diff --git a/sfBSE.rty b/sfBSE.rty index a4624de..9aa34e4 100644 --- a/sfBSE.rty +++ b/sfBSE.rty @@ -93,9 +93,12 @@ \newcommand{\vc}[1]{v_{#1}} \newcommand{\Sig}[1]{\Sigma_{#1}} \newcommand{\SigGW}[1]{\Sigma^{GW}_{#1}} +\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}} \newcommand{\Z}[1]{Z_{#1}} \newcommand{\MO}[1]{\phi_{#1}} \newcommand{\ERI}[2]{(#1|#2)} +\newcommand{\rbra}[1]{(#1|} +\newcommand{\rket}[1]{|#1)} \newcommand{\sERI}[2]{[#1|#2]} %% bold in Table @@ -108,10 +111,6 @@ \newcommand{\OmRPAx}[1]{\Omega_{#1}^{\text{RPAx}}} \newcommand{\OmBSE}[1]{\Omega_{#1}^{\text{BSE}}} -\newcommand{\spinup}{\downarrow} -\newcommand{\spindw}{\uparrow} -\newcommand{\singlet}{\uparrow\downarrow} -\newcommand{\triplet}{\uparrow\uparrow} % Matrices \newcommand{\bO}{\mathbf{0}} @@ -157,6 +156,15 @@ \newcommand{\si}{\sigma} \newcommand{\sip}{\sigma'} +\newcommand{\up}{\downarrow} +\newcommand{\dw}{\uparrow} +\newcommand{\upup}{\uparrow\uparrow} +\newcommand{\updw}{\uparrow\downarrow} +\newcommand{\dwup}{\downarrow\uparrow} +\newcommand{\dwdw}{\downarrow\downarrow} +\newcommand{\spc}{\text{sc}} +\newcommand{\spf}{\text{sf}} + % addresses \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} diff --git a/sfBSE.tex b/sfBSE.tex index cf05480..96d9b49 100644 --- a/sfBSE.tex +++ b/sfBSE.tex @@ -45,63 +45,127 @@ Unless otherwise stated, atomic units are used, and we assume real quantities th \section{Unrestricted $GW$ formalism} \label{sec:UGW} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +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 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 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 excitations is $S^\spf = S_{\up\dw}^\spf + S_{\dw\up}^\spf = O_\up V_\dw + O_\dw V_\up$. +Let us denote as $\MO{p\si}$ the $p$th orbital of spin $\sigma$ (where $\sigma = \up$ or $\dw$). +In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, $p$, $q$, $r$, and $s$ indicate arbitrary orbitals, and $m$ labels single excitations. +It is important to understand that, in a spin-conserved excitation the hole orbital $\MO{i\si}$ and particle orbital $\MO{a\si}$ have the same spin $\si$. +A bra and ket composed by these two orbitals will be denoted as $\rbra{ia\si}$ and $\rket{ia\si}$. +In a spin-flip excitation, the hole and the particle states have different spin. +%================================ \subsection{The dynamical screening} +%================================ Within the $GW$ formalism, the dynamical screening $W(\omega)$ is computed at the RPA level using the spin-conserved neutral excitations. - +The matrix elements of $W(\omega)$ read \begin{multline} W_{pq\si,rs\sip}(\omega) = \ERI{pq\si}{rs\sip} + + \sum_m \sERI{pq\si}{m}\sERI{rs\sip}{m} \\ - + \sum_m \sERI{pq\si}{m}\sERI{rs\sip}{m} \qty[ \frac{1}{\omega - \OmRPA{m} + i \eta} - \frac{1}{\omega + \OmRPA{m} - i \eta} ] + \times \qty[ \frac{1}{\omega - \Om{m}{\spc,\RPA} + i \eta} - \frac{1}{\omega + \Om{m}{\spc,\RPA} - i \eta} ] \end{multline} - - +where the two-electron integrals are \begin{equation} \ERI{pq\si}{rs\sip} = \iint \MO{p\si}(\br) \MO{q\si}(\br) \frac{1}{\abs{\br - \br'}} \MO{r\sip}(\br') \MO{s\sip}(\br') d\br d\br' \end{equation} \begin{equation} - \sERI{pq\si}{m} = \sum_{ia\sip} \ERI{pq\si}{rs\sip} (\bX{m}{\RPA}+\bY{m}{\RPA})_{ia\sip} + \sERI{pq\si}{m} = \sum_{ia\sip} \ERI{pq\si}{rs\sip} (\bX{m}{\spc,\RPA}+\bY{m}{\spc,\RPA})_{ia\sip} \end{equation} \begin{equation} \label{eq:LR-RPA} \begin{pmatrix} - \bA{\RPA} & \bB{\RPA} \\ - -\bB{\RPA} & -\bA{\RPA} \\ + \bA{\spc,\RPA} & \bB{\spc,\RPA} \\ + -\bB{\spc,\RPA} & -\bA{\spc,\RPA} \\ \end{pmatrix} \cdot \begin{pmatrix} - \bX{m}{\RPA} \\ - \bY{m}{\RPA} \\ + \bX{m}{\spc,\RPA} \\ + \bY{m}{\spc,\RPA} \\ \end{pmatrix} = - \OmRPA{m} + \Om{m}{\spc,\RPA} \begin{pmatrix} - \bX{m}{\RPA} \\ - \bY{m}{\RPA} \\ + \bX{m}{\spc,\RPA} \\ + \bY{m}{\spc,\RPA} \\ \end{pmatrix}, \end{equation} + + +\begin{align} +\label{eq:LR-RPA-AB} + \bA{\spc} & = \begin{pmatrix} + \bA{\upup,\upup} & \bA{\upup,\dwdw} \\ + \bA{\dwdw,\upup} & \bA{\dwdw,\dwdw} \\ + \end{pmatrix} + & + \bB{\spc} & = \begin{pmatrix} + \bB{\upup,\upup} & \bB{\upup,\dwdw} \\ + \bB{\dwdw,\upup} & \bB{\dwdw,\dwdw} \\ + \end{pmatrix} +\end{align} + +\begin{align} +\label{eq:LR-RPA-AB} + \bA{\spf} & = \begin{pmatrix} + \bA{\updw,\updw} & \bO \\ + \bO & \bA{\dwup,\dwup} \\ + \end{pmatrix} + & + \bB{\spf} & = \begin{pmatrix} + \bO & \bB{\updw,\dwup} \\ + \bB{\dwup,\updw} & \bO \\ + \end{pmatrix} +\end{align} with \begin{subequations} \begin{align} - \label{eq:LR_RPA-A} - \A{ia\si,jb\sip}{\RPA} & = \delta_{ij} \delta_{ab} \delta_{\si\sip} (\e{a} - \e{i}) + 2 \ERI{ia\si}{jb\sip}, + \label{eq:LR_RPA-Asc} + \A{ia\si,jb\sip}{\spc,\RPA} & = \delta_{ij} \delta_{ab} \delta_{\si\sip} (\e{a\si} - \e{i\si}) + \ERI{ia\si}{jb\sip}, \\ - \label{eq:LR_RPA-B} - \B{ia\si,jb\sip}{\RPA} & = 2 \ERI{ia\si}{bj\sip}, + \label{eq:LR_RPA-Bsc} + \B{ia\si,jb\sip}{\spc,\RPA} & = 0, \end{align} \end{subequations} + +%================================ \subsection{The $GW$ self-energy} +%================================ + +\begin{equation} +\begin{split} + \SigC{pq\si}(\omega) + & = \sum_i \sum_m \frac{\sERI{pi\si}{m} \sERI{qi\si}{m}}{\omega - \e{i\si} + \Om{m}{\spc,\RPA} - i \eta} + \\ + & + \sum_a \sum_m \frac{\sERI{pa\si}{m} \sERI{qa\si}{m}}{\omega - \e{a\si} - \Om{m}{\spc,\RPA} + i \eta} +\end{split} +\end{equation} The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-dependent quasiparticle equation \begin{equation} - \omega = \eHF{p\sigma} + \SigGW{p\sigma}(\omega) + \omega = \eHF{p\sigma} + \SigC{p\sigma}(\omega) \end{equation} + +%================================ +\subsection{The Bethe-Salpeter formalism} +%================================ + +\begin{subequations} +\begin{align} + \label{eq:LR_BSE-Asc} + \A{ia\si,jb\sip}{\spc,\BSE} & = \delta_{ij} \delta_{ab} \delta_{\si\sip} (\eGW{a\si} - \eGW{i\si}) + \ERI{ia\si}{jb\sip} - W_{ij} + \\ + \label{eq:LR_BSE-Bsc} + \B{ia\si,jb\sip}{\spc,\BSE} & = 0, +\end{align} +\end{subequations} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Computational details} +\section{Computational details} \label{sec:compdet} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%