Merge branch 'master' of git.irsamc.ups-tlse.fr:loos/SRGGW

Manuscript/SRGGW.tex

@@ -165,7 +165,7 @@ Unless otherwise stated, atomic units are used throughout.
 \label{sec:gw}
 %%%%%%%%%%%%%%%%%%%%%% 
 
-The central equation of many-body perturbation theory based on Hedin's equations is the so-called quasiparticle equation which, within the $GW$ approximation, reads
+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
 \begin{equation}
   \label{eq:quasipart_eq}
   \qty[ \bF + \bSig(\omega = \epsilon_p) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
@@ -176,9 +176,7 @@ Throughout the manuscript, the indices $p,q,r,s$ are general orbitals while $i,j
 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.}
+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.
 
 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}
@@ -212,10 +210,9 @@ and
 are bare two-electron integrals in the spin-orbital basis.
 
 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), one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a Hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$.
-\ant{The corresponding TDA screened two-electron integrals are computed using Eq.~(\ref{eq:GW_sERI}) with $\bY=0$.}
+In the Tamm-Dancoff approximation (TDA), one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a Hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$ (hence $\bY=0$).
 
-Because of the frequency dependence of the self-energy, solving exactly the quasiparticle equation \eqref{eq:quasipart_eq} is a rather complicated task. 
+As mentioned above, 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
@@ -236,11 +233,9 @@ These additional solutions with large weights are the previously mentioned intru
 
 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.
-\ant{To do so the quasiparticle energies are used to define a new RPA problem leading to updated two-electron screened integrals.
- Then the diagonal elements of the self-energy are updated as well and Eq.~\eqref{eq:G0W0} is solved again to obtain new quasiparticle energies.}
-This procedure is then iterated until convergence on the quasiparticle energies is reached.
+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 iteratively the set of quasiparticle equations \eqref{eq:G0W0} to reach convergence of the quasiparticle energies, leading to the partially self-consistent scheme named ev$GW$.
 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}
 
@@ -262,16 +257,17 @@ with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Ome
 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. 
 
 Once again, in cases where multiple solutions have large spectral weights, self-consistency can be difficult to reach at the qs$GW$ level.
-Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of th e self-energy.
+Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of the self-energy.
 Therefore, by suppressing this dependence, the static approximation relies on the fact that there is well-defined quasiparticle solutions.
 If it is not the case, the qs$GW$ self-consistent scheme inevitably oscillates between solutions with large spectral weights. \cite{Forster_2021}
 
 The satellites causing convergence problems are the above-mentioned intruder states.
 One can deal with them by introducing \textit{ad hoc} regularizers.
-\ant{The $\ii\eta$ term in the denominators of Eq.~(\ref{eq:GW_selfenergy}), which stems from a regularization of the convolution to obtain $\Sigma$ and should theoretically be set to 0,\cite{Martin_2016}  is similar to the usual imaginary-shift regularizer employed in various other theories plagued by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}.}
+For example, the $\ii\eta$ term in the denominators of Eq.~\eqref{eq:GW_selfenergy}, sometimes referred to as a broadening parameter linked to the width of the quasiparticle peak, is similar to the usual imaginary-shift regularizer employed in various other theories plagued by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}.
+However, this $\eta$ parameter stems from a regularization of the convolution to obtain $\Sigma$ and should theoretically be set to zero. \cite{Martin_2016} 
 Several other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and in particular, it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
 Nonetheless, it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
-This is the central aim of the present work.
+This is one of the aims of the present work.
 
 %%%%%%%%%%%%%%%%%%%%%%
 \section{The similarity renormalization group}
@@ -316,7 +312,8 @@ which satisfied the following condition \cite{Kehrein_2006}
 This implies that the matrix elements of the off-diagonal part decrease in a monotonic way throughout the transformation.
 Moreover, the coupling coefficients associated with the highest-energy determinants are removed first as we shall evidence in the perturbative analysis below.
 The main drawback of this generator is that it generates a stiff set of ODE which is therefore difficult to solve numerically. \cite{Hergert_2016a}
-However, here we will not tackle the full SRG problem but only consider analytical low-order perturbative expressions so we will not be affected by this problem. \cite{Evangelista_2014,Hergert_2016}
+However, here we will not tackle the full SRG problem but only consider analytical low-order perturbative expressions. 
+Hence, we will not be affected by this problem. \cite{Evangelista_2014,Hergert_2016}
 
 Let us now perform the perturbative analysis of the SRG equations.
 For $s=0$, the initial problem is
@@ -339,8 +336,8 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t
 
 By applying the SRG to $GW$, our aim is to gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle equation.
 However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
-\ant{A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms.
-Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022} }
+A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms.
+Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022}
 \begin{equation}
   \label{eq:GWlin}
   \begin{pmatrix}
@@ -434,7 +431,7 @@ where the supermatrices
 \end{align}
 \end{subequations}
 collect the 2h1p and 2p1h channels.
-Once the closed-form expressions of the low-order perturbative expansions are known, they can be inserted in Eq.~\eqref{eq:downfolded_sigma}\trashant{\eqref{eq:GWlin} before applying the downfolding process to obtain} to define a renormalized version of the quasiparticle equation.
+Once the closed-form expressions of the low-order perturbative expansions are known, they can be inserted in Eq.~\eqref{eq:downfolded_sigma} to define a renormalized version of the quasiparticle equation.
 In particular, we focus here on the second-order renormalized quasiparticle equation.
 
 %///////////////////////////%
@@ -491,13 +488,13 @@ Equation \eqref{eq:F0_C0} implies
 \begin{align}
   \bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO,
 \end{align}
-and, thanks to the \ant{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
+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
 \begin{equation}
   W_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s}
 \end{equation}
 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.
 Therefore, $W_{p,q\nu}^{(1)}(s)$ are genuine renormalized two-electron screened integrals.
-\ant{It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} for the usual electronic Hamiltonian in the context of wave-function theory within a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).}
+It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} in the context of single- and multi-reference perturbation theory (see also Ref.~\onlinecite{Hergert_2016}).
 
 %///////////////////////////%
 \subsection{Second-order matrix elements}
@@ -535,7 +532,7 @@ At $s=0$, the second-order correction vanishes while, for $s\to\infty$, it tends
   F_{pq}^{(2)}(\infty) =  \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}.
 \end{equation}
 Note that, in the limit $s\to\infty$, the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
-\ant{Therefore, the SRG flow continuously kills the dynamic part $\widetilde{\bSig}(\omega; s)$ while creating a static correction $\widetilde{\bF}(s)$.}
+\titou{Therefore, the SRG flow continuously kills the dynamic part $\widetilde{\bSig}(\omega; s)$ while creating a static correction $\widetilde{\bF}(s)$.}
 This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
 
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