done with 1st iteration of intro, starting cleaning up theory
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@ -1,5 +1,5 @@
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A 73-24-5 1.572302 0.8099 0.4735 0.1725 -0.54
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C 71-30-7 1.400624 0.6054 0.2638 -0.0257 NA
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G 73-40-5 1.884993 1.114 0.7487 0.4149 NA
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T 65-71-4 1.194189 0.3805 0.0564 -0.2136 -0.29
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U 66-22-8 1.157819 0.3428 0.0124 -0.2832 -0.22
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A 73-24-5 1.572302 0.818218 0.4735 0.1725 -0.54
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C 71-30-7 1.400624 0.613380 0.2638 -0.0257 NA
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G 73-40-5 1.884993 1.122363 0.7487 0.4149 NA
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T 65-71-4 1.194189 0.388406 0.0564 -0.2136 -0.29
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U 66-22-8 1.157819 0.350665 0.0124 -0.2832 -0.22
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@ -1,5 +1,5 @@
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A 73-24-5 -7.265806 -1.211220672617561E-002 -1.385783048793831E-002 -7.748 -6.440192825526375E-003 -6.479425433814747E-003 -7.975 -8.15 -8.33 -8.48
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C 71-30-7 -7.527844 -1.546006861838614E-002 -1.940117174135618E-002 -8.067 -7.060220080308114E-003 -7.369593245133022E-003 -8.287 -8.449 -9.512 -8.94
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G 73-40-5 -6.951933 -1.225096955803263E-002 -1.413179166956595E-002 -7.461 -6.514811691911574E-003 -6.600243767264653E-003 -7.691 -7.872 -8.034 -8.24
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T 65-71-4 -8.019884 -1.231951892771176E-002 -1.437665931379063E-002 -8.489 -6.688287305785518E-003 -6.858166603883720E-003 -8.708 -8.866 -9.081 -9.2
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U 66-22-8 -8.379481 -1.561024372083486E-002 -1.963262364674972E-002 -9.017 -8.522580585075183E-003 -9.357601174873545E-003 -9.223 -9.382 -10.125 -9.68
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A 73-24-5 -7.265806 -1.211220672617561E-002 -1.385783048793831E-002 -7.739974 -6.440192825526375E-003 -6.479425433814747E-003 -7.975 -8.15 -8.33 -8.48
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C 71-30-7 -7.527844 -1.546006861838614E-002 -1.940117174135618E-002 -8.060222 -7.060220080308114E-003 -7.369593245133022E-003 -8.287 -8.449 -9.512 -8.94
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G 73-40-5 -6.951933 -1.225096955803263E-002 -1.413179166956595E-002 -7.453525 -6.514811691911574E-003 -6.600243767264653E-003 -7.691 -7.872 -8.034 -8.24
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T 65-71-4 -8.019884 -1.231951892771176E-002 -1.437665931379063E-002 -8.481933 -6.688287305785518E-003 -6.858166603883720E-003 -8.708 -8.866 -9.081 -9.2
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U 66-22-8 -8.379481 -1.561024372083486E-002 -1.963262364674972E-002 -8.858697 -8.522580585075183E-003 -9.357601174873545E-003 -9.223 -9.382 -10.125 -9.68
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@ -72,6 +72,7 @@
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\newcommand{\RH}{R_{\ce{H2}}}
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\newcommand{\RF}{R_{\ce{F2}}}
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\newcommand{\RBeO}{R_{\ce{BeO}}}
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\newcommand{\CBS}{\text{CBS}}
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% orbital energies
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\newcommand{\nDIIS}{N^\text{DIIS}}
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@ -101,6 +102,7 @@
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\newcommand{\tB}{\Tilde{B}}
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\renewcommand{\S}[1]{S_{#1}}
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\newcommand{\G}[1]{G_{#1}}
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\newcommand{\F}[1]{F^{#1}}
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\newcommand{\Po}[1]{P_{#1}}
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\newcommand{\W}[1]{W_{#1}}
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\newcommand{\Wc}[1]{W^\text{c}_{#1}}
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@ -143,14 +145,44 @@
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\newcommand{\MO}[1]{\phi_{#1}}
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% coordinates
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\newcommand{\br}[1]{\mathbf{r}_{#1}}
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\renewcommand{\b}[1]{\mathbf{#1}}
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\renewcommand{\d}{\text{d}}
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\newcommand{\dbr}[1]{d\br{#1}}
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\renewcommand{\bra}[1]{\ensuremath{\langle #1 \vert}}
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\renewcommand{\ket}[1]{\ensuremath{\vert #1 \rangle}}
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\renewcommand{\braket}[2]{\ensuremath{\langle #1 \vert #2 \rangle}}
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% operators
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\newcommand{\hT}{\Hat{T}}
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\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
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% coordinates
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\newcommand{\br}[1]{\mathbf{r}_{#1}}
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\newcommand{\dbr}[1]{d\br{#1}}
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\newcommand{\Bas}{\mathcal{B}}
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\newcommand{\cD}{\mathcal{D}}
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\newcommand{\Ne}{N}
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\newcommand{\vne}{v_\text{ne}}
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\newcommand{\n}[2]{n_{#1}^{#2}}
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\newcommand{\E}[2]{E_{#1}^{#2}}
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\newcommand{\DE}[2]{\Delta E_{#1}^{#2}}
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\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
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\newcommand{\DbE}[2]{\Delta \Bar{E}_{#1}^{#2}}
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\newcommand{\bEc}[1]{\Bar{E}_\text{c,md}^{#1}}
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%\newcommand{\e}[2]{\varepsilon_{#1}^{#2}}
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%\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
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\newcommand{\bec}[1]{\Bar{e}^{#1}}
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\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
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%\newcommand{\W}[2]{W_{#1}^{#2}}
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\newcommand{\w}[2]{w_{#1}^{#2}}
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\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
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\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
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\newcommand{\V}[2]{V_{#1}^{#2}}
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\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
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% units
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\newcommand{\IneV}[1]{#1 eV}
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\newcommand{\InAU}[1]{#1 a.u.}
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\newcommand{\InAA}[1]{#1 \AA}
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\newcommand{\kcal}{kcal/mol}
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\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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@ -197,7 +229,7 @@ The $GW$ approximation stems from the acclaimed Hedin's equations \cite{Hedin_19
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\begin{subequations}
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\begin{align}
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\label{eq:G}
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& G(12) = G_\text{0}(12) + \int G_0(13) \Sigma(34) G(42) d(34),
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& G(12) = G_0(12) + \int G_0(13) \Sigma(34) G(42) d(34),
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\\
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\label{eq:Gamma}
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& \Gamma(123) = \delta(12) \delta(13)
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@ -225,22 +257,21 @@ Within the $GW$ approximation, one bypasses the calculation of the vertex correc
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Depending on the degree of self-consistency one is willing to perform, there exists several types of $GW$ calculations. \cite{Loos_2018}
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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 or fully self-consistent $GW$ methods have shown great promise. \cite{Ke_2011, Blase_2011, Faber_2011, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b, Koval_2014, Hung_2016, Blase_2018, Jacquemin_2017}
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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 Hyllerras \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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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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F12 methods are now routinely employed in computational chemistry and provide robust tools for electronic structure calculations where small basis sets may be used to obtain near complete basis set (CBS) limit accuracy. \cite{TewKloNeiHat-PCCP-07}
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The basis-set correction presented here follow a different avenue, and relies on the range-separated density-functional theory (RS-DFT) formalism to capture, thanks to a short-range correlation functional, the missing part of the short-range correlation effects. \cite{GinPraFerAssSavTou-JCP-18}
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As shown in recent studies on both ground- and excited-state properties, \cite{LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} similar to F12 methods, it significantly speeds up the convergence of energetics towards the CBS limit while avoiding the usage of large auxiliary basis sets.
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As shown in recent studies on both ground- and excited-state properties, \cite{LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} similar to F12 methods, it significantly speeds up the convergence of energetics towards the CBS limit while avoiding the usage of large auxiliary basis sets that are used in F12 methods to avoid the numerous three- and four-electron integrals. \cite{KonBisVal-CR-12, HatKloKohTew-CR-12, TenNog-WIREs-12, GruHirOhnTen-JCP-17, Barca_2018}
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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 on ionization potentials (IPs) obtained within {\GOWO}.
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Note that the the present basis set correction can be straightforwardly applied to other properties (e.g., electron affinities and fundamental gap), as well as other flavours of $GW$ 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{Dreuw_2005} and Bethe-Salpeter equation (BSE) formalism. \cite{Strinati_1988, Leng_2016, Blase_2018}
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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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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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The paper is organised as follows.
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In Sec.~\ref{sec:theory}, we provide details about the theory behind the present basis set correction and its adaptation to $GW$ methods.
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@ -254,91 +285,69 @@ Unless otherwise stated, atomic units are used throughout.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Many-body Green's function theory with DFT basis-set correction}
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\subsection{MBPT with DFT basis set correction}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we start by defining, for a $N$-electron system with nuclei-electron potential $v_\text{ne}(\b{r})$, the approximate ground-state energy for one-electron densities $n$ which are ``representable'' in a finite basis set ${\cal B}$
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we start by defining, for a $\Ne$-electron system with nuclei-electron potential $\vne(\b{r})$, the approximate ground-state energy for one-electron densities $\n{}{}$ which are ``representable'' in a finite basis set $\Bas$
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\begin{equation}
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E_0^{\cal B} = \min_{n \in {\cal D}^{\cal B}} \left\{ F[n] + \int v_\text{ne}(\b{r}) n(\b{r}) \d\b{r}\right\},
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\E{0}{\Bas} = \min_{\n{}{} \in \cD^\Bas} \qty{ F[n] + \int \vne(\br{}) \n{}{}(\br{}) \dbr{} },
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\label{E0B}
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\end{equation}
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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
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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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\begin{equation}
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F[n] = F^{\cal B}[n] + \bar{E}^{\cal B}[n],
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F[n] = F^{\Bas}[\n{}{}] + \bE{}{\Bas}[\n{}{}],
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\label{Fn}
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\end{equation}
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where $F^{\cal B}[n]$ is the Levy-Lieb density functional with wave functions $\Psi^{\cal B}$ expandable in the Hilbert space generated by ${\cal B}$
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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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\begin{equation}
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F^{\cal B}[n] = \min_{\Psi^{\cal B}\to n} \bra{\Psi^{\cal B}} \hat{T} + \hat{W}_\text{ee}\ket{\Psi^{\cal B}},
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F^\Bas[n] = \min_{\Psi^\Bas \to n} \mel*{\Psi^\Bas}{ \hT + \hWee{}}{\Psi^\Bas},
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\end{equation}
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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 $N$-representable one-electron Green's functions $G^{\cal B}(\b{r},\b{r}',\omega)$ representable in the basis set ${\cal B}$
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and $\bar{E}^{\Bas}[n]$ 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 contrained search over $N$-representable one-electron Green's functions $G^\Bas(\b{r},\b{r}',\omega)$ representable in the basis set $\Bas$
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\begin{equation}
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F^{\cal B}[n] = \min_{G^{\cal B}\to n} \Omega^{\cal B}[G^{\cal B}],
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F^\Bas[n] = \min_{G^\Bas\to n} \Omega^\Bas[G^\Bas],
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\label{FBn}
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\end{equation}
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where $\Omega^{\cal B}[G]$ is chosen to be a Klein-like energy functional of the Green's function (see, e.g., Refs.~\onlinecite{SteLee-BOOK-13,MarReiCep-BOOK-16,DahLee-JCP-05,DahLeeBar-IJQC-05,DahLeeBar-PRA-06})
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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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\begin{equation}
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\Omega^{\cal B}[G] = \Tr \left[\ln ( - G ) \right] - \Tr \left[ (G_\text{s}^{\cal B})^{-1} G -1 \right] + \Phi_\text{Hxc}^{\cal B}[G],
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\Omega^\Bas[G] = \Tr[\ln ( - G ) ] - \Tr[ (G_\text{s}^\Bas)^{-1} G -1 ] + \Phi_\text{Hxc}^\Bas[G],
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\label{OmegaB}
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\end{equation}
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where $(G_\text{s}^{\cal B})^{-1}$ is the projection into ${\cal B}$ of the inverse free-particle Green's function $(G_\text{s})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 )\delta(\b{r}-\b{r}')$ and we have used the notation $\Tr [A B] = 1/(2\pi i) \int_{-\infty}^{+\infty} \! \d \omega \, e^{i \omega 0^+} \! \iint \! \d \b{r} \d \b{r}' A(\b{r},\b{r}',\omega) B(\b{r}',\b{r},\omega)$. In Eq.~(\ref{OmegaB}), $\Phi_\text{Hxc}^{\cal B}[G]$ 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: $\delta \Phi_\text{Hxc}^{\cal B}[G]/\delta G(\b{r},\b{r}',\omega) = \Sigma_\text{Hxc}^{\cal B}[G](\b{r},\b{r}',\omega)$. Inserting Eqs.~(\ref{Fn}) and~(\ref{FBn}) into Eq.~(\ref{E0B}), we finally arrive at
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where $(G_\text{s}^\Bas)^{-1}$ is the projection into $\Bas$ of the inverse free-particle Green's function $(G_\text{s})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 )\delta(\b{r}-\b{r}')$ and we have used the notation $\Tr [A B] = 1/(2\pi i) \int_{-\infty}^{+\infty} \! \d \omega \, e^{i \omega 0^+} \! \iint \! \d \b{r} \d \b{r}' A(\b{r},\b{r}',\omega) B(\b{r}',\b{r},\omega)$. In Eq.~(\ref{OmegaB}), $\Phi_\text{Hxc}^\Bas[G]$ 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: $\delta \Phi_\text{Hxc}^\Bas[G]/\delta G(\b{r},\b{r}',\omega) = \Sigma_\text{Hxc}^\Bas[G](\b{r},\b{r}',\omega)$. Inserting Eqs.~(\ref{Fn}) and~(\ref{FBn}) into Eq.~(\ref{E0B}), we finally arrive at
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\begin{equation}
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E_0^{\cal B} = \min_{G^{\cal B}} \left\{ \Omega^{\cal B}[G^{\cal B}] + \int v_\text{ne}(\b{r}) n_{G^{\cal B}}(\b{r}) \d\b{r} + \bar{E}^{\cal B}[n_{G^{\cal B}}] \right\},
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E_0^\Bas = \min_{G^\Bas} \left\{ \Omega^\Bas[G^\Bas] + \int v_\text{ne}(\b{r}) n_{G^\Bas}(\b{r}) \d\b{r} + \bar{E}^\Bas[n_{G^\Bas}] \right\},
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\label{E0BGB}
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\end{equation}
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where the minimization is over $N$-representable one-electron Green's functions $G^{\cal B}(\b{r},\b{r}',\omega)$ representable in the basis set ${\cal B}$.
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where the minimization is over $N$-representable one-electron Green's functions $G^\Bas(\b{r},\b{r}',\omega)$ representable in the basis set $\Bas$.
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The stationary condition from Eq.~(\ref{E0BGB}) gives the following Dyson equation
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\begin{equation}
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(G^{\cal B})^{-1} = (G_\text{0}^{\cal B})^{-1}- \Sigma_\text{Hxc}^{\cal B}[G^{\cal B}]- \bar{\Sigma}^{\cal B}[n_{G^{\cal B}}],
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(G^\Bas)^{-1} = (G_\text{0}^\Bas)^{-1}- \Sigma_\text{Hxc}^\Bas[G^\Bas]- \bar{\Sigma}^\Bas[n_{G^\Bas}],
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\label{Dyson}
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\end{equation}
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where $(G_\text{0}^{\cal B})^{-1}$ is the basis projection of the inverse non-interacting Green's function with potential $v_\text{ne}(\b{r})$,
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$(G_\text{0})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 + v_\text{ne}(\b{r}) + \lambda)\delta(\b{r}-\b{r}')$ with the chemical potential $\lambda$, and $\bar{\Sigma}^{\cal B}$ is a frequency-independent local self-energy coming from functional derivative of the complementary basis-correction density functional
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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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$(G_\text{0})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 + v_\text{ne}(\b{r}) + \lambda)\delta(\b{r}-\b{r}')$ with the chemical potential $\lambda$, and $\bar{\Sigma}^\Bas$ is a frequency-independent local self-energy coming from functional derivative of the complementary basis-correction density functional
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\begin{equation}
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\bar{\Sigma}^{\cal B}[n](\b{r},\b{r}') = \bar{v}^{\cal B}[n](\b{r}) \delta(\b{r}-\b{r}'),
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\bar{\Sigma}^\Bas[n](\b{r},\b{r}') = \bar{v}^\Bas[n](\b{r}) \delta(\b{r}-\b{r}'),
|
||||
\end{equation}
|
||||
with $\bar{v}^{\cal B}[n](\b{r}) = \delta \bar{E}^{\cal B}[n] / \delta n(\b{r})$. The solution of the Dyson equation~(\ref{Dyson}) gives the Green's function $G^{\cal B}(\b{r},\b{r}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bar{\Sigma}^{\cal B}[n]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bar{\Sigma}^{\cal B}$. Of course, in the complete-basis-set (CBS) limit, the basis-set correction vanishes, $\bar{\Sigma}^{{\cal B}\to \text{CBS}}=0$, and the Green's function becomes exact, $G^{{\cal B}\to \text{CBS}}=G$.
|
||||
with $\bar{v}^\Bas[n](\br{}) = \delta \bE{}{\Bas}[n] / \delta \n{}{}(\br{})$. The solution of the Dyson equation~(\ref{Dyson}) gives the Green's function $G^\Bas(\b{r},\b{r}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bar{\Sigma}^\Bas[n]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bar{\Sigma}^\Bas$. Of course, in the complete-basis-set (CBS) limit, the basis-set correction vanishes, $\bar{\Sigma}^{\Bas\to \CBS} = 0$, and the Green's function becomes exact, $G^{\Bas\to \CBS}=G$.
|
||||
|
||||
%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''
|
||||
%\end{equation}
|
||||
%
|
||||
%\begin{equation}
|
||||
%n_G(r'') = i \int G(r'',r'',w) d\omega
|
||||
%\end{equation}
|
||||
%
|
||||
%
|
||||
%\begin{equation}
|
||||
%\fdv{n_G(r'')}{G(r,r',w)} = \delta(r -r') \delta (r'-r'')
|
||||
%\end{equation}
|
||||
%
|
||||
%
|
||||
%\begin{equation}
|
||||
%\begin{split}
|
||||
% \fdv{E[n_G]}{G(r,r',w)}
|
||||
% & = \int \fdv{E[n_G]}{n(r'')} \delta(r -r') \delta (r'-r'') dr''
|
||||
% \\
|
||||
% & = \fdv{E[n_G]}{n(r)} \delta(r -r')
|
||||
% \\
|
||||
% & = v[n_G](r) \delta(r -r')
|
||||
%\end{split}
|
||||
%\end{equation}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{The $GW$ Approximation}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
The Dyson equation can be written with an arbitrary reference
|
||||
\begin{equation}
|
||||
(G^{\cal B})^{-1} = (G_\text{ref}^{\cal B})^{-1}- \left( \Sigma_\text{Hxc}^{\cal B}[G^{\cal B}]- \Sigma_\text{ref}^{\cal B} \right) - \bar{\Sigma}^{\cal B}[n_{G^{\cal B}}],
|
||||
(G^\Bas)^{-1} = (G_\text{ref}^\Bas)^{-1}- \left( \Sigma_\text{Hxc}^\Bas[G^\Bas]- \Sigma_\text{ref}^\Bas \right) - \bar{\Sigma}^\Bas[n_{G^\Bas}],
|
||||
\end{equation}
|
||||
where $(G_\text{ref}^{\cal B})^{-1} = (G_\text{0}^{\cal B})^{-1} - \Sigma_\text{ref}^{\cal B}$. For example, if the reference is Hartree-Fock (HF), $\Sigma_\text{ref}^{\cal B}(\b{r},\b{r}') = \Sigma_\text{Hx,HF}^{\cal B}(\b{r},\b{r}')$ is the HF nonlocal self-energy, and if the reference is Kohn-Sham, $\Sigma_\text{ref}^{\cal B}(\b{r},\b{r}') = v_\text{Hxc}^{\cal B}(\b{r}) \delta(\b{r}-\b{r}')$ is the local Hxc potential.
|
||||
where $(G_\text{ref}^\Bas)^{-1} = (G_\text{0}^\Bas)^{-1} - \Sigma_\text{ref}^\Bas$. For example, if the reference is Hartree-Fock (HF), $\Sigma_\text{ref}^\Bas(\b{r},\b{r}') = \Sigma_\text{Hx,HF}^\Bas(\b{r},\b{r}')$ is the HF nonlocal self-energy, and if the reference is Kohn-Sham, $\Sigma_\text{ref}^\Bas(\b{r},\b{r}') = v_\text{Hxc}^\Bas(\b{r}) \delta(\b{r}-\b{r}')$ is the local Hxc potential.
|
||||
|
||||
|
||||
Here, we provide self-contained summary of the main equations and quantities behind {\GOWO} and {\evGW}.
|
||||
Here, we provide self-contained summary of the main equations and quantities behind {\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 is conveniently split in its hole (h) and particle (p) contributions
|
||||
\begin{equation}
|
||||
\label{eq:SigC}
|
||||
@ -391,64 +400,23 @@ with
|
||||
B_{ia,jb} & = 2 (ia|bj),
|
||||
\end{align}
|
||||
and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
|
||||
The one-electron energies $\epsilon_p$ in \eqref{eq:SigCh}, \eqref{eq:SigCp} and \eqref{eq:RPA} are either the HF or the $GW$ quasiparticle energies.
|
||||
The one-electron energies $\epsilon_p$ in \eqref{eq:SigCh}, \eqref{eq:SigCp} and \eqref{eq:RPA} are either the HF or KS orbital energies.
|
||||
Equation \eqref{eq:LR} also provides the neutral excitation energies $\Om{x}$.
|
||||
|
||||
In practice, there exist two ways of determining the {\GOWO} QP energies. \cite{Hybertsen_1985a, vanSetten_2013}
|
||||
In its ``graphical'' version, they are provided by one of the many solutions of the (non-linear) QP equation
|
||||
The {\GOWO} QP energies are provided by one of the many solutions of the (non-linear) QP equation \cite{Hybertsen_1985a, vanSetten_2013}
|
||||
\begin{equation}
|
||||
\label{eq:QP-G0W0}
|
||||
\omega = \eHF{p} + \Re[\SigC{p}(\omega)].
|
||||
\omega = \epsilon_{p} + \Re[\SigC{p}(\omega)].
|
||||
\end{equation}
|
||||
In this case, special care has to be taken in order to select the ``right'' solution, known as the QP solution.
|
||||
In particular, it is usually worth calculating its renormalization weight (or factor), $\Z{p}(\eHF{p})$, where
|
||||
In this case, special care has to be taken in order to select the ``right'' solution, known as the QP solution. \cite{Veril_2019}
|
||||
In particular, it is usually worth calculating its renormalization weight (or factor)
|
||||
\begin{equation}
|
||||
\label{eq:Z}
|
||||
\Z{p}(\omega) = \qty[ 1 - \pdv{\Re[\SigC{p}(\omega)]}{\omega} ]^{-1}.
|
||||
\Z{p} = \qty[ 1 - \left. \pdv{\Re[\SigC{p}(\omega)]}{\omega} \right|_{\omega = \epsilon_{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 a well-behaved case (belonging to the weakly correlated regime), the QP weight is much larger than the sum of the satellite weights, and of the order of $0.7$-$0.9$.
|
||||
In a well-behaved case, the QP weight is much larger than the sum of the satellite weights, and of the order of $0.7$-$0.9$.
|
||||
|
||||
Within the linearized version of {\GOWO}, one assumes that
|
||||
\begin{equation}
|
||||
\label{eq:SigC-lin}
|
||||
\SigC{p}(\omega) \approx \SigC{p}(\eHF{p}) + (\omega - \eHF{p}) \left. \pdv{\SigC{p}(\omega)}{\omega} \right|_{\omega = \eHF{p}},
|
||||
\end{equation}
|
||||
that is, the self-energy behaves linearly in the vicinity of $\omega = \eHF{p}$.
|
||||
Substituting \eqref{eq:SigC-lin} into \eqref{eq:QP-G0W0} yields
|
||||
\begin{equation}
|
||||
\label{eq:QP-G0W0-lin}
|
||||
\eGOWO{p} = \eHF{p} + \Z{p}(\eHF{p}) \Re[\SigC{p}(\eHF{p})].
|
||||
\end{equation}
|
||||
Unless otherwise stated, in the remaining of this paper, the {\GOWO} QP energies are determined via the linearized method.
|
||||
|
||||
In the case of {\evGW}, the QP energy, $\eGW{p}$, are obtained via Eq.~\eqref{eq:QP-G0W0}, which has to be solved self-consistently due to the QP energy dependence of the self-energy [see Eq.~\eqref{eq:SigC}]. \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011}
|
||||
At least in the weakly correlated regime where a clear QP solution exists, we believe that, within {\evGW}, the self-consistent algorithm should select the solution of the QP equation \eqref{eq:QP-G0W0} with the largest renormalization weight $\Z{p}(\eGW{p})$.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\subsection{Basis Set Correction}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%The present basis set correction is a two-level correction.
|
||||
%First, one has to correct the neutral excitations $\Om{x}$ from the RPA calculation.
|
||||
%The corrected matrix elements read
|
||||
%\begin{align}
|
||||
%\label{eq:RPA}
|
||||
% \tA_{ia,jb} & = \A{ia,jb} + (ia|\fc|jb),
|
||||
% &
|
||||
% \tB_{ia,jb} & = \B{ia,jb} + (ia|\fc|bj),
|
||||
%\end{align}
|
||||
%where the elements $\A{ia,jb}$ and $\B{ia,jb}$ are given by Eq.~\eqref{eq:RPA}.
|
||||
%\begin{equation}
|
||||
% \fc(\br{1},\br{2})= \frac{\delta^2 \Ec}{\delta n(\br{1})\delta n(\br{2})}
|
||||
%\end{equation}
|
||||
%In a second time, we correct the GW energy
|
||||
%\begin{equation}
|
||||
% \tSigC{p} = \SigC{p} + (p|\Vc|p)
|
||||
%\end{equation}
|
||||
%with
|
||||
%\begin{equation}
|
||||
% \Vc(\br{}) = \fdv{\Ec}{n(\br{})}
|
||||
%\end{equation}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Computational details}
|
||||
@ -459,7 +427,7 @@ Unless otherwise stated, all the {\GOWO} calculations have been performed with M
|
||||
The HF, PBE and PBE0 calculations as well as the srLDA and srPBE basis set corrections have been computed with Quantum Package, \cite{QP2} which by default uses the SG-2 quadrature grid for the numerical integrations.
|
||||
Frozen-core (FC) calculations are systematically performed.
|
||||
The FC density-based correction is used consistently with the FC approximation in the {\GOWO} calculations.
|
||||
The {\GOWO} quasiparticle energies have been obtained ``graphically'', i.e., by solving the non-linear, frequency-dependent quasiparticle equation (without linearization).
|
||||
The {\GOWO} quasiparticle energies have been obtained ``graphically'', \textit{i.e.}, by solving the non-linear, frequency-dependent quasiparticle equation (without linearization).
|
||||
Moreover, the infinitesimal $\eta$ has been set to zero.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
@ -470,6 +438,13 @@ 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}
|
||||
@ -586,7 +561,7 @@ The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purp
|
||||
{\GOWO}@PBE\fnm[1] & def2-SVP & 7.27[-0.88] & 7.53[-0.92] & 6.95[-0.92] & 8.02[-0.85] & 8.38[-1.00] \\
|
||||
{\GOWO}@PBE+srLDA\fnm[1] & def2-SVP & 7.60[-0.55] & 7.95[-0.50] & 7.29[-0.59] & 8.36[-0.51] & 8.80[-0.58] \\
|
||||
{\GOWO}@PBE+srPBE\fnm[1] & def2-SVP & 7.64[-0.51] & 8.06[-0.39] & 7.34[-0.54] & 8.41[-0.45] & 8.91[-0.47] \\
|
||||
{\GOWO}@PBE\fnm[2] & def2-TZVP & 7.75[-0.40] & 8.07[-0.38] & 7.46[-0.41] & 8.49[-0.38] & 9.02[-0.37] \\
|
||||
{\GOWO}@PBE\fnm[1] & def2-TZVP & 7.74[-0.41] & 8.06[-0.39] & 7.45[-0.42] & 8.48[-0.38] & 8.86[-0.52] \\
|
||||
{\GOWO}@PBE+srLDA\fnm[1] & def2-TZVP & 7.92[-0.23] & 8.26[-0.19] & 7.64[-0.23] & 8.67[-0.20] & 9.25[-0.13] \\
|
||||
{\GOWO}@PBE+srPBE\fnm[1] & def2-TZVP & 7.92[-0.23] & 8.27[-0.18] & 7.64[-0.23] & 8.68[-0.19] & 9.27[-0.11] \\
|
||||
{\GOWO}@PBE\fnm[2] & def2-QZVP & 7.98[-0.18] & 8.29[-0.16] & 7.69[-0.18] & 8.71[-0.16] & 9.22[-0.16] \\
|
||||
@ -639,10 +614,10 @@ The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purp
|
||||
|
||||
%%% FIG 2 %%%
|
||||
\begin{figure*}
|
||||
\includegraphics[width=\linewidth]{DNA}
|
||||
\includegraphics[width=\linewidth]{DNA_IP}
|
||||
\caption{
|
||||
Error (in eV) with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values fort the IPs of the five canonical nucleobases (adenine, cytosine, thymine, guanine and uracil) computed at the {\GOWO}@PBE level of theory for various basis sets.
|
||||
\label{fig:DNA}
|
||||
\label{fig:DNA_IP}
|
||||
}
|
||||
\end{figure*}
|
||||
|
||||
|
||||
10451
Notebooks/GW100.nb
10451
Notebooks/GW100.nb
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