Merge branch 'master' into overleaf-2020-07-01-0750

BSEdyn.bib

@@ -1,13 +1,24 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-06-22 20:38:15 +0200 
+%% Created for Pierre-Francois Loos at 2020-06-26 09:45:06 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Petersilka_1996,
+	Author = {M. Petersilka and U. J. Gossmann and and E. K. U. Gross},
+	Date-Added = {2020-06-26 09:43:33 +0200},
+	Date-Modified = {2020-06-26 09:45:05 +0200},
+	Doi = {10.1103/PhysRevLett.76.1212},
+	Journal = {Phys. Rev. Lett.},
+	Pages = {1212},
+	Title = {Excitation Energies From Time-Dependent Density-Functional Theory},
+	Volume = {76},
+	Year = {1996}}
+
 @article{Nielsen_1980,
 	Author = {Egon S. Nielsen and Poul Jorgensen},
 	Date-Added = {2020-06-22 20:37:32 +0200},
@@ -17,7 +28,8 @@
 	Pages = {6238},
 	Title = {Transition moments and dynamic polarizabilities in a second order polarization propagator approach},
 	Volume = {73},
-	Year = {1980}}
+	Year = {1980},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.440119}}
 
 @article{Oddershede_1977,
 	Author = {Jens Oddershede and Poul Jorgensen},
@@ -28,7 +40,8 @@
 	Pages = {1541},
 	Title = {An order analysis of the particle--hole propagator},
 	Volume = {66},
-	Year = {1977}}
+	Year = {1977},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.434118}}
 
 @phdthesis{Huix-Rotllant_PhD,
 	Author = {M. Huix-Rotllant},

Notes/BSEdyn-notes.tex

@@ -41,13 +41,13 @@ In particular, using a simple two-level model, we analyze, for each kernel, the
 \section{Linear response theory}
 \label{sec:LR}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-Linear response theory is a powerful approach that allows to directly access the optical excitations $\omega_s$ of a given electronic system (such as a molecule) and their corresponding oscillator strengths $f_s$ [extracted from their eigenvectors $\T{(\bX_s \bY_s)}$] via the response of the system to a weak electromagnetic field. \cite{Casida_1995}
+Linear response theory is a powerful approach that allows to directly access the optical excitations $\omega_s$ of a given electronic system (such as a molecule) and their corresponding oscillator strengths $f_s$ [extracted from their eigenvectors $\T{(\bX_s \bY_s)}$] via the response of the system to a weak electromagnetic field. \cite{Oddershede_1977,Casida_1995,Petersilka_1996}
 From a practical point of view, these quantities are obtained by solving non-linear, frequency-dependent Casida-like equations in the space of single excitations and de-excitations \cite{Casida_1995}
 \begin{equation} \label{eq:LR}
 	\begin{pmatrix}
 		\bR^{\sigma}(\omega_s)		&	\bC^{\sigma}(\omega_s)
 		\\
-		-\bC^{\sigma}(-\omega_s)	&	-\bR^{\sigma}(-\omega_s)
+		-\bC^{\sigma}(-\omega_s)^*	&	-\bR^{\sigma}(-\omega_s)^*
 	\end{pmatrix}
 	\cdot 
 	\begin{pmatrix}
@@ -64,8 +64,8 @@ From a practical point of view, these quantities are obtained by solving non-lin
 	\end{pmatrix}		
 \end{equation}
 where the explicit expressions of the resonant and coupling blocks, $\bR^{\sigma}(\omega)$ and $\bC^{\sigma}(\omega)$, depend on the spin manifold ($\sigma =$ $\updw$ for singlets and $\sigma =$ $\upup$ for triplets) and the level of approximation that one employs.
-Neglecting the coupling block [\ie, $\bC^{\sigma}(\omega) = 0$] between the resonant and anti-resonants parts, $\bR(\omega)$ and $-\bR(-\omega)$, is known as the Tamm-Dancoff approximation (TDA).
-The non-linear eigenvalue problem defined in Eq.~\eqref{eq:LR} has particle-hole symmetry which means that it is invariant via the transformation $\omega \to -\omega$.
+Neglecting the coupling block [\ie, $\bC^{\sigma}(\omega) = 0$] between the resonant and anti-resonants parts, $\bR^{\sigma}(\omega)$ and $-\bR^{\sigma}(-\omega)^*$, is known as the Tamm-Dancoff approximation (TDA).
+In the absence of symmetry breaking, \cite{Dreuw_2005} the non-linear eigenvalue problem defined in Eq.~\eqref{eq:LR} has particle-hole symmetry which means that it is invariant via the transformation $\omega \to -\omega$.
 Therefore, without loss of generality, we will restrict our analysis to positive frequencies. 
 
 In the one-electron basis of (real) spatial orbitals $\lbrace \MO{p} \rbrace$, we will assume that the elements of the matrices defined in Eq.~\eqref{eq:LR} have the following generic forms: \cite{Dreuw_2005}
@@ -434,6 +434,7 @@ As mentioned in Ref.~\onlinecite{Romaniello_2009b}, spurious solutions appears d
 Indeed, diagonalizing the exact Hamiltonian \eqref{eq:H-exact} produces only two singlet solutions corresponding to the singly- and doubly-excited states, and one triplet state (see Sec.~\ref{sec:exact}). 
 Therefore, there is one spurious solution for the singlet manifold ($\omega_{2}^{\dBSE,\updw}$) and two spurious solutions for the triplet manifold ($\omega_{2}^{\dBSE,\upup}$ and $\omega_{3}^{\dBSE,\upup}$).
 It is worth mentioning that, around $\omega = \omega_1^{\dBSE,\sigma}$, the slope of the curves depicted in Fig.~\ref{fig:dBSE} is small, while the two other solutions, $\omega_2^{\dBSE,\sigma}$ and $\omega_3^{\dBSE,\sigma}$, stem from poles and consequently the slope is very large around these frequency values.
+\titou{T2: add comment on how one can detect fake solutions?}
 
 %%% TABLE I %%%
 \begin{table*}
@@ -723,7 +724,7 @@ What have we learned here?
 %%%%%%%%%%%%%%%%%%%%%%%%
 \acknowledgements{
 The author thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
-He also thanks Xavier Blase for numerous insightful discussions on dynamical kernels.}
+He also thanks Xavier Blase and Juliette Authier for numerous insightful discussions on dynamical kernels.}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 % BIBLIOGRAPHY