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Pierre-Francois Loos 2021-01-22 14:47:03 +01:00
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15
Manuscript/sfBSE-control.bib
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@ -1,15 +0,0 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2021-01-19 16:50:57 +0100
%% Saved with string encoding Unicode (UTF-8)
@control{achemso-control,
ctrl-article-title = {yes},
ctrl-chapter-title = {yes},
ctrl-etal-firstonly = {yes},
ctrl-etal-number = {30}}

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Manuscript/sfBSE.tex
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@ -3,7 +3,6 @@
\usepackage[version=4]{mhchem} \usepackage[version=4]{mhchem}
\usepackage{natbib} \usepackage{natbib}
\bibliographystyle{achemso} \bibliographystyle{achemso}
\AtBeginDocument{\nocite{achemso-control}}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc} \usepackage[T1]{fontenc}
@ -259,7 +258,7 @@ Unless otherwise stated, atomic units are used.
Let us consider an electronic system consisting of $n = n_\up + n_\dw$ electrons (where $n_\up$ and $n_\dw$ are the number of spin-up and spin-down electrons, respectively) and $N$ one-electron basis functions. Let us consider an electronic system consisting of $n = n_\up + n_\dw$ electrons (where $n_\up$ and $n_\dw$ are the number of spin-up and spin-down electrons, respectively) and $N$ one-electron basis functions.
The number of spin-up and spin-down occupied orbitals are $O_\up = n_\up$ and $O_\dw = n_\dw$, respectively, and, assuming the absence of linear dependencies in the one-electron basis set, there is $V_\up = N - O_\up$ and $V_\dw = N - O_\dw$ spin-up and spin-down virtual (\ie, unoccupied) orbitals. The number of spin-up and spin-down occupied orbitals are $O_\up = n_\up$ and $O_\dw = n_\dw$, respectively, and, assuming the absence of linear dependencies in the one-electron basis set, there is $V_\up = N - O_\up$ and $V_\dw = N - O_\dw$ spin-up and spin-down virtual (\ie, unoccupied) orbitals.
The number of spin-conserved (sc) single excitations is then $S^\spc = S_{\up\up}^\spc + S_{\dw\dw}^\spc = O_\up V_\up + O_\dw V_\dw$, while the number of spin-flip (sf) excitations is $S^\spf = S_{\up\dw}^\spf + S_{\dw\up}^\spf = O_\up V_\dw + O_\dw V_\up$. The number of spin-conserved (sc) single excitations is then $S^\spc = S_{\up\up}^\spc + S_{\dw\dw}^\spc = O_\up V_\up + O_\dw V_\dw$, while the number of spin-flip (sf) excitations is $S^\spf = S_{\up\dw}^\spf + S_{\dw\up}^\spf = O_\up V_\dw + O_\dw V_\up$.
Let us denote as $\MO{p_\sig}(\br)$ the $p$th (spin)orbital with spin $\sig$ (where $\sig =$ $\up$ or $\dw$) and $\e{p_\sig}{}$ its one-electron energy. Let us denote as $\MO{p_\sig}(\br)$ the $p$th spatial orbital associated with the spin-$\sig$ electrons (where $\sig =$ $\up$ or $\dw$) and $\e{p_\sig}{}$ its one-electron energy.
It is important to understand that, in a spin-conserved excitation the hole orbital $\MO{i_\sig}$ and particle orbital $\MO{a_\sig}$ have the same spin $\sig$. It is important to understand that, in a spin-conserved excitation the hole orbital $\MO{i_\sig}$ and particle orbital $\MO{a_\sig}$ have the same spin $\sig$.
In a spin-flip excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{a_\bsig}$, have opposite spins, $\sig$ and $\bsig$. In a spin-flip excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{a_\bsig}$, have opposite spins, $\sig$ and $\bsig$.
We assume real quantities throughout this manuscript, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, $p$, $q$, $r$, and $s$ indicate arbitrary orbitals, and $m$ labels single excitations. We assume real quantities throughout this manuscript, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, $p$, $q$, $r$, and $s$ indicate arbitrary orbitals, and $m$ labels single excitations.
@ -269,7 +268,7 @@ Moreover, we consider systems with collinear spins and a spin-independent Hamilt
\subsection{The dynamical screening} \subsection{The dynamical screening}
%================================ %================================
The pillar of Green's function many-body perturbation theory is the (time-ordered) one-body Green's function, which has poles at the charged excitations (i.e., ionization potentials and electron affinities) of the system. \cite{ReiningBook} The pillar of Green's function many-body perturbation theory is the (time-ordered) one-body Green's function, which has poles at the charged excitations (i.e., ionization potentials and electron affinities) of the system. \cite{ReiningBook}
The spin-$\sig$ component of the one-body Green's function reads \cite{ReiningBook,Bruneval_2016a} The spin-$\sig$ component of the one-body Green's function reads \cite{ReiningBook,Bruneval_2016}
\begin{equation} \begin{equation}
\label{eq:G} \label{eq:G}
G^{\sig}(\br_1,\br_2;\omega) G^{\sig}(\br_1,\br_2;\omega)
@ -504,7 +503,7 @@ The purpose of the underlying $GW$ calculation is to provide quasiparticle energ
\label{sec:BSE} \label{sec:BSE}
%================================ %================================
Within the so-called static approximation of BSE, the Dyson equation that links the generalized four-point susceptibility $L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)$ and the BSE kernel $\Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6)$ is \cite{ReiningBook,Bruneval_2016a} Within the so-called static approximation of BSE, the Dyson equation that links the generalized four-point susceptibility $L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)$ and the BSE kernel $\Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6)$ is \cite{ReiningBook,Bruneval_2016}
\begin{multline} \begin{multline}
L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega) L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)
= L_{0}^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega) = L_{0}^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)
@ -1055,7 +1054,7 @@ Additional graphs comparing (SF-)TD-BLYP, (SF-)TD-B3LYP, and (SF-)dBSE with EOM-
%The data that supports the findings of this study are available within the article and its supplementary material. %The data that supports the findings of this study are available within the article and its supplementary material.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{sfBSE,sfBSE-control} \bibliography{sfBSE}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\end{document} \end{document}