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Manuscript/srDFT_SC.tex
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@ -497,9 +497,8 @@ Such a property is also important in the context of covalent bond breaking where
\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain $S_z$ invariance} \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 $\varepsilon_{\text{c,PBE}}(\argepbe)$ (see equation \eqref{eq:def_ecmdpbe}).
Of course, the dependency on the spin polarisation of the functionals can be thought as an aterfact 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 reformulating the spin polarisation of a single Slater determinant with the on-top pair density and the total density. In other terms, the spin density dependence of the correlation functionals usually try to mimic the on-top pair density of a given system.
A possible way to eliminate the $S_z$ dependency would be to simply set $\zeta(\br{})=0$, but this would lower the accuracy of the usual PBE correlation functional $\varepsilon_{\text{c,PBE}}(\argepbe)$. Based on this reasoning, a similar approach has been used in the context of multi configurational DFT approaches where an effective spin density is introduced
Therefore, we use the effective spin polarisation introduced by Scuseria and co-workers\cite{GarBulHenScu-PCCP-15} which depends on the on-top pair density and the total density of a general multi configurational wave function $\psibasis$:
\begin{equation} \begin{equation}
\label{eq:def_effspin} \label{eq:def_effspin}
\tilde{\zeta}(n,\ntwo_{\psibasis}) = \tilde{\zeta}(n,\ntwo_{\psibasis}) =
@ -508,6 +507,8 @@ Therefore, we use the effective spin polarisation introduced by Scuseria and co-
0 & \text{otherwise.} 0 & \text{otherwise.}
\end{cases} \end{cases}
\end{equation} \end{equation}
A possible way to eliminate the $S_z$ dependency would be to simply set $\zeta(\br{})=0$, but this would lower the accuracy of the usual PBE correlation functional $\varepsilon_{\text{c,PBE}}(\argepbe)$.
Therefore, we use the effective spin polarisation introduced by Scuseria and co-workers\cite{GarBulHenScu-PCCP-15} which depends on the on-top pair density and the total density of a general multi configurational wave function $\psibasis$:
If the density $n$ and on-top pair density $\ntwo_{\psibasis}$ are obtained from a single HF determinant, the definition \eqref{eq:def_effspin} is equivalent to the usual one If the density $n$ and on-top pair density $\ntwo_{\psibasis}$ are obtained from a single HF determinant, the definition \eqref{eq:def_effspin} is equivalent to the usual one
\begin{equation} \begin{equation}
\tilde{\zeta}(n^{\text{HF}},\ntwohf) = n_{\alpha}^{\text{HF}} - n_{\beta}^{\text{HF}}, \tilde{\zeta}(n^{\text{HF}},\ntwohf) = n_{\alpha}^{\text{HF}} - n_{\beta}^{\text{HF}},