153 lines
7.0 KiB
TeX
153 lines
7.0 KiB
TeX
\documentclass[10pt]{letter}
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\usepackage{UPS_letterhead,xcolor,mhchem,ragged2e,hyperref}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\definecolor{darkgreen}{HTML}{009900}
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\begin{document}
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\begin{letter}%
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{To the Editors of the Journal of Chemical Physics,}
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\opening{Dear Editors,}
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\justifying
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Please find attached a revised version of the manuscript entitled
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\begin{quote}
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\textit{``Connections between many-body perturbation and coupled-cluster theories''}.
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\end{quote}
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We thank the reviewers for their constructive comments and to support publication of the present manuscript.
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Our detailed responses to their comments can be found below.
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For convenience, changes are highlighted in red in the revised version of the manuscript.
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We look forward to hearing from you.
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\closing{Sincerely, the authors.}
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\newpage
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%%% REVIEWER 1 %%%
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\noindent \textbf{\large Authors' answer to Reviewer \#1}
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{In this manuscript, Quintero-Monsebaiz et al. expand upon previously established connections between coupled-cluster (CC) theory and the random-phase approximation.
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Specifically, they aim to connect CC theory to the Bethe-Salpeter equation (BSE) for neutral excitations and to the $GW$ approximation for charged (IP/EA) excitations, both of which have RPA screening at their heart.
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The manuscript is well-written and contributes to a growing body of literature comparing, theoretically and numerically, the performance of wavefunction and Green's function-based many-body theories.
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J. Chem. Phys. is a good venue for this work, and I am happy to recommend publication, although I have a few suggestions for the authors to consider.
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}
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\\
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\alert{
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bla bla bla
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}
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\begin{enumerate}
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\item
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{How should I understand the BSE correlation energy and the plasmon trace formula given in Eq.~(9)?
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The BSE is an approximation to the two-particle Green's function, which completely determines the two-particle reduced density matrix.
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From this, I could obtain the total energy by contraction with one- and two-electron Hamiltonian matrix elements.
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Can this viewpoint be related precisely to the trace formula (9)?
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If so, I think a few words to make this connection would be valuable to the reader.
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It's not obvious (to me at least) that all methods that allow a Tamm-Dancoff approximation can rigorously be related to a correlation energy expression in the form of the plasmon trace formula.}
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\\
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\alert{
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bla bla bla
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}
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\item
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{Concerning the discussion below Eq.~(23): I note that the ``$GW$ pre-treatment'' renormalizes the one-electron energies via more than mosaic diagrams.
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The mosaic diagrams are the only ones that are captured at the CCSD level, but in general, the $GW$ approximation also has a forward-time-ordered screened exchange interaction mediated by particle-hole pairs (requiring more than CCSD).
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}
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\\
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\alert{
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bla bla bla
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}
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\item
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{In general, Eq.~(24) defines all poles of the one-particle Green's function, although the surrounding discussion suggests that only quasiparticle excitations are found in this manner.
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(I'm sure the authors appreciate this point, as made clear elsewhere in the manuscript).
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}
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\\
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\alert{
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bla bla bla
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}
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\item
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{I quite like the discussion beginning with Eq.~(29), but it makes me wonder about the major findings of the present manuscript.
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Specifically, I think this manuscript exploits the following numerical observation.
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Consider any eigenvalue problem and partition it into two spaces,
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\begin{equation}
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\begin{pmatrix}
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\boldsymbol{A} & \boldsymbol{V} \\
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\boldsymbol{V}^T & \boldsymbol{B} \\
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\end{pmatrix}
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\begin{pmatrix}
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\boldsymbol{X} \\
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\boldsymbol{Y} \\
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\end{pmatrix}
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=
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\begin{pmatrix}
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\boldsymbol{X} \\
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\boldsymbol{Y} \\
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\end{pmatrix}
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\boldsymbol{\Omega}
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\end{equation}
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Assuming existence of the inverse, this can be trivially rewritten as an eigenvalue problem for a matrix in the $\boldsymbol{A}$ subspace,
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\begin{equation}
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\boldsymbol{A} + \boldsymbol{V} \boldsymbol{T}
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\end{equation}
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where $\boldsymbol{T} = \boldsymbol{Y} \boldsymbol{X}^{-1}$. Of course, this is not useful without $\boldsymbol{T}$. So what determines $\boldsymbol{T}$ without access to all the eigenvectors contained in $\boldsymbol{X}$, $\boldsymbol{Y}$?
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We can rewrite the two equations contained in (1) as
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\begin{gather}
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\boldsymbol{A} + \boldsymbol{V} \boldsymbol{T} = \boldsymbol{X} \boldsymbol{\Omega} \boldsymbol{X}^{-1}
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\boldsymbol{V}^T + \boldsymbol{B} \boldsymbol{T} = \boldsymbol{T} \boldsymbol{X} \boldsymbol{\Omega} \boldsymbol{X}^{-1}
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\end{gather}
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or, upon combining the above equations,
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\begin{equation}
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\boldsymbol{V}^T + \boldsymbol{B} \boldsymbol{T} - \boldsymbol{T} \boldsymbol{A} - \boldsymbol{T} \boldsymbol{V} \boldsymbol{T} = \boldsymbol{0}
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\end{equation}
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The above is some nonlinear equation that can be solved for the unknown $\boldsymbol{T}$.
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It looks superficially like a CCSD equation (which could be solved, e.g., by iteration), but clearly there is no formal connection.
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Does the fact that any linear eigenvalue equation can be partitioned to produce a nonlinear eigenvalue equation, or equivalently recast into the determination of the solution to a system of nonlinear equations, imply any formal connection with CC theory (as the section title implies)?
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Or is it a mathematically interesting observation that indicates an alternative route toward the determination of eigenvalues from a partitioned matrix?
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As far as I can tell, the above general observation is applied in two distinct settings to derive the results in the present manuscript.
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For the BSE, the partitioning is used to eliminate the ``deexcitation'' space, similar to its elimination in the RPA problem.
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For the $GW$ approximation, the partitioning is used to eliminate the 2h1p/2p1h space.
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However, I reemphasize that in both cases, the ``CC-like'' equations that result are analogous to ground-state CC theory, and so the formal connection between the theories is tenuous.
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}
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\\
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\alert{
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bla bla bla
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}
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\item
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{A confusion I have in my own discussion above is related to a question I have about the final Eq.~(34) of the present work.
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Where was the approximation made that indicates that this only finds the $N$ principle excitation energies?
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I agree that $\epsilon + \Sigma$ is only $N \times N$ so it should only have $N$ eigenvalues.
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But all equations were exact, assuming existence of the inverse.
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What am I missing here?
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If the authors appreciate this confusion, then some discussion in the manuscript would be welcome.}
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\\
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\alert{
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bla bla bla
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}
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\end{enumerate}
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%%% REVIEWER 2 %%%
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\noindent \textbf{\large Authors' answer to Reviewer \#2}
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{This manuscript builds on recent analytic work, drawing connections and deepening understanding between different electronic structure methods, specifically for excitations.
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The work is self-contained, coherent and concise.
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I cannot fault the presentation of the analytic results, their correctness, or the interpretation and understanding that can be drawn from them.
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I would recommend acceptance without reservation.}
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\\
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\alert{
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We thank the reviewer for these kind comments and supporting publication of the present Communication.
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}
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\end{letter}
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\end{document}
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