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Pierre-Francois Loos 2020-02-18 12:32:15 +01:00
parent 37437c2f4d
commit 5869fd3790
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8
Manuscript/SI/eDFT-SI.tex
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@ -959,11 +959,11 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
\\
TDLDA & & $\Ex{(1)}$ & 0.0000 & 0.0001 & 0.0000 & -0.0001 & -0.0009 & -0.0107 & -0.0539 \\
\\
eHF & $(0,0)$ & $\E{(0)}$ & 0.1918 & 0.1902 & 0.1873 & 0.1817 & 0.1713 & 0.1538 & 0.1273 \\
& & $\E{(1)}$ & 0.1810 & 0.1792 & 0.1758 & 0.1694 & 0.1579 & 0.1392 & 0.1122 \\
& & $\E{(2)}$ & 0.2532 & 0.2514 & 0.2470 & 0.2381 & 0.2220 & 0.1969 & 0.1584 \\
eHF & $(0,0)$ & $\E{(0)}$ & 0.1240 & 0.1224 & 0.1194 & 0.1135 & 0.1027 & 0.0842 & 0.0570 \\
& & $\E{(1)}$ & 0.2162 & 0.2150 & 0.2125 & 0.2081 & 0.2006 & 0.1894 & 0.1711 \\
& & $\E{(2)}$ & 0.2437 & 0.2419 & 0.2376 & 0.2291 & 0.2137 & 0.1916 & 0.1650 \\
& & $\Ex{(1)}$ & 0.2473 & 0.2458 & 0.2425 & 0.2366 & 0.2263 & 0.2099 & 0.1825 \\
& & $\Ex{(2)}$ & 0.3196 & 0.3180 & 0.3137 & 0.3054 & 0.2904 & 0.2676 & 0.2286 \\
& & $\Ex{(2)}$ & 0.1796 & 0.1780 & 0.1746 & 0.1685 & 0.1576 & 0.1406 & 0.1202 \\
\\
eHF & $(1/3,1/3)$ & $\E{(0)}$ & 0.1918 & 0.1902 & 0.1873 & 0.1817 & 0.1713 & 0.1538 & 0.1273 \\
& & $\E{(1)}$ & 0.1810 & 0.1792 & 0.1758 & 0.1694 & 0.1579 & 0.1392 & 0.1122 \\

6
Manuscript/eDFT.tex
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@ -676,7 +676,7 @@ These errors will be removed when computing individual energies according to Eq.
Turning to the density-functional ensemble correlation energy, the following eLDA will be employed:
\beq\label{eq:eLDA_corr_fun}
\E{c}{\bw}[\n{}{}] = \int \n{}{}(\br{}) \e{c}{\bw}[\n{}{}(\br{})] d\br{},
\E{c}{\bw}[\n{}{}] = \int \n{}{}(\br{}) \e{c}{\bw}(\n{}{}(\br{})) d\br{},
\eeq
where the correlation energy per particle is \textit{weight dependent}.
Its construction from a finite uniform electron gas model is discussed in detail in Sec.~\ref{sec:eDFA}.
@ -686,7 +686,7 @@ Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} le
\E{}{(I)}
& \approx \Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
\\
& + \int \e{c}{\bw}[\n{\bGam{\bw}}{}(\br{})] \n{\bGam{(I)}}{}(\br{}) d\br{}
& + \int \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
\\
&
+ \int \n{\bGam{\bw}}{}(\br{}) \qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{\bw}}{}(\br{}) ]
@ -812,7 +812,7 @@ In particular, $\be{c}{(0,0)}(\n{}{}) = \e{c}{\text{LDA}}(\n{}{})$.
Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
Finally, we note that, by construction,
\begin{equation}
\left. \pdv{\be{c}{\bw}[\n{}{}]}{\ew{J}}\right|_{\n{}{} = \n{}{\bw}(\br{})} = \be{c}{(J)}[\n{}{\bw}(\br{})] - \be{c}{(0)}[\n{}{\bw}(\br{})].
\left. \pdv{\be{c}{\bw}(\n{}{})}{\ew{J}}\right|_{\n{}{} = \n{\bGam{\bw}}{}(\br{})} = \be{c}{(J)}(\n{\bGam{\bw}}{}(\br{})) - \be{c}{(0)}(\n{\bGam{\bw}}{}(\br{})).
\end{equation}
%\titou{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}}

37106
Notebooks/eDFT_FUEG.nb
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