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Pierre-Francois Loos 2020-12-05 15:35:15 +01:00
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Cover_Letter/CoverLetter.tex
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@ -25,7 +25,7 @@ Throughout this review, we present illustrative and pedagogical examples based o
Due to the genuine interdisciplinary nature of the present article and its pedagogical aspect, we believe that it will be of interest to a wide audience within the physics and chemistry communities. Due to the genuine interdisciplinary nature of the present article and its pedagogical aspect, we believe that it will be of interest to a wide audience within the physics and chemistry communities.
We hope that the editors and the reviewers of \textit{JPCM} will find this topical review enjoyable and educative. We hope that the editors and the reviewers of \textit{JPCM} will find this topical review enjoyable and educative.
We suggest Paola Gori-Giorgi, Jeppe Olsen, So Hirata, Peter Knowles, and Kieron Burke as potential referees. We suggest Paola Gori-Giorgi, Jeppe Olsen, Peter Surjan, So Hirata, Peter Knowles, and Kieron Burke as potential referees.
We look forward to hearing from you soon. We look forward to hearing from you soon.
\closing{Sincerely, the authors.} \closing{Sincerely, the authors.}

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Manuscript/EPAWTFT.tex
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@ -895,7 +895,7 @@ for the two states using the ground-state RHF orbitals is identical.
\includegraphics[width=\linewidth]{fig5} \includegraphics[width=\linewidth]{fig5}
\caption{ \caption{
Radius of convergence $r_c$ for the RMP ground state (red), the UMP ground state (blue), and the UMP excited state (orange) Radius of convergence $r_c$ for the RMP ground state (red), the UMP ground state (blue), and the UMP excited state (orange)
series as functions of the ratio $U/t$. series \titou{of the Hubbard dimer} as functions of the ratio $U/t$.
\label{fig:RadConv}} \label{fig:RadConv}}
\end{figure} \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -1368,7 +1368,7 @@ radius of convergence (see Fig.~\ref{fig:RadConv}).
\includegraphics[height=0.23\textheight]{fig9a} \includegraphics[height=0.23\textheight]{fig9a}
\includegraphics[height=0.23\textheight]{fig9b} \includegraphics[height=0.23\textheight]{fig9b}
\caption{\label{fig:PadeRMP} \caption{\label{fig:PadeRMP}
RMP ground-state energy as a function of $\lambda$ obtained using various \titou{truncated Taylor series and approximants} RMP ground-state energy as a function of $\lambda$ \titou{in the Hubbard dimer} obtained using various \titou{truncated Taylor series and approximants}
at $U/t = 3.5$ (left) and $U/t = 4.5$ (right).} at $U/t = 3.5$ (left) and $U/t = 4.5$ (right).}
\end{figure*} \end{figure*}
%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%
@ -1421,7 +1421,7 @@ Despite this limitation, the successive diagonal Pad\'e approximants (\ie, $d_A
often define a convergent perturbation series in cases where the Taylor series expansion diverges. often define a convergent perturbation series in cases where the Taylor series expansion diverges.
\begin{table}[b] \begin{table}[b]
\caption{RMP ground-state energy estimate at $\lambda = 1$ provided by various truncated Taylor \caption{RMP ground-state energy estimate at $\lambda = 1$ \titou{of the Hubbard dimer} provided by various truncated Taylor
series and Pad\'e approximants at $U/t = 3.5$ and $4.5$. series and Pad\'e approximants at $U/t = 3.5$ and $4.5$.
We also report the distance of the closest pole to the origin $\abs{\lc}$ provided by the diagonal Pad\'e approximants. We also report the distance of the closest pole to the origin $\abs{\lc}$ provided by the diagonal Pad\'e approximants.
\label{tab:PadeRMP}} \label{tab:PadeRMP}}
@ -1467,7 +1467,7 @@ a convergent series.
\begin{figure}[t] \begin{figure}[t]
\includegraphics[width=\linewidth]{fig10} \includegraphics[width=\linewidth]{fig10}
\caption{\label{fig:QuadUMP} \caption{\label{fig:QuadUMP}
UMP energies as a function of $\lambda$ obtained using various \titou{approximants} at $U/t = 3$.} UMP energies \titou{in the Hubbard dimer} as a function of $\lambda$ obtained using various \titou{approximants} at $U/t = 3$.}
\end{figure} \end{figure}
%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%
@ -1494,7 +1494,7 @@ function $E(\lambda)$ via a generalised version of the square-root singularity
expression \cite{Mayer_1985,Goodson_2011,Goodson_2019} expression \cite{Mayer_1985,Goodson_2011,Goodson_2019}
\begin{equation} \begin{equation}
\label{eq:QuadApp} \label{eq:QuadApp}
E(\lambda) = \frac{1}{2 Q(\lambda)} \qty[ P(\lambda) \pm \sqrt{P^2(\lambda) - 4 Q(\lambda) R(\lambda)} ], \titou{E_{[d_P/d_Q,d_R]}}(\lambda) = \frac{1}{2 Q(\lambda)} \qty[ P(\lambda) \pm \sqrt{P^2(\lambda) - 4 Q(\lambda) R(\lambda)} ],
\end{equation} \end{equation}
with the polynomials with the polynomials
\begin{align} \begin{align}
@ -1538,7 +1538,7 @@ The remedy for this problem involves applying a suitable transformation of the c
\begin{table}[b] \begin{table}[b]
\caption{Estimate for the distance of the closest singularity (pole or branch point) to the origin $\abs{\lc}$ \caption{Estimate for the distance of the closest singularity (pole or branch point) to the origin $\abs{\lc}$
in the UMP energy function provided by various \titou{truncated Taylor series and approximants} at $U/t = 3$ and $7$. in the UMP energy function \titou{of the Hubbard dimer} provided by various \titou{truncated Taylor series and approximants} at $U/t = 3$ and $7$.
The truncation degree of the Taylor expansion $n$ of $E(\lambda)$ and the number of branch The truncation degree of the Taylor expansion $n$ of $E(\lambda)$ and the number of branch
points $n_\text{bp} = \max(2d_p,d_q+d_r)$ generated by the quadratic approximants are also reported. points $n_\text{bp} = \max(2d_p,d_q+d_r)$ generated by the quadratic approximants are also reported.
\label{tab:QuadUMP}} \label{tab:QuadUMP}}
@ -1593,7 +1593,7 @@ The remedy for this problem involves applying a suitable transformation of the c
\end{subfigure} \end{subfigure}
\caption{% \caption{%
Comparison of the [3/2,2] and [3/0,4] quadratic approximants with the exact UMP energy surface in the complex $\lambda$ Comparison of the [3/2,2] and [3/0,4] quadratic approximants with the exact UMP energy surface in the complex $\lambda$
plane with $U/t = 3$. plane \titou{in the Hubbard dimer} with $U/t = 3$.
Both quadratic approximants correspond to the same truncation degree of the Taylor series and model the branch points Both quadratic approximants correspond to the same truncation degree of the Taylor series and model the branch points
using a radicand polynomial of the same order. using a radicand polynomial of the same order.
However, the [3/2,2] approximant introduces poles into the surface that limits it accuracy, while the [3/0,4] approximant However, the [3/2,2] approximant introduces poles into the surface that limits it accuracy, while the [3/0,4] approximant
@ -1640,7 +1640,7 @@ energy using low-order perturbation expansions.
\begin{table}[h] \begin{table}[h]
\caption{ \caption{
Estimate and associated error of the exact UMP energy at $U/t = 7$ for Estimate and associated error of the exact UMP energy \titou{of the Hubbard dimer} at $U/t = 7$ for
various approximants using up to ten terms in the Taylor expansion. various approximants using up to ten terms in the Taylor expansion.
\label{tab:UMP_order10}} \label{tab:UMP_order10}}
\begin{ruledtabular} \begin{ruledtabular}
@ -1681,15 +1681,12 @@ If the series converges, then the partial sums will tend to the exact result
The Shanks transformation attempts to generate increasingly accurate estimates of this The Shanks transformation attempts to generate increasingly accurate estimates of this
limit by defining a new series as limit by defining a new series as
\begin{equation} \begin{equation}
T(S_n) = \frac{S_{n+1} S_{n-1} - S_{n}^2}{S_{n+1} + S_{n-1} - 2 S_{n}}. T(S_n) = \frac{S_{n+1} S_{n-1} - S_{n}^2}{S_{n+1} - 2 S_{n} + S_{n-1}}.
\end{equation} \end{equation}
This series can converge faster than the original partial sums and can thus provide greater This series can converge faster than the original partial sums and can thus provide greater
accuracy using only the first few terms in the series. accuracy using only the first few terms in the series.
However, it is only designed to accelerate converging partial sums with However, it is only designed to accelerate converging partial sums with
the approximate form the approximate form $S_n \approx S + \alpha\,\beta^n$.
\begin{equation}
S_n \approx S + \alpha\,\beta^n.
\end{equation}
Furthermore, while this transformation can accelerate the convergence of a series, Furthermore, while this transformation can accelerate the convergence of a series,
there is no guarantee that this acceleration will be fast enough to significantly there is no guarantee that this acceleration will be fast enough to significantly
improve the accuracy of low-order approximations. improve the accuracy of low-order approximations.
@ -1713,7 +1710,7 @@ terms of a perturbation series, even if it diverges.
\begin{table}[th] \begin{table}[th]
\caption{ \caption{
Acceleration of the diagonal Pad\'e approximant sequence for the RMP energy Acceleration of the diagonal Pad\'e approximant sequence for the RMP energy
using the Shanks transformation. \titou{of the Hubbard dimer at $U/t = 3.5$ and $4.5$} using the Shanks transformation.
\label{tab:RMP_shank}} \label{tab:RMP_shank}}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lcccc} \begin{tabular}{lcccc}
@ -1756,7 +1753,7 @@ the cost of larger denominators is an overall slower rate of convergence.
\includegraphics[width=\linewidth]{fig12} \includegraphics[width=\linewidth]{fig12}
\caption{% \caption{%
Comparison of the scaled RMP10 Taylor expansion with the exact RMP energy as a function Comparison of the scaled RMP10 Taylor expansion with the exact RMP energy as a function
of $\lambda$ for the symmetric Hubbard dimer at $U/t = 4.5$. of $\lambda$ for the \trash{symmetric} Hubbard dimer at $U/t = 4.5$.
The two functions correspond closely within the radius of convergence. The two functions correspond closely within the radius of convergence.
\titou{T2: are we keeping this?} \titou{T2: are we keeping this?}
} }
@ -1793,7 +1790,7 @@ the contour.
Once the contour values of $E(\lambda')$ are converged, Cauchy's integral formula Eq.~\eqref{eq:Cauchy} can Once the contour values of $E(\lambda')$ are converged, Cauchy's integral formula Eq.~\eqref{eq:Cauchy} can
be invoked to compute the value at $E(\lambda=1)$ and obtain a final estimate of the exact energy. be invoked to compute the value at $E(\lambda=1)$ and obtain a final estimate of the exact energy.
The authors illustrate this protocol for the dissociation curve of \ce{LiH} and the stretched water The authors illustrate this protocol for the dissociation curve of \ce{LiH} and the stretched water
molecule to obtain encouragingly accurate results.\cite{Mihalka_2019} molecule \trash{to obtain} \titou{and obtained?} encouragingly accurate results.\cite{Mihalka_2019}
%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%
\section{Concluding Remarks} \section{Concluding Remarks}