diff --git a/BSEdyn.bib b/BSEdyn.bib index 4125427..fefecb7 100644 --- a/BSEdyn.bib +++ b/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}, diff --git a/Notes/BSEdyn-notes.tex b/Notes/BSEdyn-notes.tex index 4cefb1a..10972e0 100644 --- a/Notes/BSEdyn-notes.tex +++ b/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