2nd part of theory made clearer
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@ -103,15 +103,11 @@
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\newcommand{\Wc}[1]{W_{#1}^\text{c}}
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\newcommand{\vc}[2]{\varv_{#1}^{#2}}
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\newcommand{\pot}[2]{v_{#1}^{#2}}
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\newcommand{\Pot}[2]{V_{#1}^{#2}}
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\newcommand{\bpot}[2]{\Bar{v}_{#1}^{#2}}
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\newcommand{\bPot}[2]{\Bar{V}_{#1}^{#2}}
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\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}}
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\newcommand{\bSig}[2]{\Bar{\Sigma}_{#1}^{#2}}
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\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
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\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
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\newcommand{\tSigC}[1]{\Tilde{\Sigma}^\text{c}_{#1}}
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\newcommand{\SigCp}[1]{\Sigma^\text{p}_{#1}}
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\newcommand{\SigCh}[1]{\Sigma^\text{h}_{#1}}
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\newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}}
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\newcommand{\Z}[1]{Z_{#1}}
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\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
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@ -179,10 +175,10 @@
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\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
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\newcommand{\HF}{\text{HF}}
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\newcommand{\KS}{\text{KS}}
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\newcommand{\Hxc}{\text{Hxc}}
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\newcommand{\Hx}{\text{Hx}}
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\newcommand{\xc}{\text{xc}}
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%\newcommand{\ref}{\text{ref}}
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% units
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@ -265,6 +261,7 @@ Depending on the degree of self-consistency one is willing to perform, there exi
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The simplest and most popular variant of {\GW} is perturbative {\GW}, or {\GOWO}. \cite{Hybertsen_1985a, Hybertsen_1986}
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Although obviously starting-point dependent, it has been widely used in the literature to study solids, atoms and molecules. \cite{Bruneval_2012, Bruneval_2013, vanSetten_2015, vanSetten_2018}
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For finite systems such as atoms and molecules, partially \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011} or fully self-consistent \cite{Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b} {\GW} methods have shown great promise. \cite{Ke_2011, Blase_2011, Faber_2011, Koval_2014, Hung_2016, Blase_2018, Jacquemin_2017}
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Similar to other electron correlation methods, MBPT methods suffer from the usual slow convergence of energetic properties with respect to the size of the one-electron basis set.
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This can be tracked down to the lack of explicit electron-electron terms modeling the infamous electron-electron coalescence point (also known as Kato cusp \cite{Kat-CPAM-57}) and, more specifically, the Coulomb correlation hole around it.
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Pioneered by Hylleraas \cite{Hyl-ZP-29} in the 1930's and popularized in the 1990's by Kutzelnigg and coworkers \cite{Kut-TCA-85, NogKut-JCP-94, KutKlo-JCP-91} (and subsequently others \cite{KonBisVal-CR-12, HatKloKohTew-CR-12, TenNog-WIREs-12, GruHirOhnTen-JCP-17}), the so-called F12 methods overcome this slow convergence by employing geminal basis functions that closely resemble the correlation holes in electronic wave functions.
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@ -276,7 +273,7 @@ As shown in recent studies on both ground- and excited-state properties, \cite{L
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Explicitly correlated F12 correction schemes have been derived for second-order Green's function methods (GF2) \cite{SzaboBook, Casida_1989, Casida_1991, Stefanucci_2013, Ortiz_2013, Phillips_2014, Phillips_2015, Rusakov_2014, Rusakov_2016, Hirata_2015, Hirata_2017, Loos_2018} by Ten-no and coworkers \cite{Ohnishi_2016, Johnson_2018} and Valeev and coworkers. \cite{Pavosevic_2017, Teke_2019}
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However, to the best of our knowledge, a F12-based correction for {\GW} has not been designed yet.
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In the present manuscript, we illustrate the performance of the density-based basis set correction developed in Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18, LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} on ionization potentials (IPs) obtained within {\GOWO}.
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In the present manuscript, we illustrate the performance of the density-based basis set correction developed in Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18, LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} on ionization potentials obtained within {\GOWO}.
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Note that the the present basis set correction can be straightforwardly applied to other properties (\textit{e.g.}, electron affinities and fundamental gap), as well as other flavours of (self-consistent) {\GW} or Green's function-based methods, such as GF2 (and its higher-order variants).
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Moreover, we are currently investigating the performances of the present approach for linear response theory, in order to speed up the convergence of excitation energies obtained within the random-phase approximation (RPA) \cite{Casida_1995, Dreuw_2005} and Bethe-Salpeter equation (BSE) formalism. \cite{Strinati_1988, Leng_2016, Blase_2018}
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@ -301,18 +298,24 @@ Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we start by defining, for
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\label{eq:E0B}
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\end{equation}
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where $\cD^\Bas$ is the set of $\Ne$-representable densities which can be extracted from a wave function $\Psi^\Bas$ expandable in the Hilbert space generated by $\Bas$.
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In this expression, $\F{}{}[n] = \min_{\Psi \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\Psi}$ is the exact Levy-Lieb universal density functional, where $\hT$ and $\hWee{}$ are the kinetic and electron-electron interaction operators, which is then decomposed as
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In this expression,
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\begin{equation}
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\F{}{}[n] = \min_{\Psi \rightsquigarrow \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\Psi}
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\end{equation}
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is the exact Levy-Lieb universal density functional, where the notation $\wf{}{} \rightsquigarrow \n{}{}$ in Eq.~\eqref{eq:E0B} states that $\wf{}{}$ yields the one-electron density $\n{}{}$.
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$\hT$ and $\hWee{}$ are the kinetic and electron-electron interaction operators, which is then decomposed as
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\begin{equation}
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\F{}{}[\n{}{}] = \F{}{\Bas}[\n{}{}] + \bE{}{\Bas}[\n{}{}],
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\label{eq:Fn}
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\end{equation}
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where $\F{}{\Bas}[\n{}{}]$ is the Levy-Lieb density functional with wave functions $\Psi^\Bas$ expandable in the Hilbert space generated by $\Bas$
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where $\F{}{\Bas}[\n{}{}]$ is the Levy-Lieb density functional \cite{Lev-PNAS-79, Lev-PRA-82, Lie-IJQC-83} with wave functions $\Psi^\Bas$ expandable in the Hilbert space generated by $\Bas$
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\begin{equation}
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\F{}{\Bas}[\n{}{}] = \min_{\Psi^\Bas \to \n{}{}} \mel*{\Psi^\Bas}{ \hT + \hWee{}}{\Psi^\Bas},
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\F{}{\Bas}[\n{}{}] = \min_{\Psi^\Bas \rightsquigarrow \n{}{}} \mel*{\Psi^\Bas}{ \hT + \hWee{}}{\Psi^\Bas},
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\end{equation}
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and $\bE{}{\Bas}[\n{}{}]$ is the complementary basis-correction density functional. In the present work, instead of using wave-function methods for calculating $\F{}{\Bas}[\n{}{}]$, we reexpress it with a constrained search over $\Ne$-representable one-electron Green's functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$
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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's functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$
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\begin{equation}
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\F{}{\Bas}[\n{}{}] = \min_{\G{}{\Bas} \to \n{}{}} \Omega^\Bas[\G{}{\Bas}],
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\F{}{\Bas}[\n{}{}] = \min_{\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's 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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@ -324,11 +327,14 @@ 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 we have used the notation
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and
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\begin{equation}
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\Tr[A B] = \frac{1}{2\pi i} \int_{-\infty}^{+\infty} d\omega \, e^{i \omega 0^+} \iint \dbr{} \dbr{}' A(\br{},\br{}',\omega) B(\br{}',\br{}{},\omega).
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\Tr[A B] = \frac{1}{2\pi i} \int_{-\infty}^{+\infty} d\omega \, 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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In Eq.~\eqref{eq:OmegaB}, $\Phi_\Hxc^\Bas[\G{}{}]$ is a Hartree-exchange-correlation ($\Hxc$) functional of the Green's function such as its functional derivatives yields the Hxc self-energy in the basis
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\begin{equation}
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\fdv{\Phi_\Hxc^\Bas[\G{}{}]}{\G{}{}(\br{},\br{}',\omega)} = \Sig{\Hxc}{\Bas}[\G{}{}](\br{},\br{}',\omega).
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\end{equation}
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In Eq.~\eqref{eq:OmegaB}, $\Phi_\Hxc^\Bas[\G{}{}]$ is a Hartree-exchange-correlation ($\Hxc$) functional of the Green's functional such as its functional derivatives yields the Hxc self-energy in the basis: $\delta \Phi_\Hxc^\Bas[\G{}{}] / \delta \G{}{}(\br{},\br{}',\omega) = \Sig{\Hxc}{\Bas}[\G{}{}](\br{},\br{}',\omega)$.
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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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@ -338,25 +344,32 @@ where the minimization is over $\Ne$-representable one-electron Green's function
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The stationary condition from Eq.~\eqref{eq:E0BGB} gives the following Dyson equation
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\begin{equation}
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(\G{}{\Bas})^{-1} = (\G{0}{\Bas})^{-1}- \Sig{\text{Hxc}}{\Bas}[\G{}{\Bas}]- \bSig{}{\Bas}[\n{\G{}{\Bas}}{}],
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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 $(G_\text{0}^\Bas)^{-1}$ is the basis projection of the inverse non-interacting Green's function with potential $v_\text{ne}(\b{r})$,
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where $(\G{0}{\Bas})^{-1}$ is the basis projection of the inverse non-interacting Green's function with potential $\vne(\b{r})$
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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 functional derivative of the complementary basis-correction density functional
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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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\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 $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$. The solution of the Dyson equation \eqref{eq:Dyson} gives the Green's 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, $\bSig{}{\Bas \to \CBS} = 0$, and the Green's function becomes exact, $\G{}{\Bas\to \CBS} = \G{}{}$.
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with $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$.
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The solution of the Dyson equation \eqref{eq:Dyson} gives the Green's 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's function becomes exact, \textit{i.e.},
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\begin{align}
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\lim_{\Bas \to \CBS} \bSig{}{\Bas} & = 0,
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&
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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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\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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where $(\G{\text{ref}}{\Bas})^{-1} = (\G{0}{\Bas})^{-1} - \Sig{\text{ref}}{\Bas}$.
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For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\br{},\br{}') = \Sig{\Hx,\HF}{\Bas}(\br{},\br{}')$ is the $\HF$ nonlocal self-energy, and if the reference is Kohn-Sham, $\Sig{\text{ref}}{\Bas}(\br{},\br{}') = \pot{\Hxc}{\Bas}(\br{}) \delta(\br{}-\br{}')$ is the local $\Hxc$ potential.
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For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\br{},\br{}') = \Sig{\Hx,\HF}{\Bas}(\br{},\br{}')$ is the $\HF$ nonlocal self-energy, and if the reference is Kohn-Sham ($\KS$), $\Sig{\text{ref}}{\Bas}(\br{},\br{}') = \pot{\Hxc}{\Bas}(\br{}) \delta(\br{}-\br{}')$ is the local $\Hxc$ potential.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -365,23 +378,21 @@ For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\
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In this section, we provide the minimal set of equations required to describe {\GOWO}.
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More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.
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For a given (occupied or virtual) orbital $p$, the correlation part of the self-energy can decomposed in its hole (h) and particle (p) contributions
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For sake of generality, we consider a $\KS$ reference.
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The one-electron energies $\e{p}$ and their corresponding orbitals $\MO{p}(\br{})$ (which defines our basis set $\Bas$) are then $\KS$ energies and orbitals.
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For a given (occupied or virtual) orbital $p$, the correlation part of the self-energy read, within the {\GW} approximation,
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\begin{equation}
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\label{eq:SigC}
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\SigC{p}(\omega) = \SigCp{p}(\omega) + \SigCh{p}(\omega),
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\end{equation}
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which, within the {\GW} approximation, read
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\begin{subequations}
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\begin{align}
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\label{eq:SigCh}
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\SigCh{p}(\omega)
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& = 2 \sum_{i}^\text{occ} \sum_{x} \frac{[pi|x]^2}{\omega - \e{i} + \Om{x} - i \eta},
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\begin{split}
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\Sig{\text{c},p}{\Bas}(\omega)
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& = \mel*{\MO{p}}{\Sig{\text{c}}{\Bas}(\omega)}{\MO{p}}
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\\
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\label{eq:SigCp}
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\SigCp{p}(\omega)
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& = 2 \sum_{a}^\text{virt} \sum_{x} \frac{[pa|x]^2}{\omega - \e{a} - \Om{x} + i \eta},
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\end{align}
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\end{subequations}
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& = 2 \sum_{i}^\text{occ} \sum_{x} \frac{[pi|x]^2}{\omega - \e{i} + \Om{x} - i \eta}
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\\
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& + 2 \sum_{a}^\text{virt} \sum_{x} \frac{[pa|x]^2}{\omega - \e{a} - \Om{x} + i \eta},
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\end{split}
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\end{equation}
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where $\eta$ is a positive infinitesimal.
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The screened two-electron integrals
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\begin{equation}
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@ -417,20 +428,30 @@ with
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B_{ia,jb} & = 2 (ia|bj),
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\end{align}
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and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
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The one-electron energies $\e{p}$ in \eqref{eq:SigCh}, \eqref{eq:SigCp} and \eqref{eq:RPA} and their corresponding orbitals $\MO{p}(\br{})$ are either the HF or KS energies and orbitals depending on the chosen reference.
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Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{x}$ which are used to build the screened Coulomb potential $\W{}{}$.
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The {\GOWO} quasiparticle energies $\eGOWO{p}$ are provided by the solution of the (non-linear) quasiparticle equation \cite{Hybertsen_1985a, vanSetten_2013, Veril_2018}
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\begin{equation}
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\label{eq:QP-G0W0}
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\omega = \e{p} + \Re[\SigC{p}(\omega)].
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\omega = \e{p} - \Pot{\xc,p}{\Bas} + \Sig{\text{x},p}{\Bas} + \Re[\Sig{\text{c},p}{\Bas}(\omega)].
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\end{equation}
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with the largest renormalization weight (or factor)
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\begin{equation}
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\label{eq:Z}
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\Z{p} = \qty[ 1 - \left. \pdv{\Re[\SigC{p}(\omega)]}{\omega} \right|_{\omega = \e{p}}]^{-1}.
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\Z{p} = \qty[ 1 - \left. \pdv{\Re[\Sig{\text{c},p}{\Bas}(\omega)]}{\omega} \right|_{\omega = \e{p}}]^{-1}.
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\end{equation}
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Because of sum rules, \cite{Martin_1959, Baym_1961, Baym_1962, vonBarth_1996} the other solutions, known as satellites, share the remaining weight.
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In Eq.~\eqref{eq:QP-G0W0}, $\Sig{\text{x},p}{\Bas} = \mel*{\MO{p}}{\Sig{\text{x}}{\Bas}}{\MO{p}}$ is the (static) exchange part of the self-energy and
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\begin{equation}
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\Pot{\xc}{\Bas} = \int \pot{\xc}{\Bas}(\br{}) \MO{p}(\br{})^2 \dbr{}.
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\end{equation}
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In particular, the ionization potential (IP) and electron affinity (EA) are defined as \cite{SzaboBook}
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\begin{align}
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\IP & = -\eGOWO{\HOMO},
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&
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\EA & = -\eGOWO{\LUMO},
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\end{align}
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where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO orbital energies, respectively.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Short-range correlation functionals}
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@ -438,19 +459,22 @@ Because of sum rules, \cite{Martin_1959, Baym_1961, Baym_1962, vonBarth_1996} th
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%%%%%%%%%%%%%%%%%%%%%%%%
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The frequency-independent local self-energy $\bSig{}{\Bas}[\n{}{}](\br{},\br{}') = \bpot{}{\Bas}[\n{}{}](\br{}) \delta(\br{}-\br{}')$ originates from the functional derivative of complementary basis-correction density functionals $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$.
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Here, we employ two types of complementary, short-range correlation functionals $\bE{}{\Bas}[\n{}{}]$: a local-density approximation ($\LDA$) functional with multideterminant reference \cite{PazMorGorBac-PRB-06} and a Perdew-Burke-Ernzerhof ($\PBE$) inspired correlation functional \cite{FerGinTou-JCP-19} which interpolate between the usual PBE functional \cite{PerBurErn-PRL-96} at $\mu = 0$ and the exact large-$\mu$ behavior. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
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Here, we employ two types of complementary, short-range correlation functionals $\bE{}{\Bas}[\n{}{}]$: a local-density approximation ($\LDA$) functional with multideterminant reference \cite{PazMorGorBac-PRB-06} and a Perdew-Burke-Ernzerhof ($\PBE$) inspired correlation functional \cite{FerGinTou-JCP-19} which interpolates between the usual PBE functional \cite{PerBurErn-PRL-96} at $\mu = 0$ and the exact large-$\mu$ behavior. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
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Additionally to the one-electron density, these RS-DFT functionals requires a range-separation function $\rsmu{}{\Bas}(\br{})$ which automatically adapts to the spatial non-homogeneity of the basis-set incompleteness error.
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We refer the interested reader to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18, LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} where our procedure is thoroughly detailed and the explicit expressions of these two short-range correlation functionals are given.
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The basis set corrected {\GOWO} quasiparticle energies $\beGOWO{p}$ are thus given by
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We refer the interested reader to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18, LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} where our procedure is thoroughly detailed and the explicit expressions of these two short-range correlation functionals are provided.
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The basis set corrected {\GOWO} quasiparticle energies are thus given by
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\begin{equation}
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\beGOWO{p} = \eGOWO{p} + \bPot{}{\Bas}
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\end{equation}
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with
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\begin{equation}
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\begin{split}
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\beGOWO{p}
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& = \eGOWO{p} + \int \bSig{}{\Bas}[\n{}{}](\br{},\br{}') \MO{p}(\br{}) \MO{p}(\br{}') \dbr{} \dbr{}'
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\bPot{}{\Bas}
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& = \int \bSig{}{\Bas}[\n{}{}](\br{},\br{}') \MO{p}(\br{}) \MO{p}(\br{}') \dbr{} \dbr{}'
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\\
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& = \eGOWO{p} + \int \bpot{}{\Bas}[\n{}{}](\br{}) \MO{p}(\br{})^2 \dbr{}.
|
||||
& = \int \bpot{}{\Bas}[\n{}{}](\br{}) \MO{p}(\br{})^2 \dbr{}.
|
||||
\end{split}
|
||||
\end{equation}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Computational details}
|
||||
\label{sec:compdetails}
|
||||
@ -471,13 +495,6 @@ Moreover, the infinitesimal $\eta$ has been set to zero.
|
||||
In this section, we study a subset of atoms and molecules from the GW100 test set.
|
||||
In particular, we study the 20 smallest molecules of the GW100 set, a subset that we label as GW20.
|
||||
We also study the five canonical nucleobases (adenine, cytosine, thymine, guanine and uracil) which are also part of the GW100 test set.
|
||||
The ionization potential (IP) and electron affinity (EA) are defined as \cite{SzaboBook}
|
||||
\begin{align}
|
||||
\IP & = -\eHOMO,
|
||||
&
|
||||
\EA & = -\eLUMO,
|
||||
\end{align}
|
||||
where $\eHOMO$ and $\eLUMO$ are the HOMO and LUMO orbital energies, respectively.
|
||||
|
||||
%%% TABLE I %%%
|
||||
\begin{squeezetable}
|
||||
|
||||
@ -1,7 +1,7 @@
|
||||
%% This BibTeX bibliography file was created using BibDesk.
|
||||
%% http://bibdesk.sourceforge.net/
|
||||
|
||||
%% Created for Pierre-Francois Loos at 2019-10-09 15:35:04 +0200
|
||||
%% Created for Pierre-Francois Loos at 2019-10-12 09:05:09 +0200
|
||||
|
||||
|
||||
%% Saved with string encoding Unicode (UTF-8)
|
||||
@ -7064,15 +7064,19 @@
|
||||
|
||||
@article{Lev-PNAS-79,
|
||||
Author = {M. Levy},
|
||||
Date-Modified = {2019-10-12 09:04:26 +0200},
|
||||
Journal = {Proc. Natl. Acad. Sci. U.S.A.},
|
||||
Pages = {6062},
|
||||
Title = {Universal Variational Functionals Of Electron Densities, First-Order Density Matrices, And Natural Spin-Orbitals And Solution Of The V-Representability Problem},
|
||||
Volume = {76},
|
||||
Year = {1979}}
|
||||
|
||||
@article{Lev-PRA-82,
|
||||
Author = {M. Levy},
|
||||
Date-Modified = {2019-10-12 09:05:05 +0200},
|
||||
Journal = {Phys. Rev. A},
|
||||
Pages = {1200},
|
||||
Title = {Electron Densities In Search Of Hamiltonians},
|
||||
Volume = {26},
|
||||
Year = {1982}}
|
||||
|
||||
@ -7115,8 +7119,10 @@
|
||||
|
||||
@article{Lie-IJQC-83,
|
||||
Author = {E. H. Lieb},
|
||||
Date-Modified = {2019-10-12 09:03:38 +0200},
|
||||
Journal = {Int. J. Quantum Chem.},
|
||||
Pages = {243},
|
||||
Title = {Density Functionals For Coulomb Systems},
|
||||
Volume = {{24}},
|
||||
Year = {1983}}
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user