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@ -165,17 +165,20 @@ The central equation of many-body perturbation theory based on Hedin's equations
\label{eq:quasipart_eq}
\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 correlation part of) the $GW$ self-energy.
where $\bF$ is the Fock matrix in the orbital basis \cite{SzaboBook} and $\bSig(\omega)$ is (the correlation part of) the $GW$ self-energy.
Both are $K \times K$ matrices with $K$ the number of one-electron orbitals.
Throughout the manuscript, the indices $p,q,r,s$ are general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
The indices $\mu$ and $\nu$ are composite indices, \eg $\nu=(ia)$, referring to neutral (single) excitations.
The self-energy can be physically understood as a correction to the Hartree-Fock (HF) problem (represented by $\bF$) accounting for dynamical screening effects.
Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-consistently.
\titou{Note that $\bSig(\omega)$ is dynamical which implies that it depends on both the one-electron orbitals $\psi_p(\bx)$ and their corresponding energies $\epsilon_p$, while $\bF$ depends only on the orbitals.}
\PFL{I still don't like it.}
The matrix elements of $\bSig(\omega)$ have the following analytic expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
The matrix elements of $\bSig(\omega)$ have the following closed-form expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
\begin{equation}
\label{eq:GW_selfenergy}
\Sigma_{pq}(\omega; \eta)
\Sigma_{pq}(\omega)
= \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta}
+ \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta},
\end{equation}
@ -184,7 +187,7 @@ where $\eta$ is a positive infinitesimal and the screened two-electron integrals
\label{eq:GW_sERI}
W_{p,q\nu} = \sum_{ia}\eri{pi}{qa}\qty(\bX+\bY)_{ia,\nu},
\end{equation}
where $\bX$ and $\bY$ are the components of the eigenvectors of the particle-hole direct (\ie without exchange) RPA problem defined as
with $\bX$ and $\bY$ the components of the eigenvectors of the direct (\ie without exchange) RPA problem defined as
\begin{equation}
\label{eq:full_dRPA}
\mqty( \bA & \bB \\ -\bB & -\bA ) \mqty( \bX \\ \bY ) = \mqty( \bX \\ \bY ) \mqty( \boldsymbol{\Omega} & \bO \\ \bO & -\boldsymbol{\Omega} ),
@ -201,14 +204,12 @@ and
\begin{equation}
\braket{pq}{rs} = \iint \frac{\SO{p}(\bx_1) \SO{q}(\bx_2)\SO{r}(\bx_1) \SO{s}(\bx_2) }{\abs{\br_1 - \br_2}} d\bx_1 d\bx_2
\end{equation}
are two-electron integrals in the spin-orbital basis.
are bare two-electron integrals in the spin-orbital basis.
The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}.
\titou{In the Tamm-Dancoff approximation (TDA), which is discussed in Appendix \ref{sec:nonTDA}, one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a Hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$.}
Throughout the manuscript, the indices $p,q,r,s$ are general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
The indices $\mu$ and $\nu$ are composite indices, \eg $\nu=(ia)$, referring to neutral (single) excitations.
The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues of the RPA problen defined in Eq.~\eqref{eq:full_dRPA} and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}.
In the Tamm-Dancoff approximation (TDA), which is discussed in Appendix \ref{sec:nonTDA}, one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a Hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$.
Because of the frequency dependence of the self-energy, fully solving the quasiparticle equation \eqref{eq:quasipart_eq} is a rather complicated task.
Because of the frequency dependence of the self-energy, solving exactly the quasiparticle equation \eqref{eq:quasipart_eq} is a rather complicated task.
Hence, several approximate schemes have been developed to bypass self-consistency.
The most popular strategy is the one-shot (perturbative) $G_0W_0$ scheme, where the self-consistency is completely abandoned, and the off-diagonal elements of Eq.~\eqref{eq:quasipart_eq} are neglected.
Assuming a HF starting point, this results in $K$ quasiparticle equations that read
@ -218,21 +219,21 @@ Assuming a HF starting point, this results in $K$ quasiparticle equations that r
\end{equation}
where $\Sigma_{pp}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
The previous equations are non-linear with respect to $\omega$ and therefore 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 renormalisation factor $Z_{p,s}$
These solutions can be characterized by their spectral weight given by the renormalization factor
\begin{equation}
\label{eq:renorm_factor}
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 quasiparticle 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, hence the quasiparticle is not well-defined.
These additional solutions with large weights are the previously mentioned intruder states.
These additional solutions with large weights are the previously mentioned intruder states. \cite{Monino_2022}
One obvious flaw of the one-shot scheme mentioned above is its starting point dependence.
Indeed, in Eq.~\eqref{eq:G0W0} we choose to rely on HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham orbital energies instead.
As commonly done, one can even ``tune'' the starting point to obtain the best possible one-shot GW quasiparticle energies \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
One obvious drawback of the one-shot scheme mentioned above is its starting point dependence.
Indeed, in Eq.~\eqref{eq:G0W0} we choose to rely on HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham energies (and orbitals) instead.
As commonly done, one can even ``tune'' the starting point to obtain the best possible one-shot $GW$ quasiparticle energies. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
Alternatively, one may solve this set of quasiparticle equations self-consistently leading to the ev$GW$ scheme.
\titou{The solutions $\epsilon_p$ are used to build Eq.~\eqref{eq:G0W0} instead of the HF ones and then these equations are solved for $\omega$ again.}
This procedure is iterated until convergence for $\epsilon_p$ is reached.
This procedure is iterated until convergence on the quasiparticle energies is reached.
However, if one of the quasiparticle equations does not have a well-defined quasiparticle solution, reaching self-consistency can be challenging, if not impossible.
Even at convergence, the starting point dependence is not totally removed as the quasiparticle energies still depend on the initial set of orbitals. \cite{Marom_2012}