diff --git a/Manuscript/ufGW.tex b/Manuscript/ufGW.tex index f60fd6f..027a178 100644 --- a/Manuscript/ufGW.tex +++ b/Manuscript/ufGW.tex @@ -173,7 +173,7 @@ Atomic units are used throughout. \section{Downfolding: The non-linear $GW$ problem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -Within the {\GOWO} approximation, in order to obtain the quasiparticle energies and the corresponding satellites, one solve, for each spatial orbital $p$, the following (non-linear) quasiparticle equation +Within the {\GOWO} approximation, in order to obtain the quasiparticle energies and the corresponding satellites, one solve, for each spatial orbital $p$ \alert{and assuming real values of the frequency $\omega$}, the following (non-linear) quasiparticle equation \begin{equation} \label{eq:qp_eq} \eps{p}{\HF} + \SigC{p}(\omega) - \omega = 0 @@ -204,7 +204,7 @@ and are two-electron integrals over the HF (spatial) orbitals $\MO{p}(\br)$. Because one must compute all the RPA eigenvalues and eigenvectors to construct the self-energy \eqref{eq:SigC}, the computational cost is $\order*{\Nocc^3 \Nvir^3} = \order*{\Norb^6}$, where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals, respectively, and $\Norb = \Nocc + \Nvir$ is the total number of orbitals. -As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,s}{\GW}$ and their corresponding weights are given by the value of the following renormalization factor +As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,s}{\GW}$ \alert{(where the index $s$ is numbering solutions)} and their corresponding weights are given by the value of the following renormalization factor \begin{equation} \label{eq:Z} 0 \le Z_{p,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{\GW}} ]^{-1} \le 1 @@ -260,6 +260,7 @@ and the corresponding coupling blocks read The size of this eigenvalue problem is $1 + \Nocc^2 \Nvir + \Nocc \Nvir^2 = \order*{\Norb^3}$, and it has to be solved for each orbital that one wishes to correct. Thus, this step scales as $\order*{\Norb^9}$ with conventional diagonalization algorithms. Note, however, that the blocks $\bC{}{\text{2h1p}}$ and $\bC{}{\text{2p1h}}$ do not need to be recomputed for each orbital. +\alert{Of course, this $\order*{\Norb^9}$ scheme is purely illustrative and current state-of-the-art $GW$ implementation scales as $\order*{\Norb^3}$ thanks to efficient contour deformation and density fitting techniques. \cite{Duchemin_2019,Duchemin_2020,Duchemin_2021}} It is crucial to understand that diagonalizing $\bH^{(p)}$ [see Eq.~\eqref{eq:Hp}] is completely equivalent to solving the quasiparticle equation \eqref{eq:qp_eq}. This can be further illustrated by expanding the secular equation associated with Eq.~\eqref{eq:Hp} @@ -478,7 +479,7 @@ This project has received funding from the European Research Council (ERC) under %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Data availability statement} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -The data that supports the findings of this study are available within the article.% and its supplementary material. +The data that supports the findings of this study are available within the article and its supplementary material. %%%%%%%%%%%%%%%%%%%%%%%% \bibliography{ufGW} diff --git a/Response_Letter/Response_Letter.tex b/Response_Letter/Response_Letter.tex index beada8d..2b0e503 100644 --- a/Response_Letter/Response_Letter.tex +++ b/Response_Letter/Response_Letter.tex @@ -84,7 +84,7 @@ Please, specify it. It may seem trivial to a regular GW practitioner, it is not \\ \alert{ The evaluation of all the quantities of interest ares performed on the real axis, i.e., for real values of $\omega$. -This is now stated clearly in the revised version of the manuscript. +This is now stated clearly in the revised version of the manuscript at the very beginning of Section II. } \item @@ -102,13 +102,15 @@ We now clearly state that our $O(K^9)$ scheme is illustrative and that state-of- I do not think that it is clearly mentioned in the article when the notation appears first time.} \\ \alert{ -We now clearly state that $s$ numbers solutions. +We now clearly state that $s$ numbers solutions just above Eq.~(8). } \item {I assume that in Fig.~1 authors plot the lowest solution after the diagonalization for each orbital $p$.} \\ \alert{In Fig.~1 we plot the quasiparticle solution for each orbital, i.e, the solution with the largest spectral weight which is obtained using the $G_0W_0$ scheme. +This is, in particular, mentioned in the caption of Fig.~1. +In general, for each orbital, it exists additional (satellites) solutions with lower and higher energies than the quasiparticle solution. } \item @@ -146,6 +148,7 @@ Could authors elaborate?} {How the values of Fig.~4 depend on different choices of $\eta$ magnitude? This is crucial for assessing if a regularizer scheme is viable. } \\ \alert{Several graphs have been added in the supporting information for different values of $\eta$ and $\kappa$. +The discussion has been expanded accordingly. } \item @@ -153,7 +156,8 @@ Could authors elaborate?} Without showing such a graph it is hard to know. Also would the regularizer help when the HF eigenvalues of Homo and Lumo are known to become degenerate at the stretched distance?} \\ -\alert{We add few graphs in the supporting where we use different values for the $\eta$ and the $\kappa$ regularizer. We can see that the value $\kappa = 1$ gives a much better result than the $\eta = 1$ one. +\alert{We add few graphs in the supporting information where we use different values for the $\eta$ and the $\kappa$ regularizer. +We can see that the value $\kappa = 1$ gives a much better result than the $\eta = 1$ one. } \item @@ -164,6 +168,7 @@ How could I recognize which value is right?} \alert{ Like in other regularized methods, $\eta$ is an empirical parameter that must be chosen. There is no definite answer but, depending on the type of properties studied, the value of $\eta$ must be chosen carefully. +This point is mentioned in the manuscript (Section V). We hope to report further on this in a forthcoming paper but this requires extensive calculations. }