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sfBSE.rty
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@ -26,7 +26,8 @@
\newcommand{\Hxc}{\text{Hxc}} \newcommand{\Hxc}{\text{Hxc}}
\newcommand{\xc}{\text{xc}} \newcommand{\xc}{\text{xc}}
\newcommand{\Ha}{\text{H}} \newcommand{\Ha}{\text{H}}
\newcommand{\co}{\text{x}} \newcommand{\co}{\text{c}}
\newcommand{\x}{\text{x}}
% %
\newcommand{\Norb}{N_\text{orb}} \newcommand{\Norb}{N_\text{orb}}
@ -92,7 +93,7 @@
\newcommand{\tW}[2]{\widetilde{W}_{#1}^{#2}} \newcommand{\tW}[2]{\widetilde{W}_{#1}^{#2}}
\newcommand{\Wc}[1]{W^\text{c}_{#1}} \newcommand{\Wc}[1]{W^\text{c}_{#1}}
\newcommand{\vc}[1]{v_{#1}} \newcommand{\vc}[1]{v_{#1}}
\newcommand{\Sig}[1]{\Sigma_{#1}} \newcommand{\Sig}[2]{\Sigma_{#1}^{#2}}
\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}} \newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}} \newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
\newcommand{\SigXC}[1]{\Sigma^\text{xc}_{#1}} \newcommand{\SigXC}[1]{\Sigma^\text{xc}_{#1}}

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sfBSE.tex
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@ -40,20 +40,19 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\alert{Here comes the introduction.} \alert{Here comes the introduction.}
Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript. Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript.
In the following, we consider systems with collinear spins and a spin-independent hamiltonian without contributions such as spin-orbit interaction.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Unrestricted $GW$ formalism} \section{Unrestricted $GW$ formalism}
\label{sec:UGW} \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. 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-up and spin-down occupied orbitals are $O_\up = n_\up$ and $O_\dw = n_\dw$, respectively, and, assuming no 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 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 $S^\spf = S_{\up\dw}^\spf + S_{\dw\up}^\spf = O_\up V_\dw + O_\dw V_\up$. 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 $\MO{p_\sig}$ the $p$th orbital of 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 (spin)orbital of spin $\sig$ (where $\sig =$ $\up$ or $\dw$) and $\e{p_\sig}{}$ its one-electron energy.
In the present context these orbitals can originate from a HF or KS calculation.
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 $\MO{i_\sig}$ 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 $\bsig$.
In the following, 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.
Moreover, we consider systems with collinear spins and a spin-independent hamiltonian without contributions such as spin-orbit interaction.
%================================ %================================
\subsection{The dynamical screening} \subsection{The dynamical screening}
@ -61,9 +60,10 @@ In a spin-flip excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{
The pillar of Green's function many-body perturbation theory is the (time-ordered) one-body Green's function, which has poles at the charged excitations (i.e., ionization potentials and electron affinities) of the system. \cite{ReiningBook} The pillar of Green's function many-body perturbation theory is the (time-ordered) one-body Green's function, which has poles at the charged excitations (i.e., ionization potentials and electron affinities) of the system. \cite{ReiningBook}
The spin-$\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_2016a}
\begin{equation} \begin{equation}
G^{\sig}(\br_1,\br_2;\omega) \label{eq:G}
= \sum_i \frac{\MO{i_\sig}(\br_1) \MO{i_\sig}(\br_2)}{\omega - \e{i_\sig}{} - i\eta} G^{\sig}(\br_1,\br_2;\omega)
+ \sum_a \frac{\MO{a_\sig}(\br_1) \MO{a_\sig}(\br_2)}{\omega - \e{a_\sig}{} + i\eta} = \sum_i \frac{\MO{i_\sig}(\br_1) \MO{i_\sig}(\br_2)}{\omega - \e{i_\sig}{} - i\eta}
+ \sum_a \frac{\MO{a_\sig}(\br_1) \MO{a_\sig}(\br_2)}{\omega - \e{a_\sig}{} + i\eta}
\end{equation} \end{equation}
where $\eta$ is a positive infinitesimal. where $\eta$ is a positive infinitesimal.
Based on the spin-up and spin-down components of $G$, one can easily compute the non-interacting polarizability (which is a sum over spins) Based on the spin-up and spin-down components of $G$, one can easily compute the non-interacting polarizability (which is a sum over spins)
@ -72,17 +72,17 @@ Based on the spin-up and spin-down components of $G$, one can easily compute the
\end{equation} \end{equation}
and subsequently the dielectric function and subsequently the dielectric function
\begin{equation} \begin{equation}
\epsilon(\br_1,\br-2;\omega) = \delta(\br_1 - \br_2) - \int \frac{\chi_0(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3 \epsilon(\br_1,\br_2;\omega) = \delta(\br_1 - \br_2) - \int \frac{\chi_0(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3
\end{equation} \end{equation}
where $\delta(\br_1 - \br_2)$ is the Dirac function. where $\delta(\br_1 - \br_2)$ is the Dirac function.
Based on this latter ingredient, one can access the dynamically-screened Coulomb potential Based on this latter ingredient, one can access the dynamically-screened Coulomb potential
\begin{equation} \begin{equation}
W(\br_1,\br_2;\omega) = \int \frac{\epsilon^{-1}(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3 W(\br_1,\br_2;\omega) = \int \frac{\epsilon^{-1}(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3
\end{equation} \end{equation}
which is spin independent as the bare Coulomb interaction $\abs{\br_1 - \br_2}^{-1}$ does not depend on spin coordinates. which is naturally spin independent as the bare Coulomb interaction $\abs{\br_1 - \br_2}^{-1}$ does not depend on spin coordinates.
Within the $GW$ formalism, the is computed at the RPA level by considering only the manifold of the spin-conserved neutral excitation. Within the $GW$ formalism, the dynamical screening is computed at the random-phase approximation (RPA) level by considering only the manifold of the spin-conserved neutral excitations.
In the orbital basis, the spectral representation of $W$ reads In the orbital basis, the spectral representation of $W$ is
\begin{multline} \begin{multline}
\label{eq:W} \label{eq:W}
W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp}
@ -92,14 +92,14 @@ In the orbital basis, the spectral representation of $W$ reads
\end{multline} \end{multline}
where the bare two-electron integrals are \cite{Gill_1994} where the bare two-electron integrals are \cite{Gill_1994}
\begin{equation} \begin{equation}
\label{eq:sERI}
\ERI{p_\sig q_\tau}{r_\sigp s_\taup} = \int \frac{\MO{p_\sig}(\br_1) \MO{q_\tau}(\br_1) \MO{r_\sigp}(\br_2) \MO{s_\taup}(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2 \ERI{p_\sig q_\tau}{r_\sigp s_\taup} = \int \frac{\MO{p_\sig}(\br_1) \MO{q_\tau}(\br_1) \MO{r_\sigp}(\br_2) \MO{s_\taup}(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2
\end{equation} \end{equation}
and the screened two-electron integrals (or spectral weights) are explicitly given by and the screened two-electron integrals (or spectral weights) are explicitly given by
\begin{equation} \begin{equation}
\label{eq:sERI}
\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} \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} \end{equation}
In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, the RPA spin-conserved neutral excitations $\Om{m}{\spc,\RPA}$ and their corresponding eigenvectors $(\bX{m}{\spc,\RPA}+\bY{m}{\spc,\RPA})$ are obtained by solving a linear response system of the form In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, the RPA spin-conserved neutral excitations $\Om{m}{\spc,\RPA}$ and their corresponding eigenvectors, $\bX{m}{\spc,\RPA}$ and $\bY{m}{\spc,\RPA}$, are obtained by solving a linear response system of the form
\begin{equation} \begin{equation}
\label{eq:LR-RPA} \label{eq:LR-RPA}
\begin{pmatrix} \begin{pmatrix}
@ -120,6 +120,7 @@ In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, the RPA spin-conserved neutral excitat
\end{equation} \end{equation}
where the expressions of the matrix elements of $\bA{}{}$ and $\bB{}{}$ are specific of the method and of the spin manifold. where the expressions of the matrix elements of $\bA{}{}$ and $\bB{}{}$ are specific of the method and of the spin manifold.
The spin structure of these matrices, though, is general The spin structure of these matrices, though, is general
\begin{subequations}
\begin{align} \begin{align}
\label{eq:LR-RPA-AB} \label{eq:LR-RPA-AB}
\bA{}{\spc} & = \begin{pmatrix} \bA{}{\spc} & = \begin{pmatrix}
@ -143,6 +144,7 @@ The spin structure of these matrices, though, is general
\bB{}{\dwup,\updw} & \bO \\ \bB{}{\dwup,\updw} & \bO \\
\end{pmatrix} \end{pmatrix}
\end{align} \end{align}
\end{subequations}
In the absence of instabilities, the linear eigenvalue problem \eqref{eq:LR-RPA} has particle-hole symmetry which means that the eigenvalues are obtained by pairs $\pm \Om{m}{}$. In the absence of instabilities, the linear eigenvalue problem \eqref{eq:LR-RPA} has particle-hole symmetry which means that the eigenvalues are obtained by pairs $\pm \Om{m}{}$.
In such a case, $(\bA{}{}-\bB{}{})^{1/2}$ is positive definite, and Eq.~\eqref{eq:LR-RPA} can be recast as a Hermitian problem of half the dimension In such a case, $(\bA{}{}-\bB{}{})^{1/2}$ is positive definite, and Eq.~\eqref{eq:LR-RPA} can be recast as a Hermitian problem of half the dimension
\begin{equation} \begin{equation}
@ -153,9 +155,11 @@ where the excitation amplitudes are
\begin{equation} \begin{equation}
\bX{}{} + \bY{}{} = \bOm{-1/2} \cdot (\bA{}{} - \bB{}{})^{1/2} \cdot \bZ{}{} \bX{}{} + \bY{}{} = \bOm{-1/2} \cdot (\bA{}{} - \bB{}{})^{1/2} \cdot \bZ{}{}
\end{equation} \end{equation}
Within the Tamm-Dancoff approximation (TDA), the coupling terms between the resonant and anti-resonant parts, $\bA{}{}$ and $-\bA{}{}$, are neglected, which consist in setting $\bB{}{} = \bO$. Within the Tamm-Dancoff approximation (TDA), the coupling terms between the resonant and anti-resonant parts, $\bA{}{}$ and $-\bA{}{}$, are neglected, which consists in setting $\bB{}{} = \bO$.
In such a case, Eq.~\eqref{eq:LR-RPA} reduces to $\bA{}{} \cdot \bX{m}{} = \Om{m}{} \bX{m}{}$. In such a case, Eq.~\eqref{eq:LR-RPA} reduces to straightforward Hermitian problem of the form:
\begin{equation}
\bA{}{} \cdot \bX{m}{} = \Om{m}{} \bX{m}{}
\end{equation}
At the RPA level, the matrix elements of $\bA{}{}$ and $\bB{}{}$ are At the RPA level, the matrix elements of $\bA{}{}$ and $\bB{}{}$ are
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
@ -194,26 +198,35 @@ for the spin-flip excitations.
Within the acclaimed $GW$ approximation, \cite{Hedin_1965,Golze_2019} the exchange-correlation (xc) part of the self-energy Within the acclaimed $GW$ approximation, \cite{Hedin_1965,Golze_2019} the exchange-correlation (xc) part of the self-energy
\begin{equation} \begin{equation}
\begin{split} \begin{split}
\Sig{}^{\text{xc},\sig}(\br_1,\br_2;\omega) \Sig{}{\xc,\sig}(\br_1,\br_2;\omega)
& = \Sig{}^{\text{x},\sig}(\br_1,\br_2) + \Sig{}^{\text{c},\sig}(\br_1,\br_2;\omega) & = \Sig{}{\x,\sig}(\br_1,\br_2) + \Sig{}{\co,\sig}(\br_1,\br_2;\omega)
\\ \\
& = \frac{i}{2\pi} \int G^{\sig}(\br_1,\br_2;\omega+\omega') W(\br_1,\br_2;\omega') e^{i \eta \omega'} d\omega' & = \frac{i}{2\pi} \int G^{\sig}(\br_1,\br_2;\omega+\omega') W(\br_1,\br_2;\omega') e^{i \eta \omega'} d\omega'
\end{split} \end{split}
\end{equation} \end{equation}
is, like the one-body Green's function, spin-diagonal, and its spectral representation reads is, like the one-body Green's function, spin-diagonal, and its spectral representation reads
\begin{gather} \begin{gather}
\SigX{p_\sig q_\sig} \Sig{p_\sig q_\sig}{\x}
= - \frac{1}{2} \sum_{i\sigp} \ERI{p_\sig i_\sigp}{i_\sigp q_\sig} = - \sum_{i} \ERI{p_\sig i_\sig}{i_\sig q_\sig}
\\ \\
\begin{split} \begin{split}
\SigC{p_\sig q_\sig}(\omega) \Sig{p_\sig q_\sig}{\co}(\omega)
& = \sum_{im} \frac{\ERI{p_\sig i_\sig}{m} \ERI{q_\sig i_\sig}{m}}{\omega - \e{i_\sig} + \Om{m}{\spc,\RPA} - i \eta} & = \sum_{im} \frac{\ERI{p_\sig i_\sig}{m} \ERI{q_\sig i_\sig}{m}}{\omega - \e{i_\sig} + \Om{m}{\spc,\RPA} - i \eta}
\\ \\
& + \sum_{am} \frac{\ERI{p_\sig a_\sig}{m} \ERI{q_\sig a_\sig}{m}}{\omega - \e{a_\sig} - \Om{m}{\spc,\RPA} + i \eta} & + \sum_{am} \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{split}
\end{gather} \end{gather}
which has been split in its exchange (x) and correlation (c) contributions. which the self-energy has been split in its exchange (x) and correlation (c) contributions.
The Dyson equation linking the Green's function and the self-energy holds separately for each spin component, and the quasiparticle energies $\eGW{p_\sig}$ are obtained by solving the frequency-dependent quasiparticle equation The Dyson equation linking the Green's function and the self-energy holds separately for each spin component
\begin{multline}
\qty[ G^{\sig}(\br_1,\br_2;\omega) ]^{-1}
= \qty[ G_{\KS}^{\sig}(\br_1,\br_2;\omega) ]^{-1}
\\
+ \Sig{}{\xc,\sig}(\br_1,\br_2;\omega) - v^{\xc}(\br_1) \delta(\br_1 - \br_2)
\end{multline}
where $G_{\KS}^{\sig}$ is the Kohn-Sham Green's function built with Kohn-Sham orbitals and one-electron energies according to Eq.~\eqref{eq:G} and $v^{\xc}(\br)$ is the Kohn-Sham local exchange-correlation potential.
The quasiparticle energies $\eGW{p_\sig}$ are obtained by solving the frequency-dependent quasiparticle equation
\begin{equation} \begin{equation}
\omega = \e{p_\sig}{} - V_{p_\sigma}^{\xc} + \SigX{p\sigma} + \SigC{p\sigma}(\omega) \omega = \e{p_\sig}{} - V_{p_\sigma}^{\xc} + \SigX{p\sigma} + \SigC{p\sigma}(\omega)
\end{equation} \end{equation}
@ -221,8 +234,7 @@ with
\begin{equation} \begin{equation}
V_{p_\sigma}^{\xc} = \int \MO{p_\sig}(\br) v^{\xc}(\br) \MO{p_\sig}(\br) d\br V_{p_\sigma}^{\xc} = \int \MO{p_\sig}(\br) v^{\xc}(\br) \MO{p_\sig}(\br) d\br
\end{equation} \end{equation}
where $v^{\xc}(\br)$ the Kohn-Sham exchange-correlation. \alert{Introduce linearization of the quasiparticle equation and different degree of self-consistency.}
\alert{Adding the Dyson equation? Introduce linearization of the quasiparticle equation and different degree of self-consistency.}
%================================ %================================
\subsection{The Bethe-Salpeter equation formalism} \subsection{The Bethe-Salpeter equation formalism}