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starting modifying theory section

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Pierre-Francois Loos 2023-01-23 21:49:44 +01:00
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Manuscript/SRGGW.tex
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@ -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