diff --git a/sfBSE.tex b/sfBSE.tex index b4eb8a6..1eb649f 100644 --- a/sfBSE.tex +++ b/sfBSE.tex @@ -68,15 +68,18 @@ The spin-$\sig$ component of the one-body Green's function reads \cite{ReiningBo 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) \begin{equation} +\label{eq:chi0} \chi_0(\br_1,\br_2;\omega) = - \frac{i}{2\pi} \sum_\sig \int G^{\sig}(\br_1,\br_2;\omega+\omega') G^{\sig}(\br_1,\br_2;\omega') d\omega' \end{equation} and subsequently the dielectric function \begin{equation} +\label{eq:eps} \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} where $\delta(\br_1 - \br_2)$ is the Dirac function. Based on this latter ingredient, one can access the dynamically-screened Coulomb potential \begin{equation} +\label{eq:W} 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} which is naturally spin independent as the bare Coulomb interaction $\abs{\br_1 - \br_2}^{-1}$ does not depend on spin coordinates. @@ -84,7 +87,7 @@ which is naturally spin independent as the bare Coulomb interaction $\abs{\br_1 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 $W$ is \begin{multline} -\label{eq:W} +\label{eq:W_spectral} 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} \\ @@ -99,7 +102,7 @@ and the screened two-electron integrals (or spectral weights) are explicitly giv \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} \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, $\bX{m}{\spc,\RPA}$ and $\bY{m}{\spc,\RPA}$, are obtained by solving a linear response system of the form +In Eqs.~\eqref{eq:W_spectral} 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} \label{eq:LR-RPA} \begin{pmatrix} @@ -197,6 +200,7 @@ for the spin-flip excitations. %================================ Within the acclaimed $GW$ approximation, \cite{Hedin_1965,Golze_2019} the exchange-correlation (xc) part of the self-energy \begin{equation} +\label{eq:Sig} \begin{split} \Sig{}{\xc,\sig}(\br_1,\br_2;\omega) & = \Sig{}{\x,\sig}(\br_1,\br_2) + \Sig{}{\co,\sig}(\br_1,\br_2;\omega) @@ -218,23 +222,52 @@ is, like the one-body Green's function, spin-diagonal, and its spectral represen \end{gather} 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 -\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} - \omega = \e{p_\sig}{} - V_{p_\sigma}^{\xc} + \SigX{p\sigma} + \SigC{p\sigma}(\omega) +\label{eq:Dyson_G} +\begin{split} + \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{split} \end{equation} -with +where $G_{\KS}^{\sig}$ is the Kohn-Sham Green's function built with Kohn-Sham orbitals $\MO{p_\sig}^{\KS}(\br)$ and one-electron energies $\e{p_\sig}^{\KS}$ according to Eq.~\eqref{eq:G} and $v^{\xc}(\br)$ is the Kohn-Sham local exchange-correlation potential. +The target quantities here are the quasiparticle energies $\eGW{p_\sig}$, \ie, the poles of $G$ [see Eq.~\eqref{eq:G}], which correspond to well-defined addition/removal energies (unlike the KS orbital energies). +Because the exchange-correlation part of the self-energy is, itself, constructed with the Green's function [see Eq.~\eqref{eq:Sig}], the present process is, by nature, self-consistent. +The same comment applies to the dynamically-screened Coulomb potential $W$ entering the definition of $\Sig{}{\xc}$ [see Eq.~\eqref{eq:Sig}] which is also constructed from $G$ [see Eqs.~\eqref{eq:chi0}, \eqref{eq:eps}, and \eqref{eq:W}]. + +%================================ +\subsection{Level of self-consistency} +%================================ +This is where $GW$ schemes differ. +In its simplest perturbative (\ie, one-shot) version, known as {\GOWO}, a single iteration is performed, and the quasiparticle energies $\eGOWO{p_\sig}$ are obtained by solving the frequency-dependent quasiparticle equation +\begin{equation} +\label{eq:QP-eq} + \omega = \e{p_\sig}{} + \Sig{p_\sig}{\xc}(\omega) - V_{p_\sig}^{\xc} +\end{equation} +where $\Sig{p_\sig}{\xc}(\omega) \equiv \Sig{p_\sig p_\sig}{\xc}(\omega)$ and its offspring quantities have been constructed at the Kohn-Sham level, and \begin{equation} V_{p_\sigma}^{\xc} = \int \MO{p_\sig}(\br) v^{\xc}(\br) \MO{p_\sig}(\br) d\br \end{equation} -\alert{Introduce linearization of the quasiparticle equation and different degree of self-consistency.} +Because, from a practical point of view, one is usually interested by the so-called quasiparticle solution (or peak), the quasiparticle equation \eqref{eq:QP-eq} is often linearized around $\omega = \e{p_\sig}^{\KS}$, yielding +\begin{equation} + \eGOWO{p_\sig} + = \e{p_\sig}^{\KS} + Z_{p_\sig} [\Sig{p_\sig p_\sig}{\xc}(\e{p_\sig}^{\KS}) - V_{p_\sig}^{\xc} ] +\end{equation} +where +\begin{equation} + Z_{p_\sig} = \qty[ 1 - \left. \pdv{\Sig{p_\sig p_\sig}{\xc}(\omega)}{\omega} \right|_{\omega = \e{p_\sig}^{\KS}} ]^{-1} +\end{equation} +is a renormalization factor which also represents the spectral weight of the quasiparticle solution. +In addition to the principal quasiparticle peak which, in a well-behaved case, contains most of the spectral weight, the frequency-dependent quasiparticle equation \eqref{eq:QP-eq} generates a finite number of satellite resonances with smaller weights. + +Within the ``eigenvalue'' self-consistent $GW$ scheme (known as ev$GW$), several iterations are performed during which only the one-electron energies entering the definition of the Green's function [see Eq.~\eqref{eq:G}] are updated by the quasiparticle energies obtained at the previous iteration (the corresponding orbitals remain evaluated at the Kohn-Sham level). + +Finally, within the quasiparticle self-consistent $GW$ scheme (qs$GW$), both the one-electron energies and the orbitals are updated until convergence is reached. +These are obtained via the diagonalization of an effective Fock matrix which includes explicitly a frequency-independent and hermitian self-energy defined as +\begin{equation} + \Tilde{\Sigma}_{p_\sig q_\sig}^{\xc} = \frac{1}{2} \qty[ \Sig{p_\sig q_\sig}{\xc}(\e{p_\sig}{}) + \Sig{q_\sig p_\sig}{\xc}(\e{p_\sig}{}) ] +\end{equation} %================================ \subsection{The Bethe-Salpeter equation formalism}