Manu: III

Manuscript/FarDFT.bib

@@ -4013,6 +4013,15 @@
 	Year = {2019},
 	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.123.016401}}
 
+@article{Gould_2019_insights,
+title={Density driven correlations in ensemble density functional theory: insights from simple excitations in atoms},
+    author={Tim Gould and Stefano Pittalis},
+    year={2020},
+    eprint={2001.09429},
+    archivePrefix={arXiv},
+    primaryClass={cond-mat.str-el}
+}
+
 @article{Gould_2013,
 	Author = {Gould, Tim and Dobson, John F.},
 	Date-Added = {2018-10-24 22:38:52 +0200},

Manuscript/FarDFT.tex

@@ -493,16 +493,42 @@ Numerical quadratures are performed with the \texttt{numgrid} library \cite{numg
 This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
 Moreover, we restrict our study to the case of a three-state ensemble (\ie, $\nEns = 3$) where the ground state ($I=0$ with weight $1 - \ew{1} - \ew{2}$), a singly-excited state ($I=1$ with weight $\ew{1}$), as well as the lowest doubly-excited state ($I=2$ with weight $\ew{2}$) are considered.
 Assuming that the singly-excited state is lower in energy than the doubly-excited state, one should have $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1 - \ew{2})/2$ to ensure the GOK variational principle.
-If the doubly-excited state is lower in energy than the singly-excited state (which can be the case as one would notice later), then one has to swap $\ew{1}$ and $\ew{2}$ in the above inequalities.
+If the doubly-excited state \manu{(whose weight is denoted $\ew{2}$
+throughout this work)} is lower in energy than the singly-excited state
+\manu{(with weight $\ew{1}$)}, which can be the case as one would notice
+later, then one has to swap $\ew{1}$ and $\ew{2}$ in the above
+inequalities. \manu{Note also that additional lower-in-energy single
+excitations may have to be included into the ensemble
+ before
+incorporating the double excitation of interest. In the present
+exploratory work, we will simply exclude them from the ensemble and
+leave the more consistent (from a GOK point of view) description of all low-lying excitations to
+future work.}
 Unless otherwise stated, we set the same weight to the two excited states (\ie, $\ew{} \equiv \ew{1} = \ew{2}$).
 In this case, the ensemble energy will be written as a single-weight quantity, $\E{}{\ew{}}$.
 The zero-weight limit (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 0$), and the equiweight ensemble (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 1/3$) are considered in the following.
 (Note that the zero-weight limit corresponds to a conventional ground-state KS calculation.)
 
-Let us mention now that we will sometimes ``violate'' the GOK variational principle in order to build our weight-dependent functionals by considering the extended range of weights $0 \le \ew{2} \le 1$.
+Let us \manu{finally} mention that we will sometimes ``violate'' the GOK
-However, let us stress that we will not compute excitation energies with these ensembles inconsistent with GOK theory.
+variational principle in order to build our weight-dependent functionals
-The pure-state limit, $\ew{1} = 0 \land \ew{2} = 1$, is nonetheless of particular interest as it is, like the (ground-state) zero-weight limit, a genuine saddle point of the restricted KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
+by considering the extended range of weights $0 \le \ew{2} \le 1$.
+The pure-state limit, $\ew{1} = 0 \land \ew{2} = 1$, is
+\trashEF{nonetheless} of particular interest as it is, like the
+(ground-state) zero-weight limit, a genuine saddle point of the
+restricted KS equations [see Eqs. (\ref{eq:min_KS_DM}) and
+(\ref{eq:eKS})], and \manu{it matches} perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
+\manu{From a GOK-DFT perspective, considering a (stationary) pure-excited-state limit 
+can be seen as a way to construct density-functional approximations to
+individual exchange and state-driven correlation within an ensemble.
+\cite{Gould_2019,Gould_2019_insights,Fromager_2020}} 
+However, \trashEF{let us stress that we will not compute excitation
+energies with these ensembles inconsistent with GOK theory}
+\manu{when it comes to compute excitation 
+energies, we will exclusively consider ensembles where the largest
+weight is assigned to the ground state.}\\
 
+\manuf{We do not completely
+follow GOK when we ignore lower-lying singles...}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results and Discussion}
@@ -1013,7 +1039,7 @@ Although the weight-dependent correlation functional developed in this paper (eV
 To better understand the reasons behind this, it would be particularly interesting to investigate the influence of the self-consistent procedure, 
 \ie, the variation in excitation energy when the \textit{exact} ensemble density (built with the exact individual densities) is used instead
 of the self-consistent one. 
-Density- and state-driven errors \cite{Gould_2019,Fromager_2020} can also be calculated to provide additional insights about the present results. 
+Density- and state-driven errors \cite{Gould_2019,Gould_2019_insights,Fromager_2020} can also be calculated to provide additional insights about the present results. 
 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.