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Manuscript/GW-srDFT.tex
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@ -143,6 +143,7 @@
\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}
\newcommand{\IUF}{Institut Universitaire de France, Paris, France}
\begin{document}
@ -161,6 +162,7 @@
\author{Julien Toulouse}
\email[Corresponding author: ]{toulouse@lct.jussieu.fr}
\affiliation{\LCT}
\affiliation{\IUF}
\begin{abstract}
%\begin{wrapfigure}[13]{o}[-1.25cm]{0.5\linewidth}
@ -168,7 +170,7 @@
% \includegraphics[width=\linewidth]{TOC}
%\end{wrapfigure}
Similar to other electron correlation methods, many-body perturbation theory methods based on Green functions, such as the so-called $GW$ approximation, suffer from the usual slow convergence of energetic properties with respect to the size of the one-electron basis set.
This displeasing feature is due to lack of explicit electron-electron terms modeling the infamous Kato's electron-electron cusp and the correlation Coulomb hole around it.
This displeasing feature is due to lack of explicit electron-electron terms modeling the infamous Kato electron-electron cusp and the correlation Coulomb hole around it.
Here, we propose a computationally efficient density-based basis set correction based on short-range correlation density functionals which significantly speeds up the convergence of energetics towards the complete basis set limit.
The performance of this density-based correction is illustrated by computing the ionization potentials of the twenty smallest atoms and molecules of the GW100 test set at the perturbative $GW$ (or $G_0W_0$) level using increasingly large basis sets.
We also compute the ionization potentials of the five canonical nucleobase (adenine, cytosine, thymine, guanine, and uracil) and show that, here again, a significant improvement is obtained.
@ -206,8 +208,8 @@ The {\GW} approximation stems from the acclaimed Hedin's equations \cite{Hedin_1
& \Sig{}{}(12) = i \int \G{}{}(13) \W{}{}(14) \Gam{}{}(324) d(34),
\end{align}
\end{subequations}
which connects the Green's function $\G{}{}$, its non-interacting version $\G{0}{}$, the irreducible vertex function $\Gam{}{}$, the irreducible polarizability $\Po{}{}$, the dynamically-screened Coulomb interaction $\W{}{}$
and the self-energy $\Sig{}{}$, where $\vc{}{}$ is the bare Coulomb interaction, $\delta(12)$ is Dirac's delta function \cite{NISTbook} and $(1)$ is a composite coordinate gathering spin, space and time variables $(\sigma_1,\br{1},t_1)$.
which connects the Green function $\G{}{}$, its non-interacting version $\G{0}{}$, the irreducible vertex function $\Gam{}{}$, the irreducible polarizability $\Po{}{}$, the dynamically-screened Coulomb interaction $\W{}{}$
and the self-energy $\Sig{}{}$, where $\vc{}{}$ is the bare Coulomb interaction, $\delta(12)$ is the Dirac delta function \cite{NISTbook} and $1$ is a composite coordinate gathering space, spin, and time variables $(\br{1},\sigma_1,t_1)$.
Within the {\GW} approximation, one bypasses the calculation of the vertex corrections by setting
\begin{equation}
\label{eq:GW}
@ -226,11 +228,11 @@ F12 methods are now routinely employed in computational chemistry and provide ro
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}
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}
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}.
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's function-based methods, such as GF2 (and its higher-order variants).
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.
In Sec.~\ref{sec:theory}, we provide details about the theory behind the present basis set correction and its adaptation to {\GW} methods.
@ -257,37 +259,37 @@ 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, 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{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 $\n{}{}$.
$\hT$ and $\hWee{}$ are the kinetic and electron-electron interaction operators.
The exact Levy-Lieb universal density functional 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 \cite{Lev-PNAS-79, Lev-PRA-82, Lie-IJQC-83} with wave functions $\wf{}{\Bas}$ expandable in the Hilbert space generated by $\Bas$
where $\F{}{\Bas}[\n{}{}]$ is the Levy-Lieb density functional with wave functions $\wf{}{\Bas}$ expandable in the Hilbert space generated by $\Bas$
\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}
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 $\Bas$
In the present work, instead of using wave-function methods for calculating $\F{}{\Bas}[\n{}{}]$, we re-express it with a constrained search over $\Ne$-representable one-electron Green functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$
\begin{equation}
\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})
where $\Omega^\Bas[G]$ is chosen to be a Klein-like energy functional of the Green function (see, \textit{e.g.}, Refs.~\onlinecite{SteLee-BOOK-13, MarReiCep-BOOK-16, DahLee-JCP-05, DahLeeBar-IJQC-05, DahLeeBar-PRA-06})
\begin{equation}
\Omega^\Bas[\G{}{}] = \Tr[\ln( - \G{}{} ) ] - \Tr[ (\Gs{\Bas})^{-1} \G{}{} - 1 ] + \Phi_\Hxc^\Bas[\G{}{}],
\label{eq:OmegaB}
\end{equation}
where $(\Gs{\Bas})^{-1}$ is the projection into $\Bas$ of the inverse free-particle Green's function
where $(\Gs{\Bas})^{-1}$ is the projection into $\Bas$ of the inverse free-particle Green function
\begin{equation}
(\Gs{})^{-1}(\br{},\br{}',\omega)= \qty(\omega + \frac{\nabla^2_{\br{}}}{2} ) \delta(\br{}-\br{}'),
\end{equation}
and
\begin{equation}
\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{}'.
\Tr[A B] = \int_{-\infty}^{+\infty} \frac{d\omega}{2\pi i} \, e^{i \omega 0^+} \iint A(\br{},\br{}',\omega) B(\br{}',\br{}{},\omega) \dbr{} \dbr{}'.
\end{equation}
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
In Eq.~\eqref{eq:OmegaB}, $\Phi_\Hxc^\Bas[\G{}{}]$ is a Hartree-exchange-correlation ($\Hxc$) functional of the Green 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}
@ -296,14 +298,14 @@ Inserting Eqs.~\eqref{eq:Fn} and \eqref{eq:FBn} into Eq.~\eqref{eq:E0B}, we fina
\E{0}{\Bas} = \min_{\G{}{\Bas}} \qty{ \Omega^\Bas[\G{}{\Bas}] + \int \vne(\br{}) \n{\G{}{\Bas}}{}(\br{}) \dbr{} + \bE{}{\Bas}[\n{\G{}{\Bas}}{}] },
\label{eq:E0BGB}
\end{equation}
where the minimization is over $\Ne$-representable one-electron Green's functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$.
where the minimization is over $\Ne$-representable one-electron Green functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$.
The stationary condition from Eq.~\eqref{eq:E0BGB} gives the following Dyson equation
\begin{equation}
(\G{}{\Bas})^{-1} = (\G{0}{\Bas})^{-1}- \Sig{\Hxc}{\Bas}[\G{}{\Bas}]- \bSig{}{\Bas}[\n{\G{}{\Bas}}{}],
\label{eq:Dyson}
\end{equation}
where $(\G{0}{\Bas})^{-1}$ is the basis projection of the inverse non-interacting Green's function with potential $\vne(\br{})$, \textit{i.e.},
where $(\G{0}{\Bas})^{-1}$ is the basis projection of the inverse non-interacting Green function with potential $\vne(\br{})$, \textit{i.e.},
\begin{equation}
(\G{0}{})^{-1}(\br{},\br{}',\omega)= \qty(\omega + \frac{\nabla_{\br{}}^2}{2} + \vne(\br{}) + \lambda) \delta(\br{}-\br{}')
\end{equation}
@ -312,8 +314,8 @@ with the chemical potential $\lambda$, and $\bSig{}{\Bas}$ is a frequency-indepe
\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 and the Green's function becomes exact, \textit{i.e.},
The solution of the Dyson equation \eqref{eq:Dyson} gives the Green function $\G{}{\Bas}(\br{},\br{}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bSig{}{\Bas}[\n{}{}]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bSig{}{\Bas}$.
Of course, in the CBS limit, the basis-set correction vanishes and the Green function becomes exact, \textit{i.e.},
\begin{align}
\lim_{\Bas \to \CBS} \bSig{}{\Bas} & = 0,
&
@ -683,7 +685,7 @@ 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's 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{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 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 or PBE0).

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Manuscript/biblio.bib
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@ -14,7 +14,8 @@
Date-Modified = {2019-10-09 15:31:02 +0200},
Journal = {J. Chem. Phys.},
Title = {Chemically Accurate Excitation Energies With Small Basis Sets},
Volume = {in press},
Volume = {151},
Pages = {144118},
Year = {2019}}
@article{LooPraSceTouGin-JPCL-19,