start theory section on Green functions

Manuscript/GW-srDFT.tex

@@ -142,7 +142,13 @@
 
 % coordinates
 \newcommand{\br}[1]{\mathbf{r}_{#1}}
+\renewcommand{\b}[1]{\mathbf{#1}}
+\renewcommand{\d}{\text{d}}
 \newcommand{\dbr}[1]{d\br{#1}}
+\renewcommand{\bra}[1]{\ensuremath{\langle #1 \vert}}
+\renewcommand{\ket}[1]{\ensuremath{\vert #1  \rangle}}
+\renewcommand{\braket}[2]{\ensuremath{\langle  #1 \vert #2  \rangle}}
+
 
 \newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France}
 \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
@@ -179,6 +185,46 @@
 \label{sec:intro}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Theory}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Many-body Green-function theory with DFT basis-set correction}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we start by defining, for a $N$-electron system with nuclei-electron potential $v_{ne}(\b{r})$, the approximate ground-state energy for one-electron densities $n$ representable in a finite basis set ${\cal B}$
+\begin{equation}
+E_0^{\cal B} = \min_{n \in {\cal D}^{\cal B}} \left\{ F[n] + \int v_{ne}(\b{r}) n(\b{r}) \d\b{r}\right\},
+\end{equation}
+where ${\cal D}^{\cal B}$ is the set of $N$-representable densities which can be extracted from a wave function $\Psi^{\cal B}$ expandable in the Hilbert space generated by ${\cal B}$. In this expression, $F[n]=\min_{\Psi\to n} \bra{\Psi} \hat{T} + \hat{W}_\text{ee}\ket{\Psi}$ is the exact Levy-Lieb universal density functional, where $\hat{T}$ and $\hat{W}_\text{ee}$ are the kinetic and electron-electron interaction operators, which is then decomposed as
+\begin{equation}
+F[n] = F^{\cal B}[n] + \bar{E}^{\cal B}[n],
+\end{equation}
+where $F^{\cal B}[n]$ is the Levy-Lieb density functional with wave functions $\Psi^{\cal B}$ expandable in the Hilbert space generated by ${\cal B}$
+\begin{equation}
+F^{\cal B}[n] = \min_{\Psi^{\cal B}\to n} \bra{\Psi^{\cal B}} \hat{T} + \hat{W}_\text{ee}\ket{\Psi^{\cal B}},
+\end{equation}
+and $\bar{E}^{\cal B}[n]$ is the complementary basis-correction density functional. In the present work, instead of using wave-function methods for calculating $F^{\cal B}[n]$, we reexpress it with a contrained search over one-electron Green functions $G^{\cal B}(\b{r},\b{r}',\omega)$ representable in the basis set ${\cal B}$
+\begin{equation}
+F^{\cal B}[n] = \min_{G^{\cal B}\to n} \Omega[G^{\cal B}],
+\end{equation}
+where $\Omega[G]$ is a universal Luttinger-Ward-like functional of the Green function
+%\begin{equation}
+%\Omega[G] = - \Tr \left[\ln ( - G^{-1} ) \right] - \Tr \left[ G_\text{s}^{-1} G -1 \right]  + \Phi[G]
+%\end{equation}
+%$\Tr [A B] = 1/(2\pi i) \int \! \d \omega \, e^{i \omega 0^+} \! \iint  \! \d \b{r} \d \b{r}' A(\b{r},\b{r}',\omega) B(\b{r}',\b{r},\omega)$
+
+\begin{equation}
+E_0^{\cal B} = \min_{G^{\cal B}} \left\{ \Omega[G^{\cal B}] + \int v_{ne}(\b{r}) n_{G^{\cal B}}(\b{r}) \d\b{r} + \bar{E}^{\cal B}[n_{G^{\cal B}}] \right\}
+\end{equation}
+ddd
+\begin{equation}
+G^{\cal B}(\omega)^{-1} = G_\text{0}^{-1}(\omega)  - \Sigma_\text{Hxc}(\omega)  - \bar{v}_\text{c}^{\cal B}[n_{G^{\cal B}}]
+\end{equation}
+
+
 From Julien:
 \begin{equation}
 \fdv{E[n_G]}{G(r,r',\omega)} = \int \fdv{E[n_G]}{n(r'')}] \fdv{n_G(r'')}{G(r,r',w)} dr''
@@ -205,9 +251,6 @@ n_G(r'')  = i \int  G(r'',r'',w) d\omega
 \end{split}
 \end{equation}
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Theory}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{The GW Approximation}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Here, we provide self-contained summary of the main equations and quantities behind {\GOWO} and {\evGW}.
@@ -350,6 +393,6 @@ This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A004
 \end{acknowledgements}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
-\bibliography{GW,GW-srDFT,GW-srDFT-control}
+\bibliography{GW-srDFT,GW-srDFT-control,biblio}
 
 \end{document}