From 36792819b650b69cc3b7bae6ef7fac0f529fcf02 Mon Sep 17 00:00:00 2001
From: bsenjean <bsenjean@gmail.com>
Date: Tue, 14 Apr 2020 19:45:20 +0200
Subject: [PATCH] Added the results for mu = 0.4 and comments in the
 conclusion.

---
 Manuscript/FarDFT.bib | 13 +++++++++++++
 Manuscript/FarDFT.tex | 22 +++++++++++++++++++---
 2 files changed, 32 insertions(+), 3 deletions(-)

diff --git a/Manuscript/FarDFT.bib b/Manuscript/FarDFT.bib
index 0dc9d51..0b84b4a 100644
--- a/Manuscript/FarDFT.bib
+++ b/Manuscript/FarDFT.bib
@@ -2146,6 +2146,19 @@
 	Volume = {143},
 	Year = {2015},
 	Bdsk-Url-1 = {https://doi.org/10.1063/1.4932595}}
+	
+@article{Toulouse_2004,
+  title={Short-range exchange-correlation energy of a uniform electron gas with modified electron--electron interaction},
+  author={Toulouse, Julien and Savin, Andreas and Flad, Heinz-J{\"u}rgen},
+  journal={Int. J. Quantum Chem.},
+  volume={100},
+  number={6},
+  pages={1047--1056},
+  year={2004},
+  publisher={Wiley Online Library},
+  url={https://doi.org/10.1002/qua.20259}
+}
+
 
 @article{Blunt_2017,
 	Author = {Blunt, N. S. and Neuscamman, Eric},
diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex
index 7e94a81..56ff262 100644
--- a/Manuscript/FarDFT.tex
+++ b/Manuscript/FarDFT.tex
@@ -648,7 +648,12 @@ For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the exci
 As expected from the linearity of the ensemble energy, the GIC-S functional coupled or not with a correlation functional yield extremely stable excitation energies as a function of the weight, with only a few tenths of eV difference between the zero- and equi-weights limits.
 Nonetheless, the excitation energy is still off by $3$ eV.
 The fundamental theoretical reason of such a poor agreement is not clear.
-The fact that HF exchange yields better excitation energies hints at the effect of self-interaction error.
+The fact that HF exchange yields better excitation energies hints at the effect of self-interaction error. For additional comparison, we provide the
+excitation energy
+calculated by short-range multiconfigurational DFT in Ref.~\cite{Senjean_2015}, using the (weight-independent) srLDA functional~\cite{Toulouse_2004}
+and setting the range-separation parameter to $\mu = 0.4$ bohr$^{-1}$. 
+The excitation energy improves by 1 eV compared
+to the weight-independent S-VWN5 functional, thus showing that treating the long-range part of the electron-electron repulsion by wavefunction theory plays a significant role.
 %\bruno{I'm a bit surprise that the ensemble correction to the correlation functional does not improve things at all... Is the derivative discontinuity, computed with this functional, almost 0 here ?}
 
 %%% TABLE IV %%%
@@ -677,11 +682,14 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
 	B3			&	LYP				&					&					&			&	5.55	\\
 	HF			&	LYP				&					&					&			&	6.68	\\
 	\hline
-	\mc{5}{l}{Accurate\fnm[2]}															&	8.69	\\
+	\mc{2}{l}{srLDA($\mu = 0.4$) \fnm[2]} & 6,39			&			6,55		&		6,47	&	\\
+	\hline
+	\mc{5}{l}{Accurate\fnm[3]}															&	8.69	\\
 \end{tabular}
 \end{ruledtabular}
 \fnt[1]{KS calculation does not converge.}
-\fnt[2]{FCI/aug-cc-pV5Z calculation performed with QUANTUM PACKAGE. \cite{QP2}}
+\fnt[2]{short-range multi-configurational DFT / aug-cc-pVQZ calculations performed in Ref.~\cite{Senjean_2015}}
+\fnt[3]{FCI/aug-cc-pV5Z calculation performed with QUANTUM PACKAGE. \cite{QP2}}
 \end{table}
 %%% %%% %%% %%%
 
@@ -780,6 +788,14 @@ Excitation energies (in hartree) associated with the lowest double excitation of
 \section{Conclusion}
 \label{sec:ccl}
 
+Although the weight-dependent functionals developed in this paper perform systematically
+better than their
+weight-independent counterparts, the improvement remains small.
+To better understand the reasons of this small improvement,
+it will be particularly interesting to investigate
+the error due to the self-consistent procedure, 
+\ie, by checking the difference in the excitation energy when the {\it exact} ensemble density (built with the exact individual densities) is used instead
+of the converged one. Density-driven as well as state-driven errors~\cite{Gould_2019,Fromager_2020} can also be calculated to provide more flavours about the results obtained in this paper. This is left for future work.
 In the light of the results obtained in this study on double excitations computed within the GOK-DFT framework, we believe that the development of more universal weight-dependent exchange and correlation functionals has a bright future, and we hope to be able to report further on this in the near future.
 
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