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Julien Toulouse 2019-10-24 16:06:49 +02:00
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20
Manuscript/GW-srDFT.tex
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@ -350,7 +350,7 @@ For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\
In this subsection, 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 the sake of simplicity, we consider closed-shell systems with a $\KS$ single-particle reference.
For the sake of simplicity, we only give the equations for closed-shell systems with a $\KS$ single-particle reference.
The one-electron energies $\e{p}$ and their corresponding (real-valued) orbitals $\MO{p}(\br{})$ (which defines the basis set $\Bas$) are then $\KS$ energies and orbitals.
Within the {\GW} approximation, the correlation part of the self-energy reads
@ -365,16 +365,16 @@ Within the {\GW} approximation, the correlation part of the self-energy reads
& + 2 \sum_{a}^\text{virt} \sum_{m} \frac{[pa|m]^2}{\omega - \e{a} - \Om{m} + i \eta},
\end{split}
\end{equation}
where $m$ corresponds to a sum over the single excitations and $\eta$ is a positive infinitesimal.
where $m$ labels excited states and $\eta$ is a positive infinitesimal.
The screened two-electron integrals
\begin{equation}
[pq|m] = \sum_{ia} (pq|ia) (\bX+\bY)_{ia}^{m}
[pq|m] = \sum_{ia} (pq|ia) (\bX_m+\bY_m)_{ia}
\end{equation}
are obtained via the contraction of the bare two-electron integrals \cite{Gill_1994}
\begin{equation}
(pq|rs) = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}',
\end{equation}
and the transition densities $(\bX+\bY)_{ia}^{m}$ originating from a (direct) random phase approximation (RPA) calculation \cite{Casida_1995, Dreuw_2005}
and the transition densities $(\bX_m+\bY_m)_{ia}$ originating from a (direct) random-phase approximation (RPA) calculation \cite{Casida_1995, Dreuw_2005}
\begin{equation}
\label{eq:LR}
\begin{pmatrix}
@ -382,20 +382,20 @@ and the transition densities $(\bX+\bY)_{ia}^{m}$ originating from a (direct) ra
-\bB & -\bA \\
\end{pmatrix}
\begin{pmatrix}
\bX \\
\bY \\
\bX_m \\
\bY_m \\
\end{pmatrix}
=
\bOm
\Om{m}
\begin{pmatrix}
\bX \\
\bY \\
\bX_m \\
\bY_m \\
\end{pmatrix},
\end{equation}
with
\begin{align}
\label{eq:RPA}
A_{ia,jb} & = \delta_{ij} \delta_{ab} (\e{a} - \e{i}) + 2 (ib|aj),
A_{ia,jb} & = \delta_{ij} \delta_{ab} (\e{a} - \e{i}) + 2 (ia|bj),
&
B_{ia,jb} & = 2 (ia|jb),
\end{align}

8
Manuscript/SI/GW-srDFT-SI.tex
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@ -155,6 +155,7 @@
\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}
@ -163,7 +164,7 @@
\author{Pierre-Fran\c{c}ois Loos}
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Bath\'elemy Pradines}
\author{Barth\'el\'emy Pradines}
\affiliation{\LCT}
\affiliation{\ISCD}
\author{Anthony Scemama}
@ -173,6 +174,7 @@
\author{Julien Toulouse}
\email[Corresponding author: ]{toulouse@lct.jussieu.fr}
\affiliation{\LCT}
\affiliation{\IUF}
\begin{abstract}
\end{abstract}
@ -182,7 +184,7 @@
\begin{figure*}
\includegraphics[width=\linewidth]{IP_G0W0HF}
\caption{
IPs (in eV) computed at the {\GOWO}@HF (black circles), {\GOWO}@HF+srLDA (red squares) and {\GOWO}@HF+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets (cc-pVDZ, cc-pVTZ, cc-pVQZ and cc-pV5Z) for the 20 smallest molecules of the GW100 set.
IPs (in eV) computed at the {\GOWO}@HF (black circles), {\GOWO}@HF+srLDA (red squares), and {\GOWO}@HF+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets (cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pV5Z) for the 20 smallest molecules of the GW100 set.
The thick black line represents the CBS value obtained by extrapolation with the three largest basis sets.
\label{fig:IP_G0W0HF}
}
@ -191,7 +193,7 @@
\begin{figure*}
\includegraphics[width=\linewidth]{IP_G0W0PBE0}
\caption{
IPs (in eV) computed at the {\GOWO}@PBE0 (black circles), {\GOWO}@PBE0+srLDA (red squares) and {\GOWO}@PBE0+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets (cc-pVDZ, cc-pVTZ, cc-pVQZ and cc-pV5Z) for the 20 smallest molecules of the GW100 set.
IPs (in eV) computed at the {\GOWO}@PBE0 (black circles), {\GOWO}@PBE0+srLDA (red squares), and {\GOWO}@PBE0+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets (cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pV5Z) for the 20 smallest molecules of the GW100 set.
The thick black line represents the CBS value obtained by extrapolation with the three largest basis sets.
\label{fig:IP_G0W0HF}
}