typo

Manuscript/srDFT_SC.tex

@@ -549,7 +549,7 @@ Therefore, following other authors, \cite{MieStoSav-MP-97,LimCarLuoMaOlsTruGag-J
  \label{eq:def_effspin-0}
 	\tilde{\zeta}(n,n_{2}) =  
    \begin{cases}
-		\sqrt{ 1 - 2 \; n_{2}/n^2 },	&	\text{if } n^2 \ge 4 n_{2},
+		\sqrt{ 1 - 2 \; n_{2}/n^2 },	&	\text{if } n^2 \ge 2 n_{2},
 		\\
 		0,								& \text{otherwise.}
    \end{cases}
@@ -557,7 +557,6 @@ Therefore, following other authors, \cite{MieStoSav-MP-97,LimCarLuoMaOlsTruGag-J
 
 An alternative way to eliminate the $S_z$ dependency is to simply set $\zeta=0$, \ie, to resort to the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, \ie, for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for sufficiently large $\mu$, it is a viable option. Indeed, the purpose of introducing the spin polarization in semilocal density-functional approximations is to mimic the exact on-top pair density, \cite{PerSavBur-PRA-95} but our functional $\ecmd(\argecmd)$ already explicitly depends on the on-top pair density [see Eqs.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}]. The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Consequently, we propose here to test the $\ecmd$ functional with \textit{a zero spin polarization}. This ensures its $S_z$ invariance and, as will be numerically demonstrated, very weakly affects the complementary density functional accuracy.
 
-
 \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, $\Psi_{\ce{A\bond{...}B}}^{\basis} = \Psi_{\ce{A}}^{\basis} \Psi_{\ce{B}}^{\basis}$.