From bcd765e3329ec80f0fe6a032c2381f511b5ad6c7 Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Sun, 17 Jan 2021 22:45:54 +0100
Subject: [PATCH] saving work in results

---
 Manuscript/Be.pdf      |  Bin 70481 -> 56592 bytes
 Manuscript/CBD_D2h.pdf |  Bin 65457 -> 65538 bytes
 Manuscript/CBD_D4h.pdf |  Bin 66414 -> 66535 bytes
 Manuscript/H2_BSE.pdf  |  Bin 30312 -> 30306 bytes
 Manuscript/sfBSE.bib   |   54 +-
 Manuscript/sfBSE.tex   |   45 +-
 Notebooks/sf-BSE.nb    | 2941 +++++++++++++---------------------------
 7 files changed, 990 insertions(+), 2050 deletions(-)

diff --git a/Manuscript/Be.pdf b/Manuscript/Be.pdf
index af619c9929e5e46fab6bde23c57907bf4701c58a..4451217e2f22ef2e23197b6091217b877182f2dd 100644
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diff --git a/Manuscript/CBD_D2h.pdf b/Manuscript/CBD_D2h.pdf
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diff --git a/Manuscript/CBD_D4h.pdf b/Manuscript/CBD_D4h.pdf
index e5e28dcca59e324b5b920fe6801c0ec82e6c35e8..e12736d2b88f2301d72dfcdec2eab1cc053e56bb 100644
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diff --git a/Manuscript/H2_BSE.pdf b/Manuscript/H2_BSE.pdf
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diff --git a/Manuscript/sfBSE.bib b/Manuscript/sfBSE.bib
index 679c444..b85a7b3 100644
--- a/Manuscript/sfBSE.bib
+++ b/Manuscript/sfBSE.bib
@@ -1,13 +1,65 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2021-01-17 21:28:04 +0100 
+%% Created for Pierre-Francois Loos at 2021-01-17 22:10:38 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Vitale_2020,
+	author = {Vitale, Eugenio and Alavi, Ali and Kats, Daniel},
+	date-added = {2021-01-17 22:10:20 +0100},
+	date-modified = {2021-01-17 22:10:37 +0100},
+	doi = {10.1021/acs.jctc.0c00470},
+	journal = {J. Chem. Theory Comput.},
+	number = {9},
+	pages = {5621-5634},
+	title = {FCIQMC-Tailored Distinguishable Cluster Approach},
+	volume = {16},
+	year = {2020},
+	Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.0c00470}}
+
+@article{Karadakov_2008,
+	author = {Karadakov, Peter B.},
+	date-added = {2021-01-17 22:07:08 +0100},
+	date-modified = {2021-01-17 22:07:26 +0100},
+	doi = {10.1021/jp8037335},
+	journal = {J. Phys. Chem. A},
+	number = {31},
+	pages = {7303-7309},
+	title = {Ground- and Excited-State Aromaticity and Antiaromaticity in Benzene and Cyclobutadiene},
+	volume = {112},
+	year = {2008},
+	Bdsk-Url-1 = {https://doi.org/10.1021/jp8037335}}
+
+@article{Li_2009,
+	author = {Li,Xiangzhu and Paldus,Josef},
+	date-added = {2021-01-17 22:06:26 +0100},
+	date-modified = {2021-01-17 22:06:47 +0100},
+	doi = {10.1063/1.3225203},
+	journal = {J. Chem. Phys.},
+	number = {11},
+	pages = {114103},
+	title = {Accounting for the exact degeneracy and quasidegeneracy in the automerization of cyclobutadiene via multireference coupled-cluster methods},
+	volume = {131},
+	year = {2009},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.3225203}}
+
+@article{Shen_2012,
+	author = {Shen,Jun and Piecuch,Piotr},
+	date-added = {2021-01-17 22:02:31 +0100},
+	date-modified = {2021-01-17 22:02:46 +0100},
+	doi = {10.1063/1.3700802},
+	journal = {J. Chem. Phys.},
+	number = {14},
+	pages = {144104},
+	title = {Combining active-space coupled-cluster methods with moment energy corrections via the CC(P;Q) methodology, with benchmark calculations for biradical transition states},
+	volume = {136},
+	year = {2012},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.3700802}}
+
 @article{Kannar_2014,
 	author = {K{\'a}nn{\'a}r, D{\'a}niel and Szalay, P{\'e}ter G.},
 	date-added = {2021-01-17 21:18:03 +0100},
diff --git a/Manuscript/sfBSE.tex b/Manuscript/sfBSE.tex
index b0d8a84..13a1cd3 100644
--- a/Manuscript/sfBSE.tex
+++ b/Manuscript/sfBSE.tex
@@ -568,13 +568,18 @@ For a given single excitation $m$, the explicit expressions of $\Delta \expval{\
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 All the  systems under investigation here are closed shell and we consider a triplet reference state.
 We then adopt the unrestricted formalism throughout this work.
-The {\GOWO} and ev$GW$ calculations performed to obtain the screened Coulomb potential and the quasiparticle energies required to compute the BSE neutral excitations are performed using an unrestricted HF (UHF) starting point, while, by construction, the corresponding qs$GW$ quantities are independent from the starting point.
-For {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation [see Eq.~\eqref{eq:G0W0_lin}].
-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.
+%The {\GOWO} and ev$GW$ calculations performed to obtain the screened Coulomb potential and the quasiparticle energies required to compute the BSE neutral excitations are performed using an unrestricted HF (UHF) starting point, while, by construction, the corresponding qs$GW$ quantities are independent from the starting point.
+%For {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation [see Eq.~\eqref{eq:G0W0_lin}].
+%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.
+The {\GOWO} calculations performed to obtain the screened Coulomb potential and the quasiparticle energies required to compute the BSE neutral excitations are performed using an unrestricted HF (UHF) starting point, and the {\GOWO} quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation [see Eq.~\eqref{eq:G0W0_lin}].
+Note that the entire set of orbitals and energies is corrected.
+Further details about our implementation of {\GOWO} 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} excitation energies.
 This is left for future work.
 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}
+Besides, {\GOWO}@UHF and ev$GW$@UHF yield similar quasiparticle energies, while {\GOWO} allows us to avoid rather laborious iterations as well as the significant additional computational effort of ev$GW$.
 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.}
 Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
@@ -646,12 +651,12 @@ Generally we can observe that all the scheme with SF-BSE used do not increase si
 									&	()		&	()				&	()				&	()				&	()		\\
 			SF-BSE@{\GOWO}			&	(0.004)	&	2.399(1.999)	&	6.191(0.023)	&	7.792(1.000)	&	9.373(0.013)	
 									&	(0.021)	&	2.286(1.994)	&	5.181(0.187)	&	6.481(1.000)	&	7.195(0.719)	\\
-			SF-BSE@{\evGW}			&	(0.004)	&	2.407(1.999)	&	6.199(0.023)	&	7.788(1.000)	&	9.388(0.013)	\\
-			\alert{SF-BSE@{\qsGW}}	&	(0.102)	&	2.532(2.000)	&	6.241(1.873)	&	7.668(1.000)	&	9.417(0.217)	\\
+%			SF-BSE@{\evGW}			&	(0.004)	&	2.407(1.999)	&	6.199(0.023)	&	7.788(1.000)	&	9.388(0.013)	\\
+%			\alert{SF-BSE@{\qsGW}}	&	(0.102)	&	2.532(2.000)	&	6.241(1.873)	&	7.668(1.000)	&	9.417(0.217)	\\
 			SF-dBSE@{\GOWO}			&			&	2.363			&	6.263			&	7.824			&	9.424	
 									&			&					&					&					&	\\
-			SF-dBSE@{\evGW}			&			&	2.369			&	6.273			&	7.820			&	9.441	\\
-			SF-dBSE@{\qsGW}			&			&	2.335			&	6.317			&	7.689			&	9.470	\\
+%			SF-dBSE@{\evGW}			&			&	2.369			&	6.273			&	7.820			&	9.441	\\
+%			SF-dBSE@{\qsGW}			&			&	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	
@@ -691,7 +696,7 @@ The lowest singly excited state $\text{B}\,{}^1 \Sigma_u^+$ has a $(1\sigma_g )(
 Because these latter two excited states interact strongly and form an avoided crossing around $R(\ce{H-H}) = 1.4$ \AA, they are usually labeled as the $\text{EF}\,{}^1 \Sigma_g^+$ state.
 Note that this avoided crossing is not visible with non-spin-flip methods, such as CIS, TD-DFT, and BSE, as these are ``blind'' to double excitations.
 Three methods, in their standard and spin-flip versions, are studied here (CIS, TD-BH\&HLYP and BSE) and are compared to the reference EOM-CCSD excitation energies (that is equivalent to FCI in the case of \ce{H2}).
-All these calculations are performed in the cc-pVQZ basis.
+All these calculations are performed in the cc-pVQZ basis, and both the spin-conserved and spin-flip calculations are performed with an unrestricted reference.
 
 The top panel of Fig.~\ref{fig:H2} shows the CIS (dotted lines) and SF-CIS (dashed lines) excitation energies as a function of $R(\ce{H-H})$.
 The EOM-CCSD reference energies are represented by solid lines.
@@ -704,14 +709,11 @@ Nonetheless, this results in a rather good qualitative agreement with an avoided
 As mentioned earlier, CIS is unable to locate any avoided crossing as it cannot access double excitations.
 However, CIS is quite accurate for the $\text{E}\,{}^1\Sigma_g^+$.
 \titou{Spin-contamination of the E state?}
-\titou{CIS or UCIS?}
-
 
 In the center panel of Fig.~\ref{fig:H2}, we report the (SF-)TD-BH\&HLYP results.
 TD-BH\&HLYP shows bad results for all the states of interest with and without spin-flip. 
 Note that \ce{H2} is a rather challenging system for (SF)-TD-DFT from a general point of view. \cite{Cohen_2008a,Cohen_2008c,Cohen_2012}
 Indeed, for the three states we have a difference in the excitation energy at the dissociation limit of several eV with and without spin-flip. 
-\titou{RKS or UKS?}
 Similar graphs for (SF-)TD-BLYP and (SF-)TD-B3LYP are reported in the {\SI}.
 
 In the bottom panel of Fig.~\ref{fig:H2} we have results for BSE calculation with and without spin-flip. 
@@ -721,7 +723,6 @@ For the $\text{E}\,{}^1\Sigma_g^+$ state BSE gives closer results to the referen
 However we can observe that for all the methods that we compared, when the spin-flip is not used standard methods can not retrieve double excitation. 
 There is no avoided crossing or perturbation in the curve for the $\text{E}\,{}^1\Sigma_g^+$ state when spin-flip is not used. 
 This is because for these methods we are in the space of single excitation and de-excitation.
-\titou{BSE@RHF?}
 A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be found in the {\SI}.
 
 %%% FIG 2 %%%
@@ -732,7 +733,7 @@ A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be foun
 \caption{
 	Excitation energies of the $\text{B}\,{}^1\Sigma_u^+$ (red), $\text{E}\,{}^1\Sigma_g^+$ (black), and $\text{E}\,{}^1\Sigma_g^+$ (blue) states (with respect to the $\text{X}\,{}^1 \Sigma_g^+$ ground state) of \ce{H2} obtained with the cc-pVQZ basis at the (SF-)CIS (top), (SF-)TD-BH\&HLYP (middle), and (SF-)BSE (bottom) levels of theory.  
 	The reference EOM-CCSD excitation energies are represented as solid lines, while the results obtained with and without spin-flip results are represented as dashed and dotted lines, respectively.
-	All the spin-flip calculations have been performed with a UHF reference.
+	All the spin-conserved and spin-flip calculations have been performed with an unrestriced reference.
 	The raw data are reported in the {\SI}.
 	\label{fig:H2}}
 \end{figure}
@@ -743,20 +744,22 @@ A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be foun
 \label{sec:CBD}
 %===============================
 
-Cyclobutadiene (CBD) is an interesting example as its electronic character of its ground state can be tune via geometrical deformation. \cite{Balkova_1994,Manohar_2008,Lefrancois_2015,Casanova_2020}
+Cyclobutadiene (CBD) is an interesting example as its electronic character of its ground state can be tune via geometrical deformation. \cite{Balkova_1994,Levchenko_2004,Manohar_2008,Karadakov_2008,Li_2009,Shen_2012,Lefrancois_2015,Casanova_2020,Vitale_2020}
 %with potential large spin contamination.
 In its $D_{2h}$ rectangular $^1 A_g$ singlet ground-state equilibrium geometry, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are non-degenerate, and the singlet ground state can be safely labeled as single-reference with well-defined doubly-occupied orbitals.
-However, in its $D_{4h}$ square-planar $^3 A_{2g}$ triplet round-state equilibrium geometry, the HOMO and LUMO are strictly degenerate, and the electronic ground state (which is still of singlet nature with $B_{1g}$ spatial symmetry, hence violating Hund's rule) is strongly multi-reference with singly occupied orbitals.
+However, in its $D_{4h}$ square-planar $^3 A_{2g}$ ground-state equilibrium geometry, the HOMO and LUMO are strictly degenerate, and the electronic ground state (which is still of singlet nature with $B_{1g}$ spatial symmetry, hence violating Hund's rule) is strongly multi-reference with singly occupied orbitals (\ie, singlet open-shell state).
 In this case, single-reference methods notoriously fail.
 Nonetheless, the lowest triplet state of symmetry $^3 A_{2g}$ remains of single-reference character and is then a perfect starting point for spin-flip calculations.
-
 The $D_{2h}$ and $D_{4h}$ optimized geometries of the $^1 A_g$ and $^3 A_{2g}$ states of CBD have been extracted from Ref.~\onlinecite{Manohar_2008} and have been obtained at the CCSD(T)/cc-pVTZ level.
 EOM-CCSD and SF-ADC calculations have been taken from Refs.~\onlinecite{Manohar_2008} and Ref.~\onlinecite{Lefrancois_2015}.
 All of them have been obtained with a UHF reference like the SF-BSE calculations performed here.
 
-Table~\ref{tab:CBD_D2h} shows results obtained for the $D_{2h}$ rectangular equilibrium geometry and Table~\ref{tab:CBD_D4h} shows results obtained for $D_{4h}$ square equilibrium geometry. These results are given with respect to the singlet ground state. 
-For each geometry three states are under investigation, for the $D_{2h}$ CBD we look at the $1 ~^3 B_{1g}$, $1~^1 B_{1g}$ and $2 ~^1 A_{1g}$ states. 
-For the $D_{4h}$ CBD we look at the $1 ~^3 A_{2g}$, $2~^1 A_{1g}$ and $1 ~^1 B_{2g}$ states. 
+Tables~\ref{tab:CBD_D2h} and \ref{tab:CBD_D4h} report excitation energies (with respect to the singlet ground state) obtained for the $D_{2h}$ and $D_{4h}$ geometries, respectively. 
+These are also represented in Fig.~\ref{fig:CBD}.
+For each geometry, three states are under investigation. 
+For the $D_{2h}$ CBD, we look at the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$ and $2\,{}^1A_{1g}$ states. 
+In this case, it is important to mention  that the $2\,{}^1A_{1g}$ state has a significant double excitation character.
+For the $D_{4h}$ CBD we look at the $1\,{}^3 A_{2g}$, $2\,{}^1 A_{1g}$ and $1\,{}^1 B_{2g}$ states. 
 Several methods using spin-flip are compared to the spin-flip version of BSE with and without dynamical corrections. 
 In Table~\ref{tab:CBD_D2h}, comparing the results of our work and the most accurate ADC level, \ie, SF-ADC(3) with SF-BSE@{\GOWO} we have a difference in the excitation energy of 0.017 eV for the $1 ~^3 B_{1g}$ state. 
 This difference grows to 0.572 eV for the $1 ~^1 B_{1g}$ state and then it is 0.212 eV for the $2 ~^1 A_{1g}$ state. 
diff --git a/Notebooks/sf-BSE.nb b/Notebooks/sf-BSE.nb
index 21e2dce..ea110bc 100644
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