E vs w

Manuscript/eDFT.tex

@@ -31,9 +31,6 @@
 \newcommand{\ie}{\textit{i.e.}}
 \newcommand{\eg}{\textit{e.g.}}
 
-% numbers
-\newcommand{\nEl}{N}
-
 % operators
 \newcommand{\hH}{\Hat{H}}
 \newcommand{\hh}{\Hat{h}}
@@ -87,8 +84,8 @@
 \newcommand{\dbERI}[2]{(#1||#2)}
 
 % Numbers
-\newcommand{\Nel}{N}
-\newcommand{\Nbas}{K}
+\newcommand{\nEl}{N}
+\newcommand{\nBas}{K}
 
 % AO and MO basis
 \newcommand{\Det}[1]{\Phi^{#1}}
@@ -820,8 +817,8 @@ Finally, we note that, by construction,
 \label{sec:comp_details}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Having defined the eLDA functional in the previous section [see Eq.~\eqref{eq:eLDA}], we now turn to its validation.
-Our testing playground for the validation of the eLDA functional is the ubiquitous ``electrons in a box'' model where $\Nel$ electrons are confined in a 1D box of length $L$, a family of systems that we call $\Nel$-boxium.
-In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ and $2 \le \Nel \le 7$. 
+Our testing playground for the validation of the eLDA functional is the ubiquitous ``electrons in a box'' model where $\nEl$ electrons are confined in a 1D box of length $L$, a family of systems that we call $\nEl$-boxium.
+In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ and $2 \le \nEl \le 7$. 
 %\titou{Comment on the quality of these density: density- and functional-driven errors?}
 
 These inhomogeneous systems have non-trivial electronic structure properties which can be tuned by varying the box length.
@@ -835,7 +832,7 @@ We use as basis functions the (orthonormal) orbitals of the one-electron system,
 		\sqrt{2/L} \sin(\mu \pi x/L),	&	\mu \text{ is even,}
 	\end{cases}
 \end{equation} 
-with $ \mu = 1,\ldots,\Nbas$ and $\Nbas = 30$ for all calculations.
+with $ \mu = 1,\ldots,\nBas$ and $\nBas = 30$ for all calculations.
 For the self-consistent calculations (such as HF, KS-DFT or KS-eDFT), the convergence threshold $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw} \bS - \bS \bGam{\bw} \bF{\bw}}}$ been set to $10^{-5}$.
 For KS-DFT and KS-eDFT calculations, a Gauss-Legendre quadrature is employed to compute the various integrals that cannot be performed in closed form.
 
@@ -864,14 +861,14 @@ Moreover, because the GIE can easily computed via Eq.~\eqref{eq:WHF} even for re
 	\includegraphics[width=\linewidth]{EvsW_n5}
 	\caption{
 	\label{fig:EvsW}
-	Weight dependence of the ensemble energy with and without ghost interaction correction (GIC) for 5-boxium with a box length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
+	Weight dependence of the ensemble energy $\E{}{(\ew{1},\ew{2})}$ with and without ghost interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
 	}
 \end{figure*}
 %%% %%% %%%
 
 
-In Fig.~\ref{fig:EvsL}, we report the excitation energies (multiplied by $L^2$) for various methods and box sizes in the case of 5-boxium (\ie, $\Nel = 5$).
-Similar graphs are obtained for the other $\Nel$ values and they can be found in the {\SI} alongside the numerical data associated with each method.
+In Fig.~\ref{fig:EvsL}, we report the excitation energies (multiplied by $L^2$) for various methods and box sizes in the case of 5-boxium (\ie, $\nEl = 5$).
+Similar graphs are obtained for the other $\nEl$ values and they can be found in the {\SI} alongside the numerical data associated with each method.
 In the weakly correlated regime (\ie, small $L$), all methods provide accurate estimates of the excitation energies.
 When the box gets larger, they start to deviate.
 For the single excitation, TDHF is extremely accurate over the whole range of $L$ values, while CIS is slightly less accurate and starts to overestimate the excitation energy by a few percent at $L=8\pi$.
@@ -889,12 +886,12 @@ This conclusion is verified for smaller and larger number of electrons (see {\SI
 	\caption{
 	\label{fig:EvsL}
 	Excitation energies (multiplied by $L^2$) associated with the single excitation $\Ex{(1)}$ (bottom) and double excitation $\Ex{(2)}$ (top) of 5-boxium for various methods and box length $L$.
-	Graphs for additional values of $\Nel$ can be found as {\SI}.
+	Graphs for additional values of $\nEl$ can be found as {\SI}.
 	}
 \end{figure}
 %%% %%% %%%
 
-Figure \ref{fig:EvsN} reports the error (in \%) in excitation energies (as compared to FCI), for the same methods, as a function of $\Nel$ and fixed $L$ (in this case $L=\pi$).
+Figure \ref{fig:EvsN} reports the error (in \%) in excitation energies (as compared to FCI), for the same methods, as a function of $\nEl$ and fixed $L$ (in this case $L=\pi$).
 The graphs associated with other $L$ values are reported as {\SI}.
 Again, the graph for $L=\pi$ is quite typical and we draw similar conclusions as in the previous paragraph: irrespectively of the number of electrons, the eLDA functional with state-averaged weights is able to accurately model single and double excitations. 
 As a rule of thumb, we see that eLDA single excitations are of the same quality as the ones obtained in the linear response formalism (such as TDHF or TDLDA), while double excitations only deviates from the FCI values by a few tenth of percent for $L=\pi$.
@@ -905,7 +902,7 @@ Even for larger boxes, the discrepancy between FCI and eLDA for double excitatio
 	\includegraphics[width=\linewidth]{EvsN_1}
 	\caption{
 	\label{fig:EvsN}
-	Error with respect to FCI in single and double excitation energies for $\Nel$-boxium for various methods and number of electrons $\Nel$ at $L=\pi$.
+	Error with respect to FCI in single and double excitation energies for $\nEl$-boxium for various methods and number of electrons $\nEl$ at $L=\pi$.
 	Graphs for additional values of $L$ can be found as {\SI}.
 	}
 \end{figure}