saving work

BSEdyn.bib

@@ -1,13 +1,25 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-05-18 22:14:10 +0200 
+%% Created for Pierre-Francois Loos at 2020-05-19 14:00:59 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Loos_2018a,
+	Author = {P. F. Loos and A. Scemama and A. Blondel and Y. Garniron and M. Caffarel and D. Jacquemin},
+	Date-Added = {2020-05-19 14:00:54 +0200},
+	Date-Modified = {2020-05-19 14:00:58 +0200},
+	Doi = {10.1021/acs.jctc.8b00406},
+	Journal = {J. Chem. Theory Comput.},
+	Pages = {4360},
+	Title = {A Mountaineering Strategy to Excited States: Highly-Accurate Reference Energies and Benchmarks},
+	Volume = {14},
+	Year = {2018},
+	Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b00406}}
+
 @article{Loos_2020b,
 	Author = {P. F. Loos and F. Lipparini and M. Boggio-Pasqua and A. Scemama and D. Jacquemin},
 	Date-Added = {2020-05-18 22:13:24 +0200},

BSEdyn.tex

@@ -235,7 +235,7 @@ Because the excitonic effect corresponds physically to the stabilization implied
 Most of BSE implementations rely on the so-called static approximation, which approximates the dynamical (\ie, frequency-dependent) BSE kernel by its static limit.
 One key consequence of this approximation is that double (and higher) excitations are completely absent from the BSE spectra. 
 Although these double excitations are usually experimentally dark (which means that they usually cannot be observed in photo-absorption spectroscopy), these states play, indirectly, a key role in many photochemistry mechanisms. \cite{Boggio-Pasqua_2007} 
-They are, moreover, a real challenge for high-level computational methods. \cite{Loos_2018,Loos_2019,Loos_2020b}
+They are, moreover, a real challenge for high-level computational methods. \cite{Loos_2018a,Loos_2019,Loos_2020b}
 
 Going beyond the static approximation is tricky and very few groups have dared to take the plunge. \cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Ma_2009,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
 Nonetheless, it is worth mentioning the seminal work of Strinati, \cite{Strinati_1988} who \titou{bla bla bla.}
@@ -387,7 +387,7 @@ This leads to reduced electron-hole screening, namely larger electron-hole stabi
 \subsection{Theory for chemists}
 %=================================
 
-For a closed-shell system in a finite basis, to compute the BSE excitation energies, one must solve the following (non-linear) dynamical (\ie, frequency-dependent) response problem 
+For a closed-shell system in a finite basis, to compute the BSE excitation energies, one must solve the following (non-linear) dynamical (\ie, frequency-dependent) response problem \cite{Strinati_1988}
 \begin{equation}
 \label{eq:LR-dyn}
 	\begin{pmatrix}
@@ -406,8 +406,9 @@ For a closed-shell system in a finite basis, to compute the BSE excitation energ
 		\bY{m}{}(\omega)	\\
 	\end{pmatrix},
 \end{equation}
-where the dynamical matrices $\bA{}(\omega)$, $\bB{}(\omega)$, $\bX{}{}(\omega)$, and $\bY{}{}(\omega)$ are all of size $\Nocc \Nvir \times \Nocc \Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively.
+where the dynamical matrices $\bA{}(\omega)$ and $\bB{}(\omega)$ are of size $\Nocc \Nvir \times \Nocc \Nvir$, where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively, and $\bX{m}{}(\omega)$, and $\bY{m}{}(\omega)$ are (eigen)vectors of length $\Nocc \Nvir$. 
 In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
+Note that, due to its non-linear nature, Eq.~\eqref{eq:LR-dyn} may provide more than one solution for each value of $m$. \cite{Romaniello_2009b,Sangalli_2011,Martin_2016}
 
 The BSE matrix elements read
 \begin{subequations}
@@ -419,6 +420,7 @@ The BSE matrix elements read
 	\B{ia,jb}{}(\omega) & =  2 \ERI{ia}{bj} -   \W{ib,aj}{}(\omega),
 \end{align}
 \end{subequations}
+\titou{singlet and triplet}
 where $\eGW{ia} = \eGW{a} - \eGW{i}$ are occupied-to-virtual differences of $GW$ quasiparticle energies, 
 \begin{equation}
 	\ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}'
@@ -436,7 +438,7 @@ where $\eta$ is a positive infinitesimal, and
 	\sERI{pq}{m} =  \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{m}{\RPA} + \bY{m}{\RPA})_{ia}
 \end{equation}
 are the spectral weights.
-In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, $\OmRPA{m}{}$ and $(\bX{m}{\RPA} + \bY{m}{\RPA})$ are direct (\ie, without exchange) RPA neutral excitations and their corresponding transition vectors computed by solving the (linear) static response problem 
+In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, $\OmRPA{m}{}$ and $(\bX{m}{\RPA} + \bY{m}{\RPA})$ are direct (\ie, without exchange) RPA neutral excitations and their corresponding transition vectors computed by solving the (static) linear response problem 
 \begin{equation}
 \label{eq:LR-stat}
 	\begin{pmatrix}
@@ -465,11 +467,11 @@ with
 	\B{ia,jb}{\RPA} & =   2 \ERI{ia}{bj},
 \end{align}
 \end{subequations}
-where the $\e{p}$'s are taken as the Hartree-Fock (HF) orbital energies in the case of $G_0W_0$ or as the $GW$ quasiparticle energies in the case of self-consistent scheme such as ev$GW$.
+where the $\e{p}$'s are taken as the HF orbital energies in the case of $G_0W_0$ or as the $GW$ quasiparticle energies in the case of self-consistent schemes such as ev$GW$.
 
-Now, let us decompose, using basis perturbation theory, the eigenproblem \eqref{eq:LR-dyn} as a zeroth-order static part and a first-order dynamic part, such that
+Now, let us decompose, using basic perturbation theory, the non-linear eigenproblem \eqref{eq:LR-dyn} as a zeroth-order static (hence linear) reference and a first-order dynamic (hence non-linear) perturbation, such that
 \begin{equation}
-\label{eq:LR-dyn}
+\label{eq:LR-PT}
 	\begin{pmatrix}
 		\bA{}(\omega)	&	\bB{}(\omega)	\\
 		-\bB{}(\omega)	&	-\bA{}(\omega)	\\
@@ -488,20 +490,20 @@ Now, let us decompose, using basis perturbation theory, the eigenproblem \eqref{
 where
 \begin{subequations}
 \begin{align}
-	\label{eq:BSE-0}
+	\label{eq:BSE-A0}
 	\A{ia,jb}{(0)} & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \ERI{ia}{jb} - \W{ij,ab}{\text{stat}},
 	\\ 
-	\label{eq:BSE-0}
+	\label{eq:BSE-B0}
 	\B{ia,jb}{(0)} & =  2 \ERI{ia}{bj} -   \W{ib,aj}{\text{stat}},
 \end{align}
 \end{subequations}
 and 
 \begin{subequations}
 \begin{align}
-	\label{eq:BSE-1}
+	\label{eq:BSE-A1}
 	\A{ia,jb}{(1)}(\omega) & = - \W{ij,ab}{}(\omega) + \W{ij,ab}{\text{stat}},
 	\\ 
-	\label{eq:BSE-1}
+	\label{eq:BSE-B1}
 	\B{ia,jb}{(1)}(\omega) & = -  \W{ib,aj}{}(\omega) + \W{ib,aj}{\text{stat}},
 \end{align}
 \end{subequations}
@@ -592,6 +594,31 @@ This is our final expression.
 %\end{figure}
 %%% %%% %%%
 
+In terms of computational consideration, because Eq.~\eqref{eq:EcBSE} requires the entire BSE singlet excitation spectrum, we perform a complete diagonalization of the $\Nocc \Nvir \times \Nocc \Nvir$ BSE linear response matrix [see Eq.~\eqref{eq:small-LR}], which corresponds to a $\order{\Nocc^3 \Nvir^3} = \order{\Norb^6}$ computational cost.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Computational details}
+\label{sec:compdet}
+%%%%%%%%%%%%%%%%%%%%%%%%
+The restricted HF formalism has been systematically employed in the present study.
+All the $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies are done using a (restricted) HF starting point.
+Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} and partially self-consistent ev$GW$ calculations are employed as starting points to compute the BSE neutral excitations.
+Within $GW$, all orbitals are corrected.
+In the case of {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation.
+For ev$GW$, the convergence creiterion has been set to $10^{-5}$.
+Further details about our implementation of {\GOWO} and evGW can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018}.
+Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
+
+For comparison purposes, we employ the reference excitation energies and geometries from Ref.~\onlinecite{Loos_2018a} also computed the PES at the second-order M{\o}ller-Plesset perturbation theory (MP2), as well as with various increasingly accurate CC methods, namely, CC2 \cite{Christiansen_1995}, CCSD, \cite{Purvis_1982} and CC3. \cite{Christiansen_1995b} 
+All the other calculations have been performed with our locally developed $GW$ software. \cite{Loos_2018b,Veril_2018}
+As one-electron basis sets, we employ the augmented Dunning family (aug-cc-pVXZ) defined with cartesian Gaussian functions. 
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Conclusion}
+\label{sec:conclusion}
+%%%%%%%%%%%%%%%%%%%%%%%%
+This is the conclusion
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section{Conclusion}