From 755be923c8a5742413043e895ac8bb489d809f64 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Thu, 24 Oct 2019 21:08:29 +0200 Subject: [PATCH] Done for T2 --- Manuscript/GW-srDFT.tex | 31 +++++++++++++++++++------------ 1 file changed, 19 insertions(+), 12 deletions(-) diff --git a/Manuscript/GW-srDFT.tex b/Manuscript/GW-srDFT.tex index 9150c49..a8755df 100644 --- a/Manuscript/GW-srDFT.tex +++ b/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}