diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 6954dcb..f6187f3 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -336,27 +336,62 @@ we finally recover from Eqs.~\eqref{eq:KS_ens_density} and \right|_{\n{}{} = \n{\opGam{\bw}}{}}. \end{split} \eeq -Note that, when $\bw=0$, the ensemble correlation functional becomes the -conventional (ground-state) correlation functional $E_{\rm c}[n]$ and -the ensemble density reduces to the ground-state one $n^{(0)}$. As a -result, -Eq.~(\ref{eq:exact_ener_level_dets}) can be simplified as follows: +Note that, when $\bw=0$, the ensemble correlation functional reduces to the +conventional (ground-state) correlation functional $E_{\rm c}[n]$. As a +result, the regular KS-DFT expression is recovered from +Eq.~(\ref{eq:exact_ener_level_dets}) for the ground-state energy: \beq +\E{}{(0)}=\mel*{\Det{(0)}}{\hH}{\Det{(0)}} + +\E{c}{}[\n{\Det{(0)}}{}], +\eeq +or, equivalently, +\beq\label{eq:gs_ener_level_gs_lim} +\begin{split} +\E{}{(0)}&=\mel*{\Det{(0)}}{\hat{H}[\n{\Det{(0)}}{}]}{\Det{(0)}} +\\ +& ++ +\E{c}{}[\n{\Det{(0)}}{}] +-\int \fdv{\E{c}{}[\n{\Det{(0)}}{}]}{\n{}{}(\br{})}\n{\Det{(0)}}{}(\br{}) +, +\end{split} +\eeq +where the ground-state-density-functional Hamiltonian reads +\beq +\hat{H}[\n{\Det{(0)}}{}]=\hH+ +\sum^N_{i=1}\fdv{\E{c}{}[\n{\Det{(0)}}{}]}{\n{}{}(\br{i})}. +\eeq +Note that, when divided by the total number $N$ of electrons, the sum of +the last two terms on the right-hand side of +Eq.~(\ref{eq:gs_ener_level_gs_lim}) match the correlation component of +Levy--Zahariev's shift in the KS potential~\cite{Levy_2014}. +The excited-state ($I>0$) energy level expressions +can also be simplified, thus leading to +\beq\label{eq:excited_ener_level_gs_lim} \begin{split} \E{}{(I)} - & = \mel*{\Det{(I)}}{\hH}{\Det{(I)}} + \E{c}{}[\n{}{(0)}] + & = \mel*{\Det{(I)}}{\hat{H}[\n{\Det{(0)}}{}]}{\Det{(I)}} + + + \left. + \pdv{\E{c}{\bw}[\n{\Det{(0)}}{}]}{\ew{I}} + \right|_{\bw=0}. \\ - & + \int \fdv{\E{c}{}[\n{}{(0)}]}{\n{}{}(\br{})} - \qty[ \n{\Det{(I)}}{}(\br{}) - \n{\opGam{\bw}}{}(\br{}) ] d\br{} - \\ - &+ - \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) - \left. - \pdv{\E{c}{\bw}[\n{}{}]}{\ew{K}} - \right|_{\n{}{} = \n{\opGam{\bw}}{}}. + & + + + \E{c}{}[\n{\Det{(0)}}{}] + -\int \fdv{\E{c}{}[\n{\Det{(0)}}{}]}{\n{}{}(\br{})}\n{\Det{(0)}}{}(\br{}) \end{split} \eeq -\titou{Manu, shall we mention that the last term in Eq.~\eqref{eq:exact_ener_level_dets} corresponds to the derivative discontinuity (DD)?} +As readily seen from Eqs.~(\ref{eq:gs_ener_level_gs_lim}) and +(\ref{eq:excited_ener_level_gs_lim}), any constant shift $\delta +\E{c}{}[\n{\Det{(0)}}{}]/\delta n({\bf r})\rightarrow \delta +\E{c}{}[\n{\Det{(0)}}{}]/\delta n({\bf r})+C$ in the correlation +potential leaves the individual energy levels unchanged. As a result, in +this context, +the correlation derivative discontinuities that is expected to appear when the +excitation process occurs~\cite{Levy_1995} are fully described by the ensemble +correlation derivatives [second term on the right-hand side of +Eq.~(\ref{eq:excited_ener_level_gs_lim})]. %%%%%%%%%%%%%%%% \subsection{One-electron reduced density matrix formulation}