Done for T2

Manuscript/GW-srDFT.tex

@@ -266,12 +266,14 @@ where $\F{}{\Bas}[\n{}{}]$ is the Levy-Lieb density functional with wave functio
 	\F{}{\Bas}[\n{}{}] = \min_{\wf{}{\Bas} \rightsquigarrow \n{}{}} \mel*{\wf{}{\Bas}}{ \hT + \hWee{}}{\wf{}{\Bas}},
 \end{equation}
 and $\bE{}{\Bas}[\n{}{}]$ is the complementary basis-correction density functional. \cite{Giner_2018}
-In the present work, instead of using wave-function methods for calculating $\F{}{\Bas}[\n{}{}]$, we use Green-function methods. We assume that there exists a functional $\Omega^\Bas[\G{}{\Bas}]$ of $\Ne$-representable one-electron Green functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$ and yielding the density $n$ which gives $\F{}{\Bas}[\n{}{}]$ at a stationary point
+In the present work, instead of using wave-function methods for calculating $\F{}{\Bas}[\n{}{}]$, we use Green-function methods. 
+We assume that there exists a functional $\Omega^\Bas[\G{}{\Bas}]$ of $\Ne$-representable one-electron Green functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$ and yielding the density $\n{}{}$ which gives $\F{}{\Bas}[\n{}{}]$ at a stationary point
 \begin{equation}
 	\F{}{\Bas}[\n{}{}] = \stat{\G{}{\Bas} \rightsquigarrow \n{}{}} \Omega^\Bas[\G{}{\Bas}].
 \label{eq:FBn}
 \end{equation}
-The reason why we use a stationary condition rather than a minimization condition is that only a stationary property is generally known for functionals of the Green function. For example, we can choose for $\Omega^\Bas[G]$ a Klein-like energy functional (see, \textit{e.g.}, Refs.~\onlinecite{Stefanucci_2013, Martin_2016, Dahlen_2005, Dahlen_2005a, Dahlen_2006})
+The reason why we use a stationary condition rather than a minimization condition is that only a stationary property is generally known for functionals of the Green function. 
+For example, we can choose for $\Omega^\Bas[\G{}{}]$ a Klein-like energy functional (see, \textit{e.g.}, Refs.~\onlinecite{Stefanucci_2013, Martin_2016, Dahlen_2005, Dahlen_2005a, Dahlen_2006})
 \begin{equation}
 	\Omega^\Bas[\G{}{}] = \Tr[\ln( - \G{}{} ) ] - \Tr[ (\Gs{\Bas})^{-1} \G{}{} - 1 ]  + \Phi_\Hxc^\Bas[\G{}{}],
 \label{eq:OmegaB}
@@ -282,7 +284,7 @@ where $(\Gs{\Bas})^{-1}$ is the projection into $\Bas$ of the inverse free-parti
 \end{equation}
 and we have introduced the trace 
 \begin{equation}
-	\Tr[A B] = \int_{-\infty}^{+\infty} \frac{d\omega}{2\pi i} \, e^{i \omega 0^+} \iint A(\br{},\br{}',\omega) B(\br{}',\br{}{},\omega)  \dbr{} \dbr{}'.
+	\Tr[A B] = \int_{-\infty}^{+\infty} \frac{d\omega}{2\pi i} e^{i \omega 0^+} \iint A(\br{},\br{}',\omega) B(\br{}',\br{}{},\omega)  \dbr{} \dbr{}'.
 \end{equation} 
 In Eq.~\eqref{eq:OmegaB}, $\Phi_\Hxc^\Bas[\G{}{}]$ is a Hartree-exchange-correlation ($\Hxc$) functional of the Green function such as its functional derivatives yields the $\Hxc$ self-energy in the basis
 \begin{equation}
@@ -296,13 +298,14 @@ Inserting Eqs.~\eqref{eq:Fn} and \eqref{eq:FBn} into Eq.~\eqref{eq:E0B}, we fina
 where the stationary point is searched over $\Ne$-representable one-electron Green functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$.
 
 The stationary condition from Eq.~\eqref{eq:E0BGB} is
-\begin{eqnarray}
-\frac{\delta}{\delta \G{}{\Bas}} \Biggl( \Omega^\Bas[\G{}{\Bas}] + \int \vne(\br{}) \n{\G{}{\Bas}}{}(\br{}) \dbr{} + \bE{}{\Bas}[\n{\G{}{\Bas}}{}] 
-\nonumber\\
-- \lambda \int \n{\G{}{\Bas}}{}(\br{}) \dbr{} \Biggl) = 0,
+\begin{multline}
+	\fdv{}{\G{}{\Bas}} \Bigg( \Omega^\Bas[\G{}{\Bas}] + \int \vne(\br{}) \n{\G{}{\Bas}}{}(\br{}) \dbr{} + \bE{}{\Bas}[\n{\G{}{\Bas}}{}] 
+	\\
+	- \lambda \int \n{\G{}{\Bas}}{}(\br{}) \dbr{} \Bigg) = 0,
 \label{eq:stat}
-\end{eqnarray}
-where $\lambda$ is the chemical potential (enforcing the electron number). It leads the following Dyson equation
+\end{multline}
+where $\lambda$ is the chemical potential (enforcing the electron number). 
+It leads the following Dyson equation
 \begin{equation}
 	(\G{}{\Bas})^{-1} = (\G{0}{\Bas})^{-1}- \Sig{\Hxc}{\Bas}[\G{}{\Bas}]- \bSig{}{\Bas}[\n{\G{}{\Bas}}{}],
 \label{eq:Dyson}
@@ -315,11 +318,15 @@ and $\bSig{}{\Bas}$ is a frequency-independent local self-energy coming from the
 \begin{equation}
 	\bSig{}{\Bas}[\n{}{}](\br{},\br{}') = \bpot{}{\Bas}[\n{}{}](\br{}) \delta(\br{}-\br{}'),
 \end{equation}
-with $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$. This is found from Eq.~\eqref{eq:stat} by using the chain rule,
+with $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$. 
+This is found from Eq.~\eqref{eq:stat} by using the chain rule,
 \begin{equation}
- \frac{\delta \bE{}{\Bas}[\n{}{}]}{\delta \G{}{}(\br{},\br{}',\omega)} = \int \frac{\delta \bE{}{\Bas}[\n{}{}]}{\delta n(\br{}'')}  \frac{\delta n(\br{}'')}{\delta \G{}{}(\br{},\br{}',\omega)} \dbr{}'',
+	\pdv{\bE{}{\Bas}[\n{}{}]}{\G{}{}(\br{},\br{}',\omega)} = \int \pdv{\bE{}{\Bas}[\n{}{}]}{\n{}{}(\br{}'')}  \frac{\delta \n{}{}(\br{}'')}{\delta \G{}{}(\br{},\br{}',\omega)} \dbr{}'',
+\end{equation}
+and 
+\begin{equation}
+	\n{}{}(\br{}) = \int_{-\infty}^{+\infty} \frac{d\omega}{2\pi i} e^{i \omega 0^+} \G{}{}(\br{},\br{},\omega).
 \end{equation}
-and $\n{}{}(\br{}) = \int_{-\infty}^{+\infty} \frac{d\omega}{2\pi i} \, e^{i \omega 0^+} \G{}{}(\br{},\br{},\omega)$.
 The solution of the Dyson equation \eqref{eq:Dyson} gives the Green function $\G{}{\Bas}(\br{},\br{}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bSig{}{\Bas}[\n{}{}]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bSig{}{\Bas}$. 
 Of course, in the CBS limit, the basis-set correction vanishes and the Green function becomes exact, \textit{i.e.},
 \begin{align}