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eginer 2019-12-25 12:26:54 +01:00
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Manuscript/srDFT_SC.tex
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@ -537,13 +537,6 @@ First, thanks to the properties in Eqs.~\eqref{eq:cbs_mu} and~\eqref{eq:lim_muin
\label{eq:lim_ebasis}
\lim_{\basis \rightarrow \text{CBS}} \efuncdenpbe{\argebasis} = 0, \quad \forall\, \psibasis,
\end{equation}
%<<<<<<< HEAD
%which guarantees an unaltered limit when reaching the CBS limit.
%Also, the $\efuncdenpbe{\argecmd}$ vanishes for systems with vanishing on-top pair density, which guarantees the good limit in the case of stretched H$_2$ and for one-electron system.
%Such a property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ (see eq. \eqref{eq:wbasis}) together with the condition \eqref{eq:lim_muinf}, ii) the fact that the $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes (see eq. \eqref{eq:lim_n2}).
%=======
%which guarantees an unaltered CBS limit.
%>>>>>>> e7e87f50643a28a43e403bc17effeaa48bb01e35
Second, the fact that $\efuncdenpbe{\argebasis}$ vanishes for systems with vanishing on-top pair density guarantees the correct limit for one-electron systems and for the stretched H$_2$ molecule. This property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ [see Eq.~\eqref{eq:wbasis}] together with the condition in Eq.~\eqref{eq:lim_muinf}, ii) the fact that $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes [see Eq.~\eqref{eq:lim_n2}].
@ -587,13 +580,9 @@ Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb
As the present work focuses on the strong-correlation regime, we propose here to investigate only approximate functionals which are $S_z$ independent and size-consistent in the case of covalent bond breaking. Therefore, the wave functions $\psibasis$ used throughout this paper are CASSCF wave functions in order to ensure size consistency of all local quantities. The difference between two flavors of functionals are only due to the type of i) spin polarization, and ii) on-top pair density.
<<<<<<< HEAD
Regarding the approximation to the \textit{exact} on-top pair density entering in eq. \eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads
=======
Regarding the spin polarization that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$-independent formulations are considered: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in Eq.~\eqref{eq:def_effspin} and calculated from the CASSCF wave function, and ii) a \textit{zero} spin polarization.
Regarding the on-top pair density entering in Eq.~\eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads
>>>>>>> 7d17cec3065dbb2c36d90f24f286b41985ef6529
\begin{equation}
\label{eq:def_n2ueg}
\ntwo^{\text{UEG}}(n,\zeta) \approx n^2\big(1-\zeta^2\big)g_0(n),