a few changes related to Pina and Arjan's comments
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@ -93,7 +93,7 @@
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\newcommand{\A}[1]{A_{#1}}
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\newcommand{\B}[1]{B_{#1}}
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\newcommand{\G}[2]{G_{#1}^{#2}}
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\newcommand{\Gs}[1]{G_\text{s}^{#1}}
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\newcommand{\Gs}[1]{G_\text{f}^{#1}}
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\newcommand{\F}[2]{F_{#1}^{#2}}
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\newcommand{\Po}[2]{P_{#1}^{#2}}
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\newcommand{\W}[2]{W_{#1}^{#2}}
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@ -113,6 +113,7 @@
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\newcommand{\hT}{\Hat{T}}
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\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
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\newcommand{\MO}[1]{\phi_{#1}}
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\newcommand{\stat}[1]{\underset{#1}{\text{stat}}}
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% Matrices
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\newcommand{\bOm}{\boldsymbol{\Omega}}
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@ -273,12 +274,12 @@ where $\F{}{\Bas}[\n{}{}]$ is the Levy-Lieb density functional with wave functio
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\F{}{\Bas}[\n{}{}] = \min_{\wf{}{\Bas} \rightsquigarrow \n{}{}} \mel*{\wf{}{\Bas}}{ \hT + \hWee{}}{\wf{}{\Bas}},
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\end{equation}
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and $\bE{}{\Bas}[\n{}{}]$ is the complementary basis-correction density functional. \cite{GinPraFerAssSavTou-JCP-18}
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In the present work, instead of using wave-function methods for calculating $\F{}{\Bas}[\n{}{}]$, we re-express it with a constrained search over $\Ne$-representable one-electron Green functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$
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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
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\begin{equation}
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\F{}{\Bas}[\n{}{}] = \min_{\G{}{\Bas} \rightsquigarrow \n{}{}} \Omega^\Bas[\G{}{\Bas}],
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\F{}{\Bas}[\n{}{}] = \stat{\G{}{\Bas} \rightsquigarrow \n{}{}} \Omega^\Bas[\G{}{\Bas}].
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\label{eq:FBn}
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\end{equation}
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where $\Omega^\Bas[G]$ is chosen to be a Klein-like energy functional of the Green function (see, \textit{e.g.}, Refs.~\onlinecite{SteLee-BOOK-13, MarReiCep-BOOK-16, DahLee-JCP-05, DahLeeBar-IJQC-05, DahLeeBar-PRA-06})
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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{SteLee-BOOK-13, MarReiCep-BOOK-16, DahLee-JCP-05, DahLeeBar-IJQC-05, DahLeeBar-PRA-06})
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\begin{equation}
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\Omega^\Bas[\G{}{}] = \Tr[\ln( - \G{}{} ) ] - \Tr[ (\Gs{\Bas})^{-1} \G{}{} - 1 ] + \Phi_\Hxc^\Bas[\G{}{}],
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\label{eq:OmegaB}
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@ -287,7 +288,7 @@ where $(\Gs{\Bas})^{-1}$ is the projection into $\Bas$ of the inverse free-parti
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\begin{equation}
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(\Gs{})^{-1}(\br{},\br{}',\omega)= \qty(\omega + \frac{\nabla^2_{\br{}}}{2} ) \delta(\br{}-\br{}'),
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\end{equation}
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and
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and we have introduced the trace
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\begin{equation}
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\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{}'.
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\end{equation}
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@ -297,25 +298,36 @@ In Eq.~\eqref{eq:OmegaB}, $\Phi_\Hxc^\Bas[\G{}{}]$ is a Hartree-exchange-correla
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\end{equation}
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Inserting Eqs.~\eqref{eq:Fn} and \eqref{eq:FBn} into Eq.~\eqref{eq:E0B}, we finally arrive at
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\begin{equation}
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\E{0}{\Bas} = \min_{\G{}{\Bas}} \qty{ \Omega^\Bas[\G{}{\Bas}] + \int \vne(\br{}) \n{\G{}{\Bas}}{}(\br{}) \dbr{} + \bE{}{\Bas}[\n{\G{}{\Bas}}{}] },
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\E{0}{\Bas} = \stat{\G{}{\Bas}} \qty{ \Omega^\Bas[\G{}{\Bas}] + \int \vne(\br{}) \n{\G{}{\Bas}}{}(\br{}) \dbr{} + \bE{}{\Bas}[\n{\G{}{\Bas}}{}] },
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\label{eq:E0BGB}
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\end{equation}
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where the minimization is over $\Ne$-representable one-electron Green functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$.
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where the stationary point is searched over $\Ne$-representable one-electron Green functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$.
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The stationary condition from Eq.~\eqref{eq:E0BGB} gives the following Dyson equation
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The stationary condition from Eq.~\eqref{eq:E0BGB} is
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\begin{eqnarray}
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\frac{\delta}{\delta \G{}{\Bas}} \Biggl( \Omega^\Bas[\G{}{\Bas}] + \int \vne(\br{}) \n{\G{}{\Bas}}{}(\br{}) \dbr{} + \bE{}{\Bas}[\n{\G{}{\Bas}}{}]
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\nonumber\\
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- \lambda \int \n{\G{}{\Bas}}{}(\br{}) \dbr{} \Biggl) = 0,
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\label{eq:stat}
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\end{eqnarray}
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where $\lambda$ is the chemical potential (enforcing the electron number). It leads the following Dyson equation
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\begin{equation}
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(\G{}{\Bas})^{-1} = (\G{0}{\Bas})^{-1}- \Sig{\Hxc}{\Bas}[\G{}{\Bas}]- \bSig{}{\Bas}[\n{\G{}{\Bas}}{}],
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\label{eq:Dyson}
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\end{equation}
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where $(\G{0}{\Bas})^{-1}$ is the basis projection of the inverse non-interacting Green function with potential $\vne(\br{})$, \textit{i.e.},
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\begin{equation}
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(\G{0}{})^{-1}(\br{},\br{}',\omega)= \qty(\omega + \frac{\nabla_{\br{}}^2}{2} + \vne(\br{}) + \lambda) \delta(\br{}-\br{}')
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(\G{0}{})^{-1}(\br{},\br{}',\omega)= \qty(\omega + \frac{\nabla_{\br{}}^2}{2} - \vne(\br{}) + \lambda) \delta(\br{}-\br{}'),
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\end{equation}
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with the chemical potential $\lambda$, and $\bSig{}{\Bas}$ is a frequency-independent local self-energy coming from the functional derivative of the complementary basis-correction density functional
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and $\bSig{}{\Bas}$ is a frequency-independent local self-energy coming from the functional derivative of the complementary basis-correction density functional
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\begin{equation}
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\bSig{}{\Bas}[\n{}{}](\br{},\br{}') = \bpot{}{\Bas}[\n{}{}](\br{}) \delta(\br{}-\br{}'),
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\end{equation}
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with $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$.
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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,
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\begin{equation}
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\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{}'',
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\end{equation}
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and $\n{}{}(\br{}) = \int_{-\infty}^{+\infty} \frac{d\omega}{2\pi i} \, e^{i \omega 0^+} \G{}{}(\br{},\br{},\omega)$.
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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 $\bSig{}{\Bas}$.
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Of course, in the CBS limit, the basis-set correction vanishes and the Green function becomes exact, \textit{i.e.},
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\begin{align}
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@ -324,7 +336,7 @@ Of course, in the CBS limit, the basis-set correction vanishes and the Green fun
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\lim_{\Bas \to \CBS} \G{}{\Bas} & = \G{}{}.
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\end{align}
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The Dyson equation \eqref{eq:Dyson} can be written with an arbitrary reference
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The Dyson equation \eqref{eq:Dyson} can also be written with an arbitrary reference
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\begin{equation}
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(\G{}{\Bas})^{-1} = (\G{\text{ref}}{\Bas})^{-1} - \qty( \Sig{\Hxc}{\Bas}[\G{}{\Bas}]- \Sig{\text{ref}}{\Bas} ) - \bSig{}{\Bas}[\n{\G{}{\Bas}}{}],
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\end{equation}
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