237 lines
8.7 KiB
TeX
237 lines
8.7 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
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\usepackage[version=4]{mhchem}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{txfonts}
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\usepackage[
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colorlinks=true,
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citecolor=blue,
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breaklinks=true
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]{hyperref}
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\urlstyle{same}
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\begin{document}
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\title{Spin-Conserved and Spin-Flip Optical Excitations From the Bethe-Salpeter Equation Formalism}
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\author{Enzo \surname{Monino}}
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\affiliation{\LCPQ}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\begin{abstract}
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\alert{Here comes the abstract.}
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%\bigskip
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%\begin{center}
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% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
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%\end{center}
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%\bigskip
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\end{abstract}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\alert{Here comes the introduction.}
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Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Unrestricted $GW$ formalism}
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\label{sec:UGW}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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.
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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.
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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 $S^\spf = S_{\up\dw}^\spf + S_{\dw\up}^\spf = O_\up V_\dw + O_\dw V_\up$.
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Let us denote as $\MO{p_\sig}$ the $p$th orbital of spin $\sig$ (where $\sig =$ $\up$ or $\dw$).
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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.
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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 $\sig$.
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A bra and ket composed by these two orbitals will be denoted as $\rbra{ia\sig}$ and $\rket{ia\sig}$.
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In a spin-flip excitation, the hole has a spin $\sig$ and the particle has the opposite spin $\bsig$.
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%================================
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\subsection{The dynamical screening}
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%================================
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Within the $GW$ formalism, the dynamical screening $W(\omega)$ is computed at the RPA level using the spin-conserved neutral excitations.
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The matrix elements of $W(\omega)$ read
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\begin{multline}
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W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp}
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+ \sum_m \ERI{p_\sig q_\sig}{m}\ERI{r_\sigp s_\sigp}{m}
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\\
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\times \qty[ \frac{1}{\omega - \Om{m}{\spc,\RPA} + i \eta} - \frac{1}{\omega + \Om{m}{\spc,\RPA} - i \eta} ]
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\end{multline}
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where the two-electron integrals are
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\begin{equation}
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\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'
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\end{equation}
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\begin{equation}
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\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}
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\end{equation}
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\begin{equation}
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\label{eq:LR-RPA}
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\begin{pmatrix}
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\bA{}{\spc,\RPA} & \bB{}{\spc,\RPA} \\
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-\bB{}{\spc,\RPA} & -\bA{}{\spc,\RPA} \\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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\bX{m}{\spc,\RPA} \\
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\bY{m}{\spc,\RPA} \\
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\end{pmatrix}
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=
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\Om{m}{\spc,\RPA}
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\begin{pmatrix}
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\bX{m}{\spc,\RPA} \\
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\bY{m}{\spc,\RPA} \\
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\end{pmatrix},
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\end{equation}
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The spin structure of these matrices are general and reads
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\begin{align}
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\label{eq:LR-RPA-AB}
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\bA{}{\spc} & = \begin{pmatrix}
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\bA{\upup,\upup}{} & \bA{\upup,\dwdw}{} \\
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\bA{\dwdw,\upup}{} & \bA{\dwdw,\dwdw}{} \\
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\end{pmatrix}
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&
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\bB{}{\spc} & = \begin{pmatrix}
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\bB{\upup,\upup}{} & \bB{\upup,\dwdw}{} \\
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\bB{\dwdw,\upup}{} & \bB{\dwdw,\dwdw}{} \\
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\end{pmatrix}
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\\
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\label{eq:LR-RPA-AB}
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\bA{}{\spf} & = \begin{pmatrix}
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\bA{\updw,\updw}{} & \bO \\
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\bO & \bA{\dwup,\dwup}{} \\
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\end{pmatrix}
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&
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\bB{}{\spf} & = \begin{pmatrix}
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\bO & \bB{\updw,\dwup}{} \\
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\bB{\dwup,\updw}{} & \bO \\
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\end{pmatrix}
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\end{align}
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with
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\begin{subequations}
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\begin{align}
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\label{eq:LR_RPA-A}
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\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}
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\\
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\label{eq:LR_RPA-B}
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\B{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} & = \ERI{i_\sig a_\tau}{j_\sigp b_\taup}
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\end{align}
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\end{subequations}
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from which we obtain, at the RPA level, the following expressions
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\begin{subequations}
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\begin{align}
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\label{eq:LR_RPA-Asc}
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\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}
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\\
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\label{eq:LR_RPA-Bsc}
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\B{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\RPA} & = \ERI{i_\sig a_\sig}{j_\sigp b_\sigp}
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\end{align}
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\end{subequations}
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for the spin-conserved excitations and
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\begin{subequations}
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\begin{align}
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\label{eq:LR_RPA-Asf}
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\A{i_\sig a_\bsig,j_\sig b_\bsig}{\spf,\RPA} & = \delta_{ij} \delta_{ab} (\e{a_\bsig} - \e{i_\sig})
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\\
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\label{eq:LR_RPA-Bsf}
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\B{i_\sig a_\bsig,j_\bsig b_\sig}{\spf,\RPA} & = 0
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\end{align}
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\end{subequations}
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for the spin-flip excitations.
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%================================
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\subsection{The $GW$ self-energy}
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%================================
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\begin{equation}
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\begin{split}
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\SigC{p_\sig q_\sig}(\omega)
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& = \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}
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\\
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& + \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}
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\end{split}
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\end{equation}
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The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-dependent quasiparticle equation
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\begin{equation}
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\omega = \eHF{p\sigma} + \SigC{p\sigma}(\omega)
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\end{equation}
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%================================
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\subsection{The Bethe-Salpeter equation formalism}
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%================================
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Defining $W^{\stat}_{p_\sig q_\sig,r_\sigp s_\sigp} = W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega = 0)$, we have
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\begin{subequations}
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\begin{align}
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\label{eq:LR_BSE-A}
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\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}
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\\
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\label{eq:LR_BSE-B}
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\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}
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\end{align}
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\end{subequations}
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from which we obtain, at the BSE level, the following expressions
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\begin{subequations}
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\begin{align}
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\label{eq:LR_BSE-Asc}
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\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}
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\\
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\label{eq:LR_BSE-Bsc}
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\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}
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\end{align}
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\end{subequations}
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for the spin-conserved excitations and
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\begin{subequations}
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\begin{align}
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\label{eq:LR_BSE-Asf}
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\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}
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\\
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\label{eq:LR_BSE-Bsf}
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\B{i_\sig a_\bsig,j_\bsig b_\sig}{\spf,\BSE} & = - W^{\stat}_{i_\sig b_\sig,j_\bsig a_\bsig}
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\end{align}
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\end{subequations}
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for the spin-flip excitations.
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%================================
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\subsection{The dynamical Bethe-Salpeter equation correction}
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%================================
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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\label{sec:compdet}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Conclusion}
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\label{sec:ccl}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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\acknowledgements{
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We would like to thank Xavier Blase and Denis Jacquemin for insightful discussions.
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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).}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Data availability statement}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The data that supports the findings of this study are available within the article and its supplementary material.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\bibliography{sf-BSE}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}
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