Manu: done (so far) with the computational details

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Emmanuel Fromager 2020-04-23 15:38:41 +02:00
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commit 6f82d5aa02

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@ -342,11 +342,21 @@ equals the exact ensemble one
n^{\bw}(\br)=\sum_{I=0}^{\nEns-1} n^{\bw}(\br)=\sum_{I=0}^{\nEns-1}
\ew{I}n_{\Psi_I}(\br). \ew{I}n_{\Psi_I}(\br).
\eeq \eeq
The minimizing density-functional KS In practice, the minimizing KS density matrix operator
wavefunctions $\left\{\Det{I}{\bw}\left[n^{\bw}\right]\right\}_{0\leq $\hgam{\bw}\left[\n{}{\bw}\right]$
I\leq M-1}$ are constructed from (weight-dependent) KS orbitals can be determined from the following KS reformulation of the
GOK variational principle, \cite{Gross_1988b,Senjean_2015}
\beq\label{eq:min_KS_DM}
\E{}{\bw} = \min_{\hGam{\bw}} \left\{\Tr[\hGam{\bw}
\left(\hT+\hVne\right)]+\E{\Ha}{}[\n{\hGam{\bw}}{}]+\E{\xc}{\bw}[\n{\hGam{\bw}}{}]\right\},
\eeq
where $\n{\hGam{\bw}}{}(\br)=\sum_{I=0}^{\nEns - 1}
\ew{I}\n{\overline{\Psi}^{(I)}}{}$ is the trial ensemble density. As a
result, the orbitals
$\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le $\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le
\nOrb}$. The latters are determined by solving the ensemble KS equation \nOrb}$ from which the KS
wavefunctions $\left\{\Det{I}{\bw}\left[n^{\bw}\right]\right\}_{0\leq
I\leq M-1}$ are constructed can be obtained by solving the following ensemble KS equation
\begin{equation} \begin{equation}
\label{eq:eKS} \label{eq:eKS}
\qty{ \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{\bw}]}{\n{}{}(\br{})}} \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}), \qty{ \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{\bw}]}{\n{}{}(\br{})}} \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
@ -491,8 +501,19 @@ This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\
Moreover, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$ with weight $1 - \ew{}$) and the first doubly-excited state ($I=1$ with weight $\ew{}$) are considered. Moreover, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$ with weight $1 - \ew{}$) and the first doubly-excited state ($I=1$ with weight $\ew{}$) are considered.
Although one should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle, we will sometimes ``violate'' this variational constraint. Although one should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle, we will sometimes ``violate'' this variational constraint.
Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a genuine saddle point of the 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} Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a genuine saddle point of the 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}
\manu{Maybe we should be more clear about what we mean with $\ew{} = 1$. \manu{I think we need to be a bit more clear about the $\ew{} = 1$ case, as
In the range $1/2\leq \ew{}\leq 1$, } it stands a little bit beyond the theory discussed previously. What you
are looking at in the range $1/2\leq \ew{}\leq 1$ are, indeed, other
stationary points (than the minimizing ones) of the density matrix
operator functional in Eq.~(\ref{eq:min_KS_DM}). I would say that we
look at these solutions for analysis purposes. I personally never looked
(formally) at these solutions and their physical meaning. One should clearly
mention that applying GOK-DFT in this range of weights would simply
consists in switching ground and first excited states if a true
minimization of the ensemble energy were performed. From this point of
view we do not violate anything.
}
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\section{Results} \section{Results}
\label{sec:res} \label{sec:res}