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%% This BibTeX bibliography file was created using BibDesk.
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Abstract = {"Recent progress in the theory and computation of electronic structure is bringing an unprecedented level of capability for research. Many-body methods are becoming essential tools vital for quantitative calculations and understanding materials phenomena in physics, chemistry, materials science and other fields. This book provides a unified exposition of the most-used tools: many-body perturbation theory, dynamical mean field theory and quantum Monte Carlo simulations. Each topic is introduced with a less technical overview for a broad readership, followed by in-depth descriptions and mathematical formulation"--},
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Manuscript/GW-srDFT.tex
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@ -32,14 +32,6 @@
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
\newcommand{\AS}[1]{\toto{(\underline{\bf AS}: #1)}}
\usepackage{hyperref}
\hypersetup{
colorlinks=true,
linkcolor=blue,
filecolor=blue,
urlcolor=blue,
citecolor=blue
}
\newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
@ -184,7 +176,7 @@ We also compute the ionization potentials of the five canonical nucleobases (ade
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%
The purpose of many-body perturbation theory (MBPT) based on Green functions is to solve the formidable many-body problem by adding the electron-electron Coulomb interaction perturbatively starting from an independent-particle model. \cite{MarReiCep-BOOK-16} In this approach, the \textit{screening} of the Coulomb interaction is an essential quantity that is responsible for a rich variety of phenomena that would be otherwise absent (such as quasiparticle satellites and lifetimes). \cite{Aryasetiawan_1998, Onida_2002, Reining_2017}
The purpose of many-body perturbation theory (MBPT) based on Green functions is to solve the formidable many-body problem by adding the electron-electron Coulomb interaction perturbatively starting from an independent-particle model. \cite{Martin_2016} In this approach, the \textit{screening} of the Coulomb interaction is an essential quantity that is responsible for a rich variety of phenomena that would be otherwise absent (such as quasiparticle satellites and lifetimes). \cite{Aryasetiawan_1998, Onida_2002, Reining_2017}
%\jt{Is it the screened Coulomb interaction which is responsible for these or more generally the Coulomb interaction? I would say the second.}
The so-called {\GW} approximation is the workhorse of MBPT and has a long and successful history in the calculation of the electronic structure of solids. \cite{Aryasetiawan_1998, Onida_2002, Reining_2017}
@ -224,17 +216,17 @@ Although obviously starting-point dependent, \cite{Bruneval_2013, Jacquemin_2016
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.
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}
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{Kato_1957}) and, more specifically, the Coulomb correlation hole around it.
Pioneered by Hylleraas \cite{Hylleraas_1929} in the 1930's and popularized in the 1990's by Kutzelnigg and coworkers \cite{Kutzelnigg_1985, Noga_1994, Kutzelnigg_1991} (and subsequently others \cite{Kong_2012, Hattig_2012, Tenno_2012a, Tenno_2012b, Gruneis_2017}), the so-called F12 methods overcome this slow convergence by employing geminal basis functions that closely resemble the correlation holes in electronic wave functions.
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{Tew_2007}
The basis-set correction presented here follow a different route, 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}
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 the 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}
The basis-set correction presented here follow a different route, 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{Giner_2018}
As shown in recent studies on both ground- and excited-state properties, \cite{Loos_2019, Giner_2019} similar to F12 methods, it significantly speeds up the convergence of energetics towards the CBS limit while avoiding the usage of the large auxiliary basis sets that are used in F12 methods to avoid the numerous three- and four-electron integrals. \cite{Kong_2012, Hattig_2012, Tenno_2012a, Tenno_2012b, Gruneis_2017, Barca_2018}
Explicitly correlated F12 correction schemes have been derived for second-order Green 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 obtained within {\GOWO}.
In the present manuscript, we illustrate the performance of the density-based basis set correction developed in Refs.~\onlinecite{Giner_2018, Loos_2019, Giner_2019} on ionization potentials obtained within {\GOWO}.
Note that the present basis set correction can be straightforwardly applied to other properties (\textit{e.g.}, electron affinities and fundamental gap), as well as other flavors of (self-consistent) {\GW} or Green function-based methods, such as GF2 (and its higher-order variants).
The paper is organized as follows.
@ -252,7 +244,7 @@ Unless otherwise stated, atomic units are used throughout.
\subsection{MBPT with DFT basis set correction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we start by defining, for a $\Ne$-electron system with nuclei-electron potential $\vne(\br{})$, the approximate ground-state energy for one-electron densities $\n{}{}$ which are ``representable'' in a finite basis set $\Bas$
Following Ref.~\onlinecite{Giner_2018}, we start by defining, for a $\Ne$-electron system with nuclei-electron potential $\vne(\br{})$, the approximate ground-state energy for one-electron densities $\n{}{}$ which are ``representable'' in a finite basis set $\Bas$
\begin{equation}
\E{0}{\Bas} = \min_{\n{}{} \in \cD^\Bas} \qty{ \F{}{}[n] + \int \vne(\br{}) \n{}{}(\br{}) \dbr{} },
\label{eq:E0B}
@ -262,7 +254,7 @@ In this expression,
\begin{equation}
\F{}{}[n] = \min_{\wf{}{} \rightsquigarrow \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}}
\end{equation}
is the exact Levy-Lieb universal density functional~\cite{Lev-PNAS-79, Lev-PRA-82, Lie-IJQC-83}, where the notation $\wf{}{} \rightsquigarrow \n{}{}$ in Eq.~\eqref{eq:E0B} states that $\wf{}{}$ yields the one-electron density $\n{}{}$.
is the exact Levy-Lieb universal density functional, \cite{Levy_1979, Levy_1982, Lieb_1983} 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.
The exact Levy-Lieb universal density functional is then decomposed as
\begin{equation}
@ -273,13 +265,13 @@ where $\F{}{\Bas}[\n{}{}]$ is the Levy-Lieb density functional with wave functio
\begin{equation}
\F{}{\Bas}[\n{}{}] = \min_{\wf{}{\Bas} \rightsquigarrow \n{}{}} \mel*{\wf{}{\Bas}}{ \hT + \hWee{}}{\wf{}{\Bas}},
\end{equation}
and $\bE{}{\Bas}[\n{}{}]$ is the complementary basis-correction density functional. \cite{GinPraFerAssSavTou-JCP-18}
and $\bE{}{\Bas}[\n{}{}]$ is the complementary basis-correction density functional. \cite{Giner_2018}
In the present work, instead of using wave-function methods for calculating $\F{}{\Bas}[\n{}{}]$, we use Green-function methods. We assume that there exists a functional $\Omega^\Bas[\G{}{\Bas}]$ of $\Ne$-representable one-electron Green functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$ and yielding the density $n$ which gives $\F{}{\Bas}[\n{}{}]$ at a stationary point
\begin{equation}
\F{}{\Bas}[\n{}{}] = \stat{\G{}{\Bas} \rightsquigarrow \n{}{}} \Omega^\Bas[\G{}{\Bas}].
\label{eq:FBn}
\end{equation}
The reason why we use a stationary condition rather than a minimization condition is that only a stationary property is generally known for functionals of the Green function. For example, we can choose for $\Omega^\Bas[G]$ a Klein-like energy functional (see, \textit{e.g.}, Refs.~\onlinecite{SteLee-BOOK-13, MarReiCep-BOOK-16, DahLee-JCP-05, DahLeeBar-IJQC-05, DahLeeBar-PRA-06})
The reason why we use a stationary condition rather than a minimization condition is that only a stationary property is generally known for functionals of the Green function. For example, we can choose for $\Omega^\Bas[G]$ a Klein-like energy functional (see, \textit{e.g.}, Refs.~\onlinecite{Stefanucci_2013, Martin_2016, Dahlen_2005, Dahlen_2005a, Dahlen_2006})
\begin{equation}
\Omega^\Bas[\G{}{}] = \Tr[\ln( - \G{}{} ) ] - \Tr[ (\Gs{\Bas})^{-1} \G{}{} - 1 ] + \Phi_\Hxc^\Bas[\G{}{}],
\label{eq:OmegaB}
@ -431,9 +423,9 @@ where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO orbital energies
%%%%%%%%%%%%%%%%%%%%%%%%
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 short-range local-density approximation ($\srLDA$) functional with multideterminant reference \cite{TouGorSav-TCA-05,PazMorGorBac-PRB-06} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) correlation functional \cite{FerGinTou-JCP-19,LooPraSceTouGin-JPCL-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} using the on-top pair density from the uniform-electron gas. \cite{LooPraSceTouGin-JPCL-19}
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 and is computed using the opposite-spin on-top pair density.
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.
Here, we employ two types of complementary, short-range correlation functionals $\bE{}{\Bas}[\n{}{}]$: a short-range local-density approximation ($\srLDA$) functional with multideterminant reference \cite{Toulouse_2005, Paziani_2006} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) correlation functional \cite{Ferte_2019, Loos_2019} which interpolates between the usual PBE functional \cite{Perdew_1996} at $\mu = 0$ and the exact large-$\mu$ behavior~\cite{Toulouse_2004, Gori-Giorgi_2006, Paziani_2006} using the on-top pair density from the uniform-electron gas. \cite{Loos_2019}
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 and is computed using the one-electron density and opposite-spin on-top pair density originating from the HF or KS orbitals obtained in the basis set $\Bas$.
We refer the interested reader to Refs.~\onlinecite{Giner_2018, Loos_2019, Giner_2019} 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 are thus given by
\begin{equation}
@ -546,11 +538,11 @@ All the geometries have been extracted from the GW100 set. \cite{vanSetten_2015}
Unless otherwise stated, all the {\GOWO} calculations have been performed with the MOLGW software developed by Bruneval and coworkers. \cite{Bruneval_2016a}
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 basis set correction~\cite{LooPraSceTouGin-JPCL-19} is used consistently with the FC approximation in the {\GOWO} calculations.
The FC density-based basis set correction~\cite{Loos_2019} is used consistently with the FC approximation in the {\GOWO} calculations.
The {\GOWO} quasiparticle energies have been obtained ``graphically'', \textit{i.e.}, by solving the non-linear, frequency-dependent quasiparticle equation \eqref{eq:QP-G0W0} (without linearization).
Moreover, the infinitesimal $\eta$ in Eq.~\eqref{eq:SigC} has been set to zero.
Compared to the conventional $\order*{\Nbas^6}$ computational cost of {\GW} (where $\Nbas$ is the number of basis functions), the present basis set correction represents a marginal additional cost as further discussed in Refs.~\onlinecite{LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19}.
Compared to the conventional $\order*{\Nbas^6}$ computational cost of {\GW} (where $\Nbas$ is the number of basis functions), the present basis set correction represents a marginal additional cost as further discussed in Refs.~\onlinecite{Loos_2019, Giner_2019}.
Note, however, that the formal $\order*{\Nbas^6}$ cost of {\GW} can be significantly reduced thanks to resolution-of-the-identity techniques \cite{vanSetten_2013, Bruneval_2016, Duchemin_2017} and other tricks. \cite{Rojas_1995, Duchemin_2019}
%%% FIG 1 %%%
@ -559,7 +551,7 @@ Note, however, that the formal $\order*{\Nbas^6}$ cost of {\GW} can be significa
\hspace{1cm}
\includegraphics[width=0.45\linewidth]{IP_G0W0PBE0_H2O}
\caption{
IP (in eV) of the water molecule computed at the {\GOWO} (black circles), {\GOWO}+srLDA (red squares), and {\GOWO}+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets \cite{Dun-JCP-89} (cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pV5Z) with two different starting points: HF (left) and PBE0 (right).
IP (in eV) of the water molecule computed at the {\GOWO} (black circles), {\GOWO}+srLDA (red squares), and {\GOWO}+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets \cite{Dunning_1989} (cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pV5Z) with two different starting points: HF (left) and PBE0 (right).
The thick black line represents the CBS value obtained by extrapolation (see text for more details).
The green area corresponds to chemical accuracy (\textit{i.e.}, error below $1$ {\kcal} or $0.043$ eV).
\label{fig:IP_G0W0_H2O}
@ -604,7 +596,7 @@ Very similar conclusions are drawn at the {\GOWO}@{\PBEO} level (see Table \ref{
For example, at the {\GOWO}@PBE0+srLDA/cc-pVQZ level, the MAD is only $0.02$ eV with a maximum error as small as $0.09$ eV.
It is worth pointing out that, for ground-state properties such as atomization and correlation energies, the density-based correction brought a more significant basis set reduction.
For example, we evidenced in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} that quintuple-$\zeta$ quality atomization and correlation energies are recovered with triple-$\zeta$ basis sets.
For example, we evidenced in Ref.~\onlinecite{Loos_2019} that quintuple-$\zeta$ quality atomization and correlation energies are recovered with triple-$\zeta$ basis sets.
Here, the overall gain seems to be less important.
The potential reasons for this could be: i) potential-based DFT corrections are usually less accurate than the ones based directly on energies, and ii) because the present scheme only corrects the basis set incompleteness error originating from the electron-electron cusp, some incompleteness remains at the HF or KS level.
@ -703,7 +695,7 @@ This is quite remarkable as the number of basis functions jumps from $371$ to $7
\section{Conclusion}
\label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%
In the present manuscript, we have shown that the density-based basis set correction developed by some of the authors in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and applied recently to ground- and excited-state properties \cite{LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} can also be successfully applied to Green function methods such as {\GW}.
In the present manuscript, we have shown that the density-based basis set correction developed by some of the authors in Ref.~\onlinecite{Giner_2018} and applied recently to ground- and excited-state properties \cite{Loos_2019, Giner_2019} can also be successfully applied to Green function methods such as {\GW}.
In particular, we have evidenced that the present basis set correction (which relies on LDA- or PBE-based short-range correlation functionals) significantly speeds up the convergence of IPs for small and larger molecules towards the CBS limit.
These findings have been observed for different {\GW} starting points (HF, PBE, and PBE0).
@ -727,6 +719,6 @@ This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the
\end{acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%
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