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revised computational details

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Pierre-Francois Loos 2020-04-24 23:09:15 +02:00
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Manuscript/FarDFT.tex
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@ -471,20 +471,13 @@ For all calculations, we use the aug-cc-pVXZ (X = D, T, Q, and 5) Dunning family
Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988b,Lindh_2001}
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 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.
\titou{To ensure the GOK variational principle, one should then have $0 \le \ew{} \le 1/2$.
However, 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}
\manu{I think we need to be a bit more clear about the $\ew{} = 1$ case, as
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.~\eqref{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.
}
Although the range $1/2 < \ew{} \leq 1$ stands a little bit beyond the theory discussed previously, we look at these solutions for analysis purposes mainly.
These solutions of the density matrix operator functional in Eq.~\eqref{eq:min_KS_DM} correspond to stationary points rather than minimising ones.
Applying GOK-DFT in this range of weights would simply consists in switching the ground and excited states if true minimisations of the ensemble energy were performed.}
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\section{Results}