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Manu: polished II B

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Emmanuel Fromager 2020-02-25 11:00:46 +01:00
parent 9e7fb1453f
commit d150248c25
1 changed files with 14 additions and 10 deletions
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Manuscript/eDFT.tex
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@ -15,7 +15,7 @@
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\manu}[1]{\textcolor{blue}{#1}}
\newcommand{\manu}[1]{\textcolor{blue}{Manu: #1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\trashEF}[1]{\textcolor{blue}{\sout{#1}}}
@ -392,19 +392,23 @@ Eq.~(\ref{eq:excited_ener_level_gs_lim})].
For implementation purposes, we will use in the rest of this work
(one-electron reduced) density matrices
as basic variables, rather than Slater determinants. If we expand the
ensemble KS spinorbitals [from which the latter are constructed] in an atomic orbital (AO) basis,
ensemble KS (spin) orbitals [from which the latter determinants are constructed] in an atomic orbital (AO) basis,
\beq
\SO{p}{}(\br{}) = \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
\eeq
then the density matrix elements obtained from the
then the density matrix of the
determinant $\Det{(K)}$ can be expressed as follows in the AO basis:
\beq
\bGam{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{},
\eeq
where the summation runs over the spinorbitals that are occupied in $\Det{(K)}$.
Note that the density of the $K$th KS state reads
where the summation runs over the spinorbitals that are occupied in
$\Det{(K)}$. 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?}
The electron density of the $K$th KS determinant can then be evaluated
as follows:
\beq
\n{\bGam{(K)}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{(K)} \AO{\nu}(\br{}).
\n{\bGam{(K)}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{(K)} \AO{\nu}(\br{}),
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Manu's derivation %%%
@ -426,8 +430,8 @@ p}}c^\sigma_{{\nu p}}
}
\fi%%%
%%%% end Manu
We can then construct the ensemble density matrix
and the ensemble density as follows:
while the ensemble density matrix
and ensemble density read
\beq
\bGam{\bw}
= \sum_{K\geq 0} \ew{K} \bGam{(K)}
@ -439,7 +443,7 @@ and
\n{\bGam{\bw}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{\bw} \AO{\nu}(\br{}),
\eeq
respectively.
The exact expression of the individual energies in Eq.~\eqref{eq:exact_ener_level_dets} can then be rewritten as
The individual energy expression in Eq.~\eqref{eq:exact_ener_level_dets} can then be rewritten as
\beq\label{eq:exact_ind_ener_rdm}
\begin{split}
\E{}{(I)}
@ -460,7 +464,7 @@ where
\bh \equiv h_{\mu\nu} = \mel*{\AO{\mu}}{\hh}{\AO{\nu}}
\eeq
denotes the one-electron integrals matrix.
The individual Hx energy is obtained from the following trace formula
The exact individual Hx energies are obtained from the following trace formula
\beq
\Tr[\bGam{(K)} \bG \bGam{(L)}]
= \sum_{\mu\nu\la\si} \eGam{\mu\nu}{(K)} \eG{\mu\nu\la\si} \eGam{\la\si}{(L)},