From 519ff3e7ef2f940cb683ed7897a2fa5e1ec2dd55 Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Sun, 10 Jan 2021 23:10:46 +0100
Subject: [PATCH] Be results

---
 Manuscript/Be.pdf    |  Bin 0 -> 73308 bytes
 Manuscript/sfBSE.bib |   27 +-
 Manuscript/sfBSE.tex |   58 +-
 Notebooks/sf-BSE.nb  | 2606 ++++++++++++++++++++++++------------------
 4 files changed, 1543 insertions(+), 1148 deletions(-)
 create mode 100644 Manuscript/Be.pdf

diff --git a/Manuscript/Be.pdf b/Manuscript/Be.pdf
new file mode 100644
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diff --git a/Manuscript/sfBSE.bib b/Manuscript/sfBSE.bib
index 35cd375..6fc2d7b 100644
--- a/Manuscript/sfBSE.bib
+++ b/Manuscript/sfBSE.bib
@@ -1,7 +1,7 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2021-01-10 18:27:42 +0100 
+%% Created for Pierre-Francois Loos at 2021-01-10 22:26:03 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
@@ -1492,23 +1492,6 @@
 	Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261401002871},
 	Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(01)00287-1}}
 
-@article{Krylov_2001b,
-	abstract = {A new approach to the bond-breaking problem is proposed. Both closed and open shell singlet states are described within a single reference formalism as spin-flipping, e.g., α→β, excitations from a triplet (Ms=1) reference state for which both dynamical and non-dynamical correlation effects are much smaller than for the corresponding singlet state. Formally, the new theory can be viewed as an equation-of-motion (EOM) model where excited states are sought in the basis of determinants conserving the total number of electrons but changing the number of α and β electrons. The results for two simplest members of the proposed hierarchy of approximations are presented.},
-	author = {Anna I. Krylov},
-	date-added = {2020-12-06 14:36:46 +0100},
-	date-modified = {2020-12-06 14:37:01 +0100},
-	doi = {https://doi.org/10.1016/S0009-2614(01)00287-1},
-	issn = {0009-2614},
-	journal = {Chem. Phys. Lett.},
-	number = {4},
-	pages = {375 - 384},
-	title = {Size-consistent wave functions for bond-breaking: the equation-of-motion spin-flip model},
-	url = {http://www.sciencedirect.com/science/article/pii/S0009261401002871},
-	volume = {338},
-	year = {2001},
-	Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261401002871},
-	Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(01)00287-1}}
-
 @article{Krylov_2002,
 	author = {Krylov,Anna I. and Sherrill,C. David},
 	date-added = {2020-12-06 14:36:32 +0100},
@@ -4204,9 +4187,9 @@
 @article{Refaely-Abramson_2012,
 	author = {Sivan Refaely-Abramson and Sahar Sharifzadeh and Niranjan Govind and Jochen Autschbach and Jeffrey B. Neaton and Roi Baer and Leeor Kronik},
 	date-added = {2020-05-18 21:40:28 +0200},
-	date-modified = {2020-05-18 21:40:28 +0200},
+	date-modified = {2021-01-10 20:59:42 +0100},
 	doi = {10.1103/PhysRevLett.109.226405},
-	journal = {Phys. Rev. X},
+	journal = {Phys. Rev. Lett.},
 	pages = {226405},
 	title = {Quasiparticle Spectra from a Nonempirical Optimally Tuned Range-Separated Hybrid Density Functional},
 	volume = {109},
@@ -8832,10 +8815,10 @@
 	Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.93.075143},
 	Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevB.93.075143}}
 
-@article{Krylov_2001c,
+@article{Krylov_2001b,
 	author = {Krylov, Anna I.},
 	date-added = {2020-01-01 21:36:51 +0100},
-	date-modified = {2020-12-06 14:37:24 +0100},
+	date-modified = {2021-01-10 22:26:03 +0100},
 	doi = {10.1016/S0009-2614(01)01316-1},
 	issn = {00092614},
 	journal = {Chem. Phys. Lett.},
diff --git a/Manuscript/sfBSE.tex b/Manuscript/sfBSE.tex
index b7e4ed7..eaaed28 100644
--- a/Manuscript/sfBSE.tex
+++ b/Manuscript/sfBSE.tex
@@ -572,13 +572,15 @@ Note that, in any case, the entire set of orbitals and energies is corrected.
 Further details about our implementation of {\GOWO}, ev$GW$, and qs$GW$ can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018,Loos_2020e,Loos_2020h}.
 Here, we do not investigate how the starting orbitals affect the BSE@{\GOWO} and BSE@ev$GW$ excitation energies.
 This is left for future work.
-However, it is worth mentioning that, for the present (small) molecular systems, HF is usually an excellent starting point. \cite{Loos_2020a,Loos_2020e,Loos_2020h}
+However, it is worth mentioning that, for the present (small) molecular systems, HF is usually a good starting point, \cite{Loos_2020a,Loos_2020e,Loos_2020h} although improvements could certainly be obtained with starting orbitals and energies computed with, for example, optimally-tuned range-separated hybrid functionals. \cite{Stein_2009,Stein_2010,Refaely-Abramson_2012,Kronik_2012}
 In the following, all linear response calculations are performed within the TDA to ensure consistency between the spin-conserved and spin-flip results.
-\titou{As one-electron basis sets, we employ Pople's 6-31G basis or the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions.}
+%\titou{As one-electron basis sets, we employ Pople's 6-31G basis or the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions.}
 Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
 
 All the static and dynamic BSE calculations have been performed with the software \texttt{QuAcK}, \cite{QuAcK} developed in our group and freely available on \texttt{github}.
-The TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and the EOM-CCSD calculation with Gaussian 09. \cite{g09}
+The SF-ADC, EOM-SF-CC and SF-TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and the EOM-CCSD calculation with Gaussian 09. \cite{g09}
+As a consistency check, we systematically perform the SF-CIS calculations with both \texttt{QuAcK} and Q-CHEM, and make sure that they yield identical excitation energies.
+Throughout this work, all spin-flip calculations have been performed with a UHF reference.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results}
@@ -589,40 +591,56 @@ The TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and
 \subsection{Beryllium atom}
 \label{sec:Be}
 %===============================
+As a first example, we consider the simple case of the beryllium atom which was considered by Krylov in two of her very first papers on spin-flip methods \cite{Krylov_2001a,Krylov_2001b} and was also considered in later studies thanks to its pedagogical value. \cite{Sears_2003,Casanova_2020} 
+Beryllium has a $^1S$ ground state with $1s^2 2s^2$ configuration. 
+The excitation energies corresponding to the first singlet and triplet single excitations $2s \to 2p$ with $P$ spatial symmetry as well as the first singlet and triplet double excitations $2s^2 \to 2p^2$ with $P$ and $D$ spatial symmetries (respectively), obtained with the 6-31G basis set are reported in Table \ref{tab:Be} and depicted in Fig.~\ref{fig:Be}.
 
 %%% TABLE I %%%
 \begin{squeezetable}
 \begin{table}
 	\caption{
 	Excitation energies [with respect to the $^1S(1s^2 2s^2)$ singlet ground state] of \ce{Be} obtained for various methods with the 6-31G basis.
+	All the spin-flip calculations have been performed with a UHF reference.
 	\label{tab:Be}}
 	\begin{ruledtabular}
 		\begin{tabular}{lcccc}
-								&	\mc{4}{c}{Excitation energies (eV)}			\\
+									&	\mc{4}{c}{Excitation energies (eV)}			\\
 									\cline{2-5}
-			Method				&	$^3P(1s^22s2p)$	&	$^1P(1s^22s2p)$	
-								&	$^3P(1s^22p^2)$	&	$^1P(1s^22p^2)$			\\
+			Method					&	$^3P(1s^22s2p)$	&	$^1P(1s^22s2p)$	
+									&	$^3P(1s^22p^2)$	&	$^1D(1s^22p^2)$			\\
 			\hline
-			SF-TD-BLYP			&	3.210	&	3.210	&	6.691	&	7.598	\\
-			SF-TD-B3LYP			&	3.332	&	4.275	&	6.864	&	7.762	\\	
-			SF-TD-BHHLYP		&	2.874	&	4.922	&	7.112	&	8.188	\\		
-			SF-CIS				&	2.111	&	6.036	&	7.480	&	8.945	\\
-			SF-BSE@{\GOWO}@UHF	&	2.399	&	6.191	&	7.792	&	9.373	\\
-			SF-BSE@{\evGW}@UHF	&	2.407	&	6.199	&	7.788	&	9.388	\\
-			SF-BSE@{\qsGW}@UHF	&	2.376	&	6.241	&	7.668	&	9.417	\\
-			SF-dBSE@{\GOWO}@UHF	&	2.363	&	6.263	&	7.824	&	9.424	\\
-			SF-dBSE@{\evGW}@UHF	&	2.369	&	6.273	&	7.820	&	9.441 	\\
-			SF-dBSE@{\qsGW}@UHF	&	2.335	&	6.317	&	7.689	&	 9.470	\\
-			SF-ADC(2)-s			&	2.433	&	6.255	&	7.745	&	9.047 	\\
-			SF-ADC(2)-x			&	2.866	&	6.581	&	7.664	&	8.612	\\
-			SF-ADC(3)			&	2.863	&	6.579	&	7.658	&	8.618 	\\
-			FCI					&	2.862	&	6.577	&	7.669	&	8.624	\\
+			SF-TD-BLYP\fnm[1]		&	3.210	&	3.210	&	6.691	&	7.598	\\
+			SF-TD-B3LYP\fnm[1]		&	3.332	&	4.275	&	6.864	&	7.762	\\	
+			SF-TD-BH\&HLYP\fnm[1]	&	2.874	&	4.922	&	7.112	&	8.188	\\		
+			SF-BSE@{\GOWO}\fnm[2]	&	2.399	&	6.191	&	7.792	&	9.373	\\
+			SF-BSE@{\evGW}\fnm[2]	&	2.407	&	6.199	&	7.788	&	9.388	\\
+			SF-BSE@{\qsGW}\fnm[2]	&	2.376	&	6.241	&	7.668	&	9.417	\\
+			SF-dBSE@{\GOWO}\fnm[2]	&	2.363	&	6.263	&	7.824	&	9.424	\\
+			SF-dBSE@{\evGW}\fnm[2]	&	2.369	&	6.273	&	7.820	&	9.441 	\\
+			SF-dBSE@{\qsGW}\fnm[2]	&	2.335	&	6.317	&	7.689	&	9.470	\\
+			SF-CIS\fnm[3]			&	2.111	&	6.036	&	7.480	&	8.945	\\
+			SF-ADC(2)-s\fnm[2]		&	2.433	&	6.255	&	7.745	&	9.047 	\\
+			SF-ADC(2)-x\fnm[2]		&	2.866	&	6.581	&	7.664	&	8.612	\\
+			SF-ADC(3)\fnm[2]		&	2.863	&	6.579	&	7.658	&	8.618 	\\
+			FCI\fnm[3]				&	2.862	&	6.577	&	7.669	&	8.624	\\
 		\end{tabular}
 	\end{ruledtabular}
+	\fnt[1]{Excitation energies extracted from Ref.~\onlinecite{Casanova_2020}.}
+	\fnt[2]{This work.}
+	\fnt[3]{Excitation energies taken from Ref.~\onlinecite{Krylov_2001a}.}
 \end{table}
 \end{squeezetable}
 %%% %%% %%% %%%
 
+%%% FIG. 1 %%%
+\begin{figure}
+	\includegraphics[width=\linewidth]{Be}
+	\caption{
+	Excitation energies [with respect to the $^1S(1s^2 2s^2)$ singlet ground state] of \ce{Be} obtained with the 6-31G basis for various levels of theory:
+	SD-TD-DFT \cite{Casanova_2020} (red), SF-BSE (blue), SF-CIS \cite{Krylov_2001a} and SF-ADC (orange), and FCI \cite{Krylov_2001a} (black). 
+	All the spin-flip calculations have been performed with a UHF reference.
+	\label{fig:Be}}
+\end{figure}
 
 %%% TABLE II %%%
 %\begin{squeezetable}
diff --git a/Notebooks/sf-BSE.nb b/Notebooks/sf-BSE.nb
index 115e38d..6766cd1 100644
--- a/Notebooks/sf-BSE.nb
+++ b/Notebooks/sf-BSE.nb
@@ -10,10 +10,10 @@
 NotebookFileLineBreakTest
 NotebookFileLineBreakTest
 NotebookDataPosition[       158,          7]
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-NotebookOptionsPosition[    242557,       4434]
-NotebookOutlinePosition[    242951,       4450]
-CellTagsIndexPosition[    242908,       4447]
+NotebookDataLength[    268110,       4885]
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