From 1e0159f1b8151adcd83f64a1de85643dd15797e1 Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Wed, 23 Oct 2019 10:15:47 +0200
Subject: [PATCH] comp cost

---
 Manuscript/GW-srDFT.bib | 35 ++++++++++++++++++++++++++++++++++-
 Manuscript/GW-srDFT.tex |  4 ++++
 2 files changed, 38 insertions(+), 1 deletion(-)

diff --git a/Manuscript/GW-srDFT.bib b/Manuscript/GW-srDFT.bib
index 0c40148..a856944 100644
--- a/Manuscript/GW-srDFT.bib
+++ b/Manuscript/GW-srDFT.bib
@@ -1,13 +1,46 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2019-10-14 10:43:41 +0200 
+%% Created for Pierre-Francois Loos at 2019-10-23 10:10:27 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Duchemin_2017,
+	Author = {I. Duchemin and J. Li and X. Blase},
+	Date-Added = {2019-10-23 10:08:33 +0200},
+	Date-Modified = {2019-10-23 10:09:31 +0200},
+	Doi = {10.1021/acs.jctc.6b01215},
+	Journal = {J. Chem. Theory Comput.},
+	Pages = {1199},
+	Title = {Hybrid and Constrained Resolution-of-Identity Techniques for Coulomb Integrals},
+	Volume = {13},
+	Year = {2017}}
+
+@article{Rojas_1995,
+	Author = {H. N. Rojas and R. W. Godby and R. J. Needs},
+	Date-Added = {2019-10-23 10:03:29 +0200},
+	Date-Modified = {2019-10-23 10:10:12 +0200},
+	Doi = {10.1103/PhysRevLett.74.1827},
+	Journal = {Phys. Rev. Lett.},
+	Pages = {1827},
+	Title = {Space-Time Method for Ab Initio Calculations of Self-Energies and Dielectric Response Functions of Solids},
+	Volume = {74},
+	Year = {1995}}
+
+@article{Duchemin_2019,
+	Author = {Ivan Duchemin and Xavier Blase},
+	Date-Added = {2019-10-23 10:00:45 +0200},
+	Date-Modified = {2019-10-23 10:01:39 +0200},
+	Doi = {10.1063/1.5090605},
+	Journal = {J. Chem. Phys.},
+	Pages = {174120},
+	Title = {Separable Resolution-of-the-Identity with All-Electron Gaussian Bases: Application to Cubic-scaling RPA},
+	Volume = {150},
+	Year = {2019}}
+
 @article{Weigend_2003a,
 	Author = {Weigend, Florian and Furche, Filipp and Ahlrichs, Reinhart},
 	Date-Added = {2019-10-14 10:42:46 +0200},
diff --git a/Manuscript/GW-srDFT.tex b/Manuscript/GW-srDFT.tex
index 7f3614c..0cb8185 100644
--- a/Manuscript/GW-srDFT.tex
+++ b/Manuscript/GW-srDFT.tex
@@ -61,6 +61,7 @@
 \newcommand{\Bas}{\mathcal{B}}
 \newcommand{\cD}{\mathcal{D}}
 \newcommand{\Ne}{N}
+\newcommand{\Nbas}{N_\text{bas}}
 
 
 % energies
@@ -537,6 +538,9 @@ The FC density-based basis set correction~\cite{LooPraSceTouGin-JPCL-19} is used
 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}.
+Note, however, that the formal $\order*{\Nbas^6}$ cost of {\GW} can be significantly decreased thanks to resolution-of-the-identity techniques \cite{vanSetten_2013, Bruneval_2016, Duchemin_2017} and other tricks. \cite{Rojas_1995, Duchemin_2019}
+
 %%% FIG 1 %%%
 \begin{figure*}
 	\includegraphics[width=0.45\linewidth]{IP_G0W0HF_H2O}