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Manu: some polishing in II B and C

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Emmanuel Fromager 2020-02-28 16:36:51 +01:00
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1 changed files with 14 additions and 15 deletions
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Manuscript/eDFT.tex
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@ -450,8 +450,15 @@ Eq.~(\ref{eq:excited_ener_level_gs_lim})].
%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%
For implementation purposes, we will use in the rest of this work For implementation purposes, we will use in the rest of this work
(one-electron reduced) density matrices (one-electron reduced) density matrices
as basic variables, rather than Slater determinants. If we expand the as basic variables, rather than Slater determinants.
ensemble KS (spin) orbitals [from which the latter determinants are constructed] in an atomic orbital (AO) basis, As the theory is applied later on to {\it spin-polarized}
systems, we drop spin indices in the density matrices, for convenience.
If we expand the
ensemble KS orbitals [from which the determinants are constructed] in an atomic orbital (AO) basis,
\beq
\MO{p}{}(\br{}) = \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
\eeq
\iffalse%%%%%%%%%%%%%%%%%%%%%%%%
\titou{\beq \titou{\beq
\SO{p}{}(\bx{}) = s(\omega) \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}), \SO{p}{}(\bx{}) = s(\omega) \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
\eeq \eeq
@ -465,16 +472,14 @@ where $\bx{}=(\omega,\br{})$ is a composite coordinate gathering spin and spatia
\beta(\omega), & \text{for spin-down electrons,} \beta(\omega), & \text{for spin-down electrons,}
\end{cases} \end{cases}
\eeq \eeq
}then the density matrix of the }
\fi%%%%%%%%%%%%%%%%%%%%%
then the density matrix of the
determinant $\Det{(K)}$ can be expressed as follows in the AO basis: determinant $\Det{(K)}$ can be expressed as follows in the AO basis:
\beq \beq
\bGam{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{}, \bGam{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{},
\eeq \eeq
where the summation runs over the spinorbitals that are occupied in $\Det{(K)}$. where the summation runs over the orbitals that are occupied in $\Det{(K)}$.
\trashPFL{Note that, as the theory is applied later on to spin-polarized
systems, we drop spin indices in the density matrices, for convenience.}
\manu{Is the latter sentence ok with you?}
\titou{I don't think we need it anymore. What do you think?}
The electron density of the $K$th KS determinant can then be evaluated The electron density of the $K$th KS determinant can then be evaluated
as follows: as follows:
\beq \beq
@ -501,7 +506,7 @@ p}}c^\sigma_{{\nu p}}
\fi%%% \fi%%%
%%%% end Manu %%%% end Manu
while the ensemble density matrix while the ensemble density matrix
and ensemble density read and the ensemble density read
\beq \beq
\bGam{\bw} \bGam{\bw}
= \sum_{K\geq 0} \ew{K} \bGam{(K)} = \sum_{K\geq 0} \ew{K} \bGam{(K)}
@ -880,12 +885,6 @@ dependencies explicitly, thus allowing for the description of derivative
discontinuities [see Eq.~\eqref{eq:excited_ener_level_gs_lim} and the discontinuities [see Eq.~\eqref{eq:excited_ener_level_gs_lim} and the
comment that follows] {\it via} the last term on the right-hand side comment that follows] {\it via} the last term on the right-hand side
of Eq.~\eqref{eq:EI-eLDA}.\\ of Eq.~\eqref{eq:EI-eLDA}.\\
\titou{In order to test the influence of the derivative discontinuity on the excitation energies, it is useful to perform ensemble HF (labeled as eHF) calculations in which the correlation effects are removed.
In this case, the individual energies are simply defined as
\beq\label{eq:EI-eHF}
\E{eHF}{(I)} \approx \Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}].
\eeq
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Density-functional approximations for ensembles} \section{Density-functional approximations for ensembles}