From 1079226d090f52750197dfab529242946aae6480 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Thu, 4 Jun 2020 17:10:10 +0200 Subject: [PATCH] minor modification --- Manuscript/FarDFT.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 7fbcae7..ac03a93 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -572,8 +572,8 @@ Eqs.~\eqref{eq:exp_ens_ener}, \eqref{eq:diff_Ew}, and \eqref{eq:dEdw}]: We enforce this type of \textit{exact} constraint (to the maximum possible extent) when optimising the parameters in Eq.~\eqref{eq:Cxw} in order to minimise the curvature of the ensemble energy. As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cxw} (red curve), the weight-dependent correction does not affect the two ghost-interaction-free limits at $\ew{1} = \ew{2} = 0$ and $\ew{1} = 0 \land \ew{2} = 1$ (\ie, the pure-state limits), as $\Cx{\ew{2}}$ reduces to $\Cx{}$ in these two limits. Indeed, it is important to ensure that the weight-dependent functional does not alter these pure-state limits, which are genuine saddle points of the KS equations, as mentioned above. -Finally, let us mention that, around $\ew{2} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear: this is the main feature that one needs to catch in order to get accurate excitation energies in the zero-weight limit which is ghost-interaction free. -\titou{Nonetheless, beyond the $\ew{2} = 0$ limit, the CC-S functional also includes quadratic terms in order to compensate the spurious curvature of the ensemble energy originating, mainly, from the Hartree term [see Eq.~\eqref{eq:Hartree}].} +\titou{Finally, let us mention that, around $\ew{2} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear: this is the main feature that one needs to catch in order to get accurate excitation energies in the zero-weight limit which is ghost-interaction free. +Nonetheless, beyond the $\ew{2} = 0$ limit, the CC-S functional also includes quadratic terms in order to compensate the spurious curvature of the ensemble energy originating, mainly, from the Hartree term [see Eq.~\eqref{eq:Hartree}].} %We shall come back to this point later on. %%% FIG 3 %%%