diff --git a/sfBSE.rty b/sfBSE.rty index 9aa34e4..1583935 100644 --- a/sfBSE.rty +++ b/sfBSE.rty @@ -127,9 +127,8 @@ \newcommand{\bdeHF}{\mathbf{\Delta\epsilon}^\text{HF}} \newcommand{\bdeGW}{\mathbf{\Delta\epsilon}^{GW}} \newcommand{\bOm}[1]{\mathbf{\Omega}^{#1}} -\newcommand{\bA}[1]{\mathbf{A}^{#1}} -\newcommand{\btA}[1]{\Tilde{\mathbf{A}}^{#1}} -\newcommand{\bB}[1]{\mathbf{B}^{#1}} +\newcommand{\bA}[2]{\mathbf{A}_{#1}^{#2}} +\newcommand{\bB}[2]{\mathbf{B}_{#1}^{#2}} \newcommand{\bX}[2]{\mathbf{X}_{#1}^{#2}} \newcommand{\bY}[2]{\mathbf{Y}_{#1}^{#2}} \newcommand{\bZ}[2]{\mathbf{Z}_{#1}^{#2}} @@ -153,8 +152,11 @@ \newcommand{\EgOpt}{\Eg^\text{opt}} \newcommand{\EB}{E_B} -\newcommand{\si}{\sigma} -\newcommand{\sip}{\sigma'} +\newcommand{\sig}{\sigma} +\newcommand{\bsig}{{\Bar{\sigma}}} +\newcommand{\sigp}{{\sigma'}} +\newcommand{\bsigp}{{\Bar{\sigma}'}} +\newcommand{\taup}{{\tau'}} \newcommand{\up}{\downarrow} \newcommand{\dw}{\uparrow} @@ -165,7 +167,6 @@ \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 96d9b49..539219a 100644 --- a/sfBSE.tex +++ b/sfBSE.tex @@ -48,11 +48,11 @@ Unless otherwise stated, atomic units are used, and we assume real quantities th 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 $\dw$). +Let us denote as $\MO{p_\sig}$ the $p$th orbital of spin $\sig$ (where $\sig =$ $\up$ 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. +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 $\sig$. +A bra and ket composed by these two orbitals will be denoted as $\rbra{ia\sig}$ and $\rket{ia\sig}$. +In a spin-flip excitation, the hole has a spin $\sig$ and the particle has the opposite spin $\bsig$. %================================ \subsection{The dynamical screening} @@ -61,25 +61,25 @@ In a spin-flip excitation, the hole and the particle states have different spin. 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} + W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} + + \sum_m \ERI{p_\sig q_\sig}{m}\ERI{r_\sigp s_\sigp}{m} \\ \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' + \ERI{p_\sig q_\tau}{r_\sigp s_\taup} = \iint \MO{p_\sig}(\br) \MO{q_\tau}(\br) \frac{1}{\abs{\br - \br'}} \MO{r_\sigp}(\br') \MO{s_\taup}(\br') d\br d\br' \end{equation} \begin{equation} - \sERI{pq\si}{m} = \sum_{ia\sip} \ERI{pq\si}{rs\sip} (\bX{m}{\spc,\RPA}+\bY{m}{\spc,\RPA})_{ia\sip} + \ERI{p_\sig q_\sig}{m} = \sum_{ia\sigp} \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} (\bX{m}{\spc,\RPA}+\bY{m}{\spc,\RPA})_{i_\sigp a_\sigp} \end{equation} \begin{equation} \label{eq:LR-RPA} \begin{pmatrix} - \bA{\spc,\RPA} & \bB{\spc,\RPA} \\ - -\bB{\spc,\RPA} & -\bA{\spc,\RPA} \\ + \bA{}{\spc,\RPA} & \bB{}{\spc,\RPA} \\ + -\bB{}{\spc,\RPA} & -\bA{}{\spc,\RPA} \\ \end{pmatrix} \cdot \begin{pmatrix} @@ -94,43 +94,61 @@ where the two-electron integrals are \end{pmatrix}, \end{equation} - +The spin structure of these matrices are general and reads \begin{align} \label{eq:LR-RPA-AB} - \bA{\spc} & = \begin{pmatrix} - \bA{\upup,\upup} & \bA{\upup,\dwdw} \\ - \bA{\dwdw,\upup} & \bA{\dwdw,\dwdw} \\ + \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} \\ + \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} \\ + \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 \\ + \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-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-A} + \A{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} & = \delta_{ij} \delta_{ab} \delta_{\sig \sigp} \delta_{\tau \taup} (\e{a_\tau} - \e{i_\sig}) + \ERI{i_\sig a_\tau}{b_\sigp j_\taup} \\ - \label{eq:LR_RPA-Bsc} - \B{ia\si,jb\sip}{\spc,\RPA} & = 0, + \label{eq:LR_RPA-B} + \B{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} & = \ERI{i_\sig a_\tau}{j_\sigp b_\taup} \end{align} \end{subequations} - +from which we obtain, at the RPA level, the following expressions +\begin{subequations} +\begin{align} + \label{eq:LR_RPA-Asc} + \A{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\RPA} & = \delta_{ij} \delta_{ab} \delta_{\sig \sigp} (\e{a_\sig} - \e{i_\sig}) + \ERI{i_\sig a_\sig}{b_\sigp j_\sigp} + \\ + \label{eq:LR_RPA-Bsc} + \B{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\RPA} & = \ERI{i_\sig a_\sig}{j_\sigp b_\sigp} +\end{align} +\end{subequations} +for the spin-conserved excitations and +\begin{subequations} +\begin{align} + \label{eq:LR_RPA-Asf} + \A{i_\sig a_\bsig,j_\sig b_\bsig}{\spf,\RPA} & = \delta_{ij} \delta_{ab} (\e{a_\bsig} - \e{i_\sig}) + \\ + \label{eq:LR_RPA-Bsf} + \B{i_\sig a_\bsig,j_\bsig b_\sig}{\spf,\RPA} & = 0 +\end{align} +\end{subequations} +for the spin-flip excitations. %================================ \subsection{The $GW$ self-energy} @@ -138,10 +156,10 @@ with \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} + \SigC{p_\sig q_\sig}(\omega) + & = \sum_i \sum_m \frac{\ERI{p_\sig i_\sig}{m} \ERI{q_\sig i_\sig}{m}}{\omega - \e{i_\sig} + \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} + & + \sum_a \sum_m \frac{\ERI{p_\sig a_\sig}{m} \ERI{q_\sig a_\sig}{m}}{\omega - \e{a_\sig} - \Om{m}{\spc,\RPA} + i \eta} \end{split} \end{equation} @@ -151,19 +169,44 @@ The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-depen \end{equation} %================================ -\subsection{The Bethe-Salpeter formalism} +\subsection{The Bethe-Salpeter equation formalism} %================================ - +Defining $W^{\stat}_{p_\sig q_\sig,r_\sigp s_\sigp} = W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega = 0)$, we have +\begin{subequations} +\begin{align} + \label{eq:LR_BSE-A} + \A{i_\sig a_\tau,j_\sigp b_\taup}{\BSE} & = \A{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} - \delta_{\sig \sigp} W^{\stat}_{i_\sig j_\sigp,b_\taup a_\tau} + \\ + \label{eq:LR_BSE-B} + \B{i_\sig a_\tau,j_\sigp b_\taup}{\BSE} & = \B{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} - \delta_{\sig \sigp} W^{\stat}_{i_\sig b_\taup,j_\sigp a_\tau} +\end{align} +\end{subequations} +from which we obtain, at the BSE level, the following expressions \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} + \A{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\BSE} & = \A{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\RPA} - \delta_{\sig \sigp} W^{\stat}_{i_\sig j_\sigp,b_\sigp a_\sig} \\ \label{eq:LR_BSE-Bsc} - \B{ia\si,jb\sip}{\spc,\BSE} & = 0, + \B{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\BSE} & = \B{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\RPA} - \delta_{\sig \sigp} W^{\stat}_{i_\sig b_\sigp,j_\sigp a_\sig} \end{align} \end{subequations} +for the spin-conserved excitations and +\begin{subequations} +\begin{align} + \label{eq:LR_BSE-Asf} + \A{i_\sig a_\bsig,j_\sig b_\bsig}{\spf,\BSE} & = \A{i_\sig a_\bsig,j_\sig b_\bsig}{\spf,\RPA} - W^{\stat}_{i_\sig j_\sig,b_\bsig a_\bsig} + \\ + \label{eq:LR_BSE-Bsf} + \B{i_\sig a_\bsig,j_\bsig b_\sig}{\spf,\BSE} & = - W^{\stat}_{i_\sig b_\sig,j_\bsig a_\bsig} +\end{align} +\end{subequations} +for the spin-flip excitations. +%================================ +\subsection{The dynamical Bethe-Salpeter equation correction} + +%================================ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Computational details} \label{sec:compdet} @@ -178,7 +221,7 @@ The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-depen %%%%%%%%%%%%%%%%%%%%%%%% \acknowledgements{ We would like to thank Xavier Blase and Denis Jacquemin for insightful discussions. -This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481). +This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).} %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%