PTEROSOR/sfBSE
10
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

getting there with the GW section

Browse Source
This commit is contained in:
Pierre-Francois Loos 2020-10-27 22:32:24 +01:00
parent cef3f20656
commit 4629c55a76
1 changed files with 47 additions and 14 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

61
sfBSE.tex
View File

@ -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}