starting messing around with notations but I need to talk to Antoine
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@ -126,9 +126,10 @@
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\newcommand{\GW}{GW}
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\newcommand{\GF}{\text{GF(2)}}
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\newcommand{\GT}{GT}
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\newcommand{\evGW}{\text{ev}$GW$}
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\newcommand{\evGW}{\text{ev}GW}
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\newcommand{\qsGW}{\text{qs}GW}
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\newcommand{\qs}{\text{qs}}
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\newcommand{\SRGGW}{\text{SRG-}GW}
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\newcommand{\SRGqsGW}{\text{SRG-qs}GW}
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\newcommand{\GOWO}{G_0W_0}
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%%% Notations %%%
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@ -170,20 +170,20 @@ Unless otherwise stated, atomic units are used throughout.
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The central equation of many-body perturbation theory based on Hedin's equations is the so-called dynamical and non-hermitian quasiparticle equation which, within the $GW$ approximation, reads
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\begin{equation}
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\label{eq:quasipart_eq}
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\qty[ \bF + \bSig(\omega = \epsilon_p) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
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\qty[ \bF + \bSig^{\GW}(\omega = \epsilon_p) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
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\end{equation}
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where $\bF$ is the Fock matrix in the orbital basis \cite{SzaboBook} and $\bSig(\omega)$ is (the correlation part of) the $GW$ self-energy.
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where $\bF$ is the Fock matrix in the orbital basis \cite{SzaboBook} and $\bSig^{\GW}(\omega)$ is (the correlation part of) the $GW$ self-energy.
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Both are $K \times K$ matrices with $K$ the number of one-electron orbitals.
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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.
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The indices $\mu$ and $\nu$ are composite indices, \eg $\nu=(ia)$, referring to neutral (single) excitations.
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The indices $\mu$ and $\nu$ are composite indices, that is, $\nu=(ia)$, referring to neutral (single) excitations.
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The self-energy can be physically understood as a correction to the Hartree-Fock (HF) problem (represented by $\bF$) accounting for dynamical screening effects.
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Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-consistently but the dynamical and non-hermitian nature of $\bSig(\omega)$, as well as its functional form, makes it much more challenging to solve from a practical point of view.
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Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-consistently but the dynamical and non-hermitian nature of $\bSig^{\GW}(\omega)$, as well as its functional form, makes it much more challenging to solve from a practical point of view.
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The matrix elements of $\bSig(\omega)$ have the following closed-form expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
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The matrix elements of $\bSig^{\GW}(\omega)$ have the following closed-form expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
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\begin{equation}
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\label{eq:GW_selfenergy}
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\Sigma_{pq}(\omega)
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\Sigma_{pq}^{\GW}(\omega)
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= \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta}
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+ \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta},
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\end{equation}
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@ -220,14 +220,14 @@ The most popular strategy is the one-shot (perturbative) $G_0W_0$ scheme, where
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Assuming a HF starting point, this results in $K$ quasiparticle equations that read
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\begin{equation}
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\label{eq:G0W0}
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\epsilon_p^{\HF} + \Sigma_{pp}(\omega) - \omega = 0,
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\epsilon_p^{\HF} + \Sigma_{pp}^{\GW}(\omega) - \omega = 0,
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\end{equation}
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where $\Sigma_{pp}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
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where $\Sigma_{pp}^{\GW}(\omega)$ are the diagonal elements of $\bSig^{\GW}$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
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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).
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These solutions can be characterized by their spectral weight given by the renormalization factor
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\begin{equation}
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\label{eq:renorm_factor}
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0 \leq Z_{p,s} = \qty[ 1 - \eval{\pdv{\Sigma_{pp}(\omega)}{\omega}}_{\omega=\epsilon_{p,s}} ]^{-1} \leq 1.
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0 \leq Z_{p,s} = \qty[ 1 - \eval{\pdv{\Sigma_{pp}^{\GW}(\omega)}{\omega}}_{\omega=\epsilon_{p,s}} ]^{-1} \leq 1.
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\end{equation}
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The solution with the largest weight is referred to as the quasiparticle while the others are known as satellites (or shake-up transitions).
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However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights, hence the quasiparticle is not well-defined.
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@ -241,19 +241,19 @@ Alternatively, one may solve iteratively the set of quasiparticle equations \eqr
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However, if one of the quasiparticle equations does not have a well-defined quasiparticle solution, reaching self-consistency can be challenging, if not impossible.
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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}
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In order to update both the orbitals and their corresponding energies, one must consider the off-diagonal elements in $\bSig(\omega)$.
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To avoid solving the non-Hermitian and dynamic quasiparticle equation defined in Eq.~\eqref{eq:quasipart_eq}, one can resort to the qs$GW$ scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.
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Then, the qs$GW$ equations are solved via a standard self-consistent field procedure similar to the HF algorithm where $\bF$ is replaced by $\bF + \bSig^{\qs}$.
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Various choices for $\bSig^\qs$ are possible but the most popular is the following Hermitian approximation
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In order to update both the orbitals and their corresponding energies, one must consider the off-diagonal elements in $\bSig^{\GW}(\omega)$.
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To avoid solving the non-Hermitian and dynamic quasiparticle equation defined in Eq.~\eqref{eq:quasipart_eq}, one can resort to the qs$GW$ scheme in which $\bSig^{\GW}(\omega)$ is replaced by a static approximation $\bSig^{\qsGW}$.
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Then, the qs$GW$ equations are solved via a standard self-consistent field procedure similar to the HF algorithm where $\bF$ is replaced by $\bF + \bSig^{\qsGW}$.
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Various choices for $\bSig^{\qsGW}$ are possible but the most popular is the following Hermitian approximation
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\begin{equation}
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\label{eq:sym_qsgw}
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\Sigma_{pq}^\qs = \frac{1}{2}\Re[\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) ].
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\Sigma_{pq}^{\qsGW} = \frac{1}{2}\Re[\Sigma_{pq}^{\GW}(\epsilon_p) + \Sigma_{pq}^{\GW}(\epsilon_q) ].
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\end{equation}
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which was first introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007} before being derived as the effective Hamiltonian that minimizes the length of the gradient of the Klein functional for non-interacting Green's functions by Ismail-Beigi. \cite{Ismail-Beigi_2017}
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The corresponding matrix elements are
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\begin{equation}
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\label{eq:sym_qsGW}
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\Sigma_{pq}^{\text{qs}}(\eta) = \frac{1}{2} \sum_{r\nu} \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \eta^2} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \eta^2} ) W_ {p,r\nu} W_{q,r\nu}.
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\Sigma_{pq}^{\qsGW}(\eta) = \frac{1}{2} \sum_{r\nu} \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \eta^2} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \eta^2} ) W_ {p,r\nu} W_{q,r\nu}.
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\end{equation}
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with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level).
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One of the main results of the present manuscript is the derivation, from first principles, of an alternative static Hermitian form for the $GW$ self-energy.
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@ -379,7 +379,7 @@ The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitio
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\begin{equation}
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\begin{split}
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\label{eq:downfolded_sigma}
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\bSig(\omega)
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\bSig^{\GW}(\omega)
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& = \bW^{\hhp} \qty(\omega \bI - \bC^{\hhp})^{-1} (\bW^{\hhp})^\dag
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\\
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& + \bW^{\pph} \qty(\omega \bI - \bC^{\pph})^{-1} (\bW^{\pph})^\dag,
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@ -492,7 +492,7 @@ Equation \eqref{eq:F0_C0} implies
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\end{align}
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and, thanks to the diagonal structure of $\bF^{(0)}$ (which is a consequence of the HF starting point) and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields
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\begin{equation}
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W_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s}
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W_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) \titou{e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s}}
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\end{equation}
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At $s=0$ the elements $W_{p,q\nu}^{(1)}(0)$ are equal to the screened two-electron integrals defined in Eq.~\eqref{eq:GW_sERI}, while for $s\to\infty$, they tend to zero.
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Therefore, $W_{p,q\nu}^{(1)}(s)$ are genuine renormalized two-electron screened integrals.
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@ -506,7 +506,7 @@ The second-order renormalized quasiparticle equation is given by
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\begin{equation}
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\label{eq:GW_renorm}
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% \qty[ \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) ] \bX = \omega \bX,
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\qty[ \widetilde{\bF}(s) + \widetilde{\bSig}(\omega = \epsilon_p; s) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
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\qty[ \widetilde{\bF}(s) + \widetilde{\bSig}^{\GW}(\omega = \epsilon_p; s) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
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\end{equation}
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with a regularized Fock matrix of the form
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\begin{equation}
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@ -515,13 +515,13 @@ with a regularized Fock matrix of the form
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and a regularized dynamical self-energy
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\begin{equation}
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\label{eq:srg_sigma}
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\widetilde{\bSig}(\omega; s) = \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1} (\bV^{(1)}(s))^{\dagger},
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\widetilde{\bSig}^{\GW}(\omega; s) = \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1} (\bV^{(1)}(s))^{\dagger},
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\end{equation}
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with elements
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\begin{equation}
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\label{eq:SRG-GW_selfenergy}
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\begin{split}
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\widetilde{\bSig}_{pq}(\omega; s)
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\widetilde{\bSig}_{pq}^{\GW}(\omega; s)
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&= \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu}} e^{-(\Delta_{pi\nu}^2 + \Delta_{qi\nu}^2) s} \\
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&+ \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu}}e^{-(\Delta_{pa\nu}^2 + \Delta_{qa\nu}^2) s}.
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\end{split}
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@ -544,7 +544,7 @@ which can be solved by simple integration along with the initial condition $\bF^
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\centering
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\includegraphics[width=\linewidth]{flow}
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\caption{
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\ant{Schematic} evolution of the quasiparticle equation as a function of the flow parameter $s$ in the case of the dynamic SRG-$GW$ flow (magenta) and the static SRG-qs$GW$ flow (cyan). \ANT{Maybe we should replace dynamic by full?}
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Schematic evolution of the quasiparticle equation as a function of the flow parameter $s$ in the case of the dynamic SRG-$GW$ flow (magenta) and the static SRG-qs$GW$ flow (cyan). \ANT{Maybe we should replace dynamic by full?}
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\label{fig:flow}}
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\end{figure}
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%%% %%% %%% %%%
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@ -560,9 +560,9 @@ For $s\to\infty$, it tends towards the following static limit
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\end{equation}
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while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero, \ie,
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\begin{equation}
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\lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = \bO.
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\lim_{s\to\infty} \widetilde{\bSig}^{\GW}(\omega; s) = \bO.
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\end{equation}
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Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$.
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Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}^{\GW}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$.
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As illustrated in Fig.~\ref{fig:flow}, this transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
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%%% FIG 2 %%%
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@ -582,10 +582,16 @@ Because the large-$s$ limit of Eq.~\eqref{eq:GW_renorm} is purely static and her
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Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, as $s\to\infty$, self-consistency is once again quite difficult to achieve, if not impossible.
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However, one can define a more flexible new static self-energy, which will be referred to as SRG-qs$GW$ in the following, by discarding the dynamic part in Eq.~\eqref{eq:GW_renorm}.
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This yields a $s$-dependent static self-energy which matrix elements read
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\begin{multline}
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\begin{equation}
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\label{eq:SRG_qsGW}
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\Sigma_{pq}^{\text{SRG}}(s) = F_{pq}^{(2)}(s) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu} \\ \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ],
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\end{multline}
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\begin{split}
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\Sigma_{pq}^{\SRGqsGW}(s)
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& = F_{pq}^{(2)}(s)
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\\
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& = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu}
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\qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ],
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\end{split}
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\end{equation}
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Note that the SRG static approximation is naturally Hermitian as opposed to the usual case [see Eq.~(\ref{eq:sym_qsGW})] where it is enforced by brute-force symmetrization.
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Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every denominator of the self-energy.
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Yet, these approximations are closely related because for $\eta=0$ and $s\to\infty$ they share the same diagonal terms.
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