Manuscript/srDFT_SC.tex @@ -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.