2nd part of theory made clearer

Manuscript/GW-srDFT.tex

@@ -103,15 +103,11 @@
 \newcommand{\Wc}[1]{W_{#1}^\text{c}}
 \newcommand{\vc}[2]{\varv_{#1}^{#2}}
 \newcommand{\pot}[2]{v_{#1}^{#2}}
+\newcommand{\Pot}[2]{V_{#1}^{#2}}
 \newcommand{\bpot}[2]{\Bar{v}_{#1}^{#2}}
+\newcommand{\bPot}[2]{\Bar{V}_{#1}^{#2}}
 \newcommand{\Sig}[2]{\Sigma_{#1}^{#2}}
 \newcommand{\bSig}[2]{\Bar{\Sigma}_{#1}^{#2}}
-\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
-\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
-\newcommand{\tSigC}[1]{\Tilde{\Sigma}^\text{c}_{#1}}
-\newcommand{\SigCp}[1]{\Sigma^\text{p}_{#1}}
-\newcommand{\SigCh}[1]{\Sigma^\text{h}_{#1}}
-\newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}}
 \newcommand{\Z}[1]{Z_{#1}}
 \newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
 
@@ -179,10 +175,10 @@
 \newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
 
 \newcommand{\HF}{\text{HF}}
+\newcommand{\KS}{\text{KS}}
 \newcommand{\Hxc}{\text{Hxc}}
 \newcommand{\Hx}{\text{Hx}}
 \newcommand{\xc}{\text{xc}}
-%\newcommand{\ref}{\text{ref}}
 
 
 % units
@@ -265,6 +261,7 @@ Depending on the degree of self-consistency one is willing to perform, there exi
 The simplest and most popular variant of {\GW} is perturbative {\GW}, or {\GOWO}. \cite{Hybertsen_1985a, Hybertsen_1986}
 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}
 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} 
+
 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.
 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.
 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.
@@ -276,7 +273,7 @@ As shown in recent studies on both ground- and excited-state properties, \cite{L
 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}
 However, to the best of our knowledge, a F12-based correction for {\GW} has not been designed yet.
 
-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}. 
+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}. 
 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).
 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} 
 
@@ -301,18 +298,24 @@ Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we start by defining, for
 \label{eq:E0B}
 \end{equation}
 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 $\Bas$. 
-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
+In this expression, 
+\begin{equation}
+	\F{}{}[n] = \min_{\Psi \rightsquigarrow \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\Psi}
+\end{equation}
+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 $\n{}{}$.
+$\hT$ and $\hWee{}$ are the kinetic and electron-electron interaction operators, which is then decomposed as
 \begin{equation}
 	\F{}{}[\n{}{}] = \F{}{\Bas}[\n{}{}] + \bE{}{\Bas}[\n{}{}],
 \label{eq:Fn}
 \end{equation}
-where $\F{}{\Bas}[\n{}{}]$ is the Levy-Lieb density functional with wave functions $\Psi^\Bas$ expandable in the Hilbert space generated by $\Bas$
+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 $\Bas$
 \begin{equation}
-	\F{}{\Bas}[\n{}{}] = \min_{\Psi^\Bas \to \n{}{}} \mel*{\Psi^\Bas}{ \hT + \hWee{}}{\Psi^\Bas},
+	\F{}{\Bas}[\n{}{}] = \min_{\Psi^\Bas \rightsquigarrow \n{}{}} \mel*{\Psi^\Bas}{ \hT + \hWee{}}{\Psi^\Bas},
 \end{equation}
-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 $\Bas$
+and $\bE{}{\Bas}[\n{}{}]$ is the complementary basis-correction density functional. \cite{GinPraFerAssSavTou-JCP-18}
+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 $\Bas$
 \begin{equation}
-	\F{}{\Bas}[\n{}{}] = \min_{\G{}{\Bas} \to \n{}{}} \Omega^\Bas[\G{}{\Bas}],
+	\F{}{\Bas}[\n{}{}] = \min_{\G{}{\Bas} \rightsquigarrow \n{}{}} \Omega^\Bas[\G{}{\Bas}],
 \label{eq:FBn}
 \end{equation}
 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})
@@ -324,11 +327,14 @@ where $(\Gs{\Bas})^{-1}$ is the projection into $\Bas$ of the inverse free-parti
 \begin{equation}
 	(\Gs{})^{-1}(\br{},\br{}',\omega)= \qty(\omega + \frac{\nabla^2_{\br{}}}{2} ) \delta(\br{}-\br{}'),
 \end{equation}
-	 and we have used the notation 
+and
 \begin{equation}
-	\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).
+	\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{}'.
 \end{equation} 
-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: $\delta \Phi_\Hxc^\Bas[\G{}{}] / \delta \G{}{}(\br{},\br{}',\omega) = \Sig{\Hxc}{\Bas}[\G{}{}](\br{},\br{}',\omega)$.
+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
+\begin{equation}
+	\fdv{\Phi_\Hxc^\Bas[\G{}{}]}{\G{}{}(\br{},\br{}',\omega)} = \Sig{\Hxc}{\Bas}[\G{}{}](\br{},\br{}',\omega).
+\end{equation}
 Inserting Eqs.~\eqref{eq:Fn} and \eqref{eq:FBn} into Eq.~\eqref{eq:E0B}, we finally arrive at
 \begin{equation}
 	\E{0}{\Bas} = \min_{\G{}{\Bas}} \qty{ \Omega^\Bas[\G{}{\Bas}] + \int \vne(\br{}) \n{\G{}{\Bas}}{}(\br{}) \dbr{} + \bE{}{\Bas}[\n{\G{}{\Bas}}{}] },
@@ -338,25 +344,32 @@ where the minimization is over $\Ne$-representable one-electron Green's function
 
 The stationary condition from Eq.~\eqref{eq:E0BGB} gives the following Dyson equation
 \begin{equation}
-	(\G{}{\Bas})^{-1} = (\G{0}{\Bas})^{-1}- \Sig{\text{Hxc}}{\Bas}[\G{}{\Bas}]- \bSig{}{\Bas}[\n{\G{}{\Bas}}{}],
+	(\G{}{\Bas})^{-1} = (\G{0}{\Bas})^{-1}- \Sig{\Hxc}{\Bas}[\G{}{\Bas}]- \bSig{}{\Bas}[\n{\G{}{\Bas}}{}],
 \label{eq:Dyson}
 \end{equation}
-where $(G_\text{0}^\Bas)^{-1}$ is the basis projection of the inverse non-interacting Green's function with potential $v_\text{ne}(\b{r})$,
+where $(\G{0}{\Bas})^{-1}$ is the basis projection of the inverse non-interacting Green's function with potential $\vne(\b{r})$
 \begin{equation}
-(\G{0}{})^{-1}(\br{},\br{}',\omega)= \qty(\omega + \frac{\nabla_{\br{}}^2}{2} + \vne(\br{}) + \lambda) \delta(\br{}-\br{}')
+	(\G{0}{})^{-1}(\br{},\br{}',\omega)= \qty(\omega + \frac{\nabla_{\br{}}^2}{2} + \vne(\br{}) + \lambda) \delta(\br{}-\br{}')
 \end{equation}
-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
+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
 \begin{equation}
 	\bSig{}{\Bas}[\n{}{}](\br{},\br{}') = \bpot{}{\Bas}[\n{}{}](\br{}) \delta(\br{}-\br{}'),
 \end{equation}
-with $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$. 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, $\G{}{\Bas\to \CBS} = \G{}{}$.
+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 and the Green's function becomes exact, \textit{i.e.},
+\begin{align}
+	\lim_{\Bas \to \CBS} \bSig{}{\Bas} & = 0,
+	 &
+	\lim_{\Bas \to \CBS} \G{}{\Bas} & = \G{}{}.
+\end{align}
 
 The Dyson equation \eqref{eq:Dyson} can be written with an arbitrary reference
 \begin{equation}
 	(\G{}{\Bas})^{-1} = (\G{\text{ref}}{\Bas})^{-1} - \qty( \Sig{\Hxc}{\Bas}[\G{}{\Bas}]- \Sig{\text{ref}}{\Bas} )  - \bSig{}{\Bas}[\n{\G{}{\Bas}}{}],
 \end{equation}
 where $(\G{\text{ref}}{\Bas})^{-1} = (\G{0}{\Bas})^{-1} - \Sig{\text{ref}}{\Bas}$. 
-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.
+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.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -365,23 +378,21 @@ For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\
 
 In this section, we provide the minimal set of equations required to describe {\GOWO}.
 More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.
-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
+For sake of generality, we consider a $\KS$ reference.
+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.
+
+For a given (occupied or virtual) orbital $p$, the correlation part of the self-energy read, within the {\GW} approximation, 
 \begin{equation}
 \label{eq:SigC}
-	\SigC{p}(\omega) = \SigCp{p}(\omega) + \SigCh{p}(\omega),
+\begin{split}
-\end{equation}
+	\Sig{\text{c},p}{\Bas}(\omega)
-which, within the {\GW} approximation, read
+	& = \mel*{\MO{p}}{\Sig{\text{c}}{\Bas}(\omega)}{\MO{p}}
-\begin{subequations}
-\begin{align}
-\label{eq:SigCh}
-	\SigCh{p}(\omega)
-	& = 2 \sum_{i}^\text{occ} \sum_{x} \frac{[pi|x]^2}{\omega - \e{i} + \Om{x} - i \eta},
 	\\
-\label{eq:SigCp}
+	& = 2 \sum_{i}^\text{occ} \sum_{x} \frac{[pi|x]^2}{\omega - \e{i} + \Om{x} - i \eta}
-	\SigCp{p}(\omega)
+	\\
-	& = 2 \sum_{a}^\text{virt} \sum_{x} \frac{[pa|x]^2}{\omega - \e{a} - \Om{x} + i \eta},
+	& + 2 \sum_{a}^\text{virt} \sum_{x} \frac{[pa|x]^2}{\omega - \e{a} - \Om{x} + i \eta},
-\end{align}
+\end{split}
-\end{subequations}
+\end{equation}
 where $\eta$ is a positive infinitesimal.
 The screened two-electron integrals
 \begin{equation}
@@ -417,20 +428,30 @@ with
 	B_{ia,jb} & = 2 (ia|bj),
 \end{align}
 and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
-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.
 Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{x}$ which are used to build the screened Coulomb potential $\W{}{}$.
 
 The {\GOWO} quasiparticle energies $\eGOWO{p}$ are provided by the solution of the (non-linear) quasiparticle equation \cite{Hybertsen_1985a, vanSetten_2013, Veril_2018}
 \begin{equation}
 \label{eq:QP-G0W0}
-	\omega =  \e{p} + \Re[\SigC{p}(\omega)].
+	\omega =  \e{p} - \Pot{\xc,p}{\Bas} + \Sig{\text{x},p}{\Bas} + \Re[\Sig{\text{c},p}{\Bas}(\omega)].
 \end{equation}
 with the largest renormalization weight (or factor)
 \begin{equation}
 \label{eq:Z}
-	\Z{p} = \qty[ 1 - \left. \pdv{\Re[\SigC{p}(\omega)]}{\omega} \right|_{\omega = \e{p}}]^{-1}.
+	\Z{p} = \qty[ 1 - \left. \pdv{\Re[\Sig{\text{c},p}{\Bas}(\omega)]}{\omega} \right|_{\omega = \e{p}}]^{-1}.
 \end{equation}
 Because of sum rules, \cite{Martin_1959, Baym_1961, Baym_1962, vonBarth_1996} the other solutions, known as satellites, share the remaining weight.
+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
+\begin{equation}
+	\Pot{\xc}{\Bas} = \int \pot{\xc}{\Bas}(\br{}) \MO{p}(\br{})^2 \dbr{}.
+\end{equation}
+In particular, the ionization potential (IP) and electron affinity (EA) are defined as \cite{SzaboBook}
+\begin{align}
+  \IP   & = -\eGOWO{\HOMO},
+	&
+  \EA   & = -\eGOWO{\LUMO},
+\end{align}
+where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO orbital energies, respectively.
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Short-range correlation functionals}
@@ -438,19 +459,22 @@ Because of sum rules, \cite{Martin_1959, Baym_1961, Baym_1962, vonBarth_1996} th
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 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 $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$. 
-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}
+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}
 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.
-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.
+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.
-The basis set corrected {\GOWO} quasiparticle energies $\beGOWO{p}$ are thus given by
+The basis set corrected {\GOWO} quasiparticle energies are thus given by
+\begin{equation}
+	\beGOWO{p} = \eGOWO{p} + \bPot{}{\Bas}
+\end{equation}
+with
 \begin{equation}
 \begin{split}
-	\beGOWO{p} 
+	\bPot{}{\Bas}
-	& = \eGOWO{p} + \int \bSig{}{\Bas}[\n{}{}](\br{},\br{}') \MO{p}(\br{}) \MO{p}(\br{}') \dbr{} \dbr{}'
+	& = \int \bSig{}{\Bas}[\n{}{}](\br{},\br{}') \MO{p}(\br{}) \MO{p}(\br{}') \dbr{} \dbr{}'
 	\\ 
-	& = \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}

Manuscript/biblio.bib

@@ -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}}