From c04055a4c8feb41118fdafecaa3800207a2a9557 Mon Sep 17 00:00:00 2001 From: pfloos Date: Mon, 23 Jan 2023 21:49:44 +0100 Subject: [PATCH] starting modifying theory section --- Manuscript/SRGGW.tex | 45 +++++++++++++++++++++++--------------------- 1 file changed, 24 insertions(+), 21 deletions(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 2bfc537..2a75ec1 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -94,7 +94,7 @@ The $GW$ method approximates the self-energy $\Sigma$ which relates the exact in \label{eq:dyson} G(1,2) = G_0(1,2) + \int d(34) G_0(1,3)\Sigma(3,4) G(4,2), \end{equation} -where $1 = (\sigma_1, \br_1, t_1)$ is a composite coordinate gathering spin, space, and time variables. +where $1 = (\bx_1, t_1)$ is a composite coordinate gathering spin-space and time variables. The self-energy encapsulates all the Hartree-exchange-correlation effects which are not taken into account in the reference system. %Throughout this manuscript the references are chosen to be the Hartree-Fock (HF) ones so that the self-energy only account for the missing correlation. Approximating $\Sigma$ as the first-order term of its perturbative expansion with respect to the screened Coulomb potential $W$ yields the so-called $GW$ approximation \cite{Hedin_1965,Martin_2016} @@ -148,33 +148,38 @@ This section starts by \label{sec:gw} %%%%%%%%%%%%%%%%%%%%%% +\PFL{Antoine, please move the various expressions related to the $GW$ quantities in this section.} + +\titou{Here and in the following, we assume a Hartree-Fock (HF) starting point.} The central equation of many-body perturbation theory based on Hedin's equations is the so-called quasi-particle equation \begin{equation} \label{eq:quasipart_eq} - \left[ \bF + \bSig(\omega = \epsilon_p) \right] \psi_p = \epsilon_p \psi_p, + \qty[ \bF + \bSig(\omega = \epsilon_p) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx), \end{equation} -where $\bF$ is the Fock matrix, \cite{SzaboBook} and $\bSig(\omega)$ is the self-energy, both are $K \times K$ matrices with $K$ the number of one-body basis functions. -The self-energy can be physically understood as a dynamical \titou{screening} correction to the Hartree-Fock (HF) problem represented by $\bF$. -Similarly to the HF case, this equation needs to be solved self-consistently. -Note that $\bSig$ is dynamical, \ie it depends on both the eigenvalues $\epsilon_p$ and eigenvectors $\psi_p$ while $\bF$ depends only on the eigenvectors. +where $\bF$ is the Fock matrix \cite{SzaboBook} and $\bSig(\omega)$ is \titou{(the correlation part of)} the self-energy. +Both are $K \times K$ matrices with $K$ the number of one-electron orbitals. +The self-energy can be physically understood as a dynamical \titou{screening} correction to the HF problem represented by $\bF$. +Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-consistently. +Note that $\bSig$ is dynamical, \titou{\ie} it depends on the one-electron orbitals $\psi_p(\bx)$ and their corresponding energies $\epsilon_p$, while $\bF$ depends only on the orbitals. -Because of this $\omega$ dependence, fully solving this equation is a rather complicated task, hence several approximate solving schemes has been developed. -The most popular one is probably the one-shot scheme, known as $G_0W_0$ if the self-energy is the $GW$ one, in which the off-diagonal elements of Eq.~\eqref{eq:quasipart_eq} are neglected and the self-consistency is abandoned. -In this case, there are $K$ quasi-particle equations that read +Because of this frequency dependence, fully solving this equation is a rather complicated task. +Hence, several approximate schemes have been developed to bypass self-consistency. +The most popular one is the one-shot (perturbative) scheme, known as $G_0W_0$, where the self-consistency is completely abandoned, and the off-diagonal elements of Eq.~\eqref{eq:quasipart_eq} are neglected. +In this case, one gets $K$ quasi-particle equations that read tr\begin{equation} \label{eq:G0W0} - \epsilon_p^{\HF} + \Sigma_{p}(\omega) - \omega = 0, + \epsilon_p^{\HF} + \Sigma_{pp}(\omega) - \omega = 0, \end{equation} -where $\Sigma_{p}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies. -The previous equation is non-linear with respect to $\omega$ and therefore can have multiple solutions $\epsilon_{p,s}$ for a given $p$. -These solutions can be characterized by their spectral weight defined as the renormalization factor $Z_{p,s}$ +where $\Sigma_{pp}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies. +The previous equation is non-linear with respect to $\omega$ and therefore can have multiple solutions $\epsilon_{p,s}$ for a given $p$ (where the index $s$ is numbering solutions). +These solutions can be characterized by their spectral weight given by the renormalization factor $Z_{p,s}$ \begin{equation} \label{eq:renorm_factor} - 0 \leq Z_{p,s} = \left[ 1 - \pdv{\Sigma_{p}(\omega)}{\omega}\bigg|_{\omega=\epsilon_{p,s}} \right]^{-1} \leq 1. + 0 \leq Z_{p,s} = \qty[ 1 - \eval{\pdv{\Sigma_{pp}(\omega)}{\omega}}_{\omega=\epsilon_{p,s}} ]^{-1} \leq 1. \end{equation} -The solution with the largest weight is referred to as the quasi-particle solution while the others are known as satellites or shake-up solutions. -However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights and the quasi-particle solution is not well-defined. -In fact, these cases are related to the discontinuities and convergence problems discussed earlier because the additional solutions with large weights are the previously mentioned intruder states. +The solution with the largest weight is referred to as the quasi-particle while the others are known as satellites (or shake-up transitions). +However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights and the quasi-particle is not well-defined. +In fact, these cases are related to the discontinuities and convergence problems discussed earlier (see Sec.~\ref{sec:intro}) because the additional solutions with large weights are the previously mentioned intruder states. One obvious flaw of the one-shot scheme mentioned above is its starting point dependence. Indeed, in Eq.~\eqref{eq:G0W0} we chose to use the HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham orbitals for example. @@ -220,7 +225,6 @@ The upfolded $GW$ quasi-particle equation is the following (\bV^{\text{2h1p}})^{\mathrm{T}} & \bC^{\text{2h1p}} & \bO \\ (\bV^{\text{2p1h}})^{\mathrm{T}} & \bO & \bC^{\text{2p1h}} \\ \end{pmatrix} - \cdot \begin{pmatrix} \bX \\ \bY^{\text{2h1p}} \\ @@ -232,7 +236,6 @@ The upfolded $GW$ quasi-particle equation is the following \bY^{\text{2h1p}} \\ \bY^{\text{2p1h}} \\ \end{pmatrix} - \cdot \boldsymbol{\epsilon}, \end{equation} where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasi-particle and satellite energies, the 2h1p and 2p1h matrix elements are @@ -427,9 +430,9 @@ where $\bW^{(0)}(s)= \bV^{(0)}(s) \bU$. The matrix elements of $\bU$ and $\bD^{(0)}$ are \begin{align} U_{(p,v),(q,w)} &= \delta_{pq} \bX_{v,w} \\ - D_{(p,v),(q,w)}^{(0)} &= \left(\epsilon_p + \text{sign}(\epsilon_p-\epsilon_F)\Omega_v\right)\delta_{pq}\delta_{vw} + D_{(p,v),(q,w)}^{(0)} &= \left(\epsilon_p + \text{sign}(\epsilon_p-\epsilon_\text{F})\Omega_v\right)\delta_{pq}\delta_{vw} \end{align} -where $\epsilon_F$ is the Fermi level. +where $\epsilon_\text{F}$ is the Fermi level. Note that the matrix $\bU$ is also used in the downfolding process of Eq.~\eqref{eq:GWlin}. \cite{Bintrim_2021} Thanks to the diagonal structure of $\bF^{(0)}$ and $\bD^{(0)}$, Eq.~\eqref{eq:eqdiffW0} can be easily solved and give