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We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory.
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We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points.
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After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree--Fock approximation and Rayleigh--Schr\"odinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities.
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In particular, we highlight the seminal work \trashHB{of several research groups} on the convergence behaviour of perturbative series obtained within M{\o}ller--Plesset perturbation theory, and its links with quantum phase transitions.
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In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within M{\o}ller--Plesset perturbation theory, and its links with quantum phase transitions.
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We also discuss several resummation techniques (such as Pad\'e and quadratic approximants) that can improve the overall accuracy of the M{\o}ller--Plesset perturbative series in both convergent and divergent cases.
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Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.
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\end{abstract}
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@ -148,7 +148,7 @@ Each of these points is illustrated using the Hubbard dimer at half filling, whi
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% SPIKE THE READER
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Perturbation theory isn't usually considered in the complex plane.
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Normally it is applied using real numbers as one of very few available tools for
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describing realistic quantum systems \trashHB{where exact solutions of the Schr\"odinger equation are impossible \titou{to find?}}.\cite{Dirac_1929}
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describing realistic quantum systems.
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In particular, time-independent Rayleigh--Schr\"odinger perturbation theory\cite{RayleighBook,Schrodinger_1926}
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has emerged as an instrument of choice among the vast array of methods developed for this purpose.%
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\cite{SzaboBook,JensenBook,CramerBook,HelgakerBook,ParrBook,FetterBook,ReiningBook}
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@ -416,7 +416,7 @@ Expanding the wave function and energy as power series in $\lambda$ as
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\label{eq:E_expansion}
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\end{align}
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\end{subequations}
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solving the corresponding perturbation equations up to a given order \titou{$n$}, and
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solving the corresponding perturbation equations up to a given order $n$, and
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setting $\lambda = 1$ then yields approximate solutions to Eq.~\eqref{eq:SchrEq}.
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% MATHEMATICAL REPRESENTATION
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@ -465,7 +465,7 @@ Like the exact system in Sec.~\ref{sec:example}, the perturbation energy $E(\lam
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a ``one-to-many'' function with the output elements representing an approximation to both the ground and excited states.
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The most common singularities on $E(\lambda)$ therefore correspond to non-analytic EPs in the complex
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$\lambda$ plane where two states become degenerate.
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Later we will demonstrate how the choice of reference \hugh{wave function} \trashHB{Hamiltonian} controls the position of these EPs, and
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Later we will demonstrate how the choice of reference Hamiltonian controls the position of these EPs, and
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ultimately determines the convergence properties of the perturbation series.
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%===========================================%
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@ -620,7 +620,7 @@ with the corresponding UHF ground-state energy (Fig.~\ref{fig:HF_real})
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\begin{equation}
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E_\text{UHF} \equiv E_\text{HF}(\ta_\text{UHF}, \tb_\text{UHF}) = - \frac{2t^2}{U}.
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\end{equation}
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Time-reversal symmetry dictates that this UHF wave function must be degenerate with its spin-flipped \titou{pair?}, obtained
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Time-reversal symmetry dictates that this UHF wave function must be degenerate with its spin-flipped counterpart, obtained
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by swapping $\ta_{\text{UHF}}$ and $\tb_{\text{UHF}}$ in Eqs.~\eqref{eq:ta_uhf} and \eqref{eq:tb_uhf}.
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This type of symmetry breaking is also called a spin-density wave in the physics community as the system
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``oscillates'' between the two symmetry-broken configurations. \cite{GiulianiBook}
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@ -676,7 +676,7 @@ a ground-state wave function can be ``morphed'' into an excited-state wave funct
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via a stationary path of HF solutions.
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This novel approach to identifying excited-state wave functions demonstrates the fundamental
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role of \textit{quasi}-EPs in determining the behaviour of the HF approximation.
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Furthermore, the complex-scaled Fock operator can be used routinely \titou{to} construct analytic
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Furthermore, the complex-scaled Fock operator can be used routinely to construct analytic
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continuations of HF solutions beyond the points where real HF solutions
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coalesce and vanish.\cite{Burton_2019b}
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@ -889,7 +889,7 @@ In contrast, for $U = 4.5t$ one finds $\rc < 1$, and the RMP series becomes dive
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The corresponding Riemann surfaces for $U = 3.5\,t$ and $4.5\,t$ are shown in Figs.~\ref{subfig:RMP_3.5} and
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\ref{subfig:RMP_4.5}, respectively, with the single EP at $\lep$ (black dot) and the radius of convergence indicated
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by the vertical cylinder of unit radius.
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For the divergent case, the $\lep$ \antoine{(\sout{the} $\lep$)} lies inside this cylinder of convergence, while in the convergent case $\lep$ lies
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For the divergent case, $\lep$ lies inside this cylinder of convergence, while in the convergent case $\lep$ lies
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outside this cylinder.
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In both cases, the EP connects the ground state with the doubly-excited state, and thus the convergence behaviour
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for the two states using the ground-state RHF orbitals is identical.
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@ -901,7 +901,7 @@ for the two states using the ground-state RHF orbitals is identical.
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\includegraphics[width=\linewidth]{fig5}
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\caption{
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Radius of convergence $r_c$ for the RMP ground state (red), the UMP ground state (blue), and the UMP excited state (orange)
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series \titou{of the Hubbard dimer} as functions of the ratio $U/t$.
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series of the Hubbard dimer as functions of the ratio $U/t$.
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\label{fig:RadConv}}
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -966,7 +966,7 @@ for larger $U/t$ as the radius of convergence becomes increasingly close to one
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As the UHF orbitals break the spin symmetry, new coupling terms emerge between the electronic states that
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cause fundamental changes to the structure of EPs in the complex $\lambda$-plane.
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For example, while the RMP energy shows only one EP between the ground and
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doubly-excited states (Fig.~\ref{fig:RMP}), the UMP energy has two (\antoine{pairs of}) EPs: one connecting the ground state with the
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doubly-excited states (Fig.~\ref{fig:RMP}), the UMP energy has two pairs of complex-conjugate EPs: one connecting the ground state with the
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singly-excited open-shell singlet, and the other connecting this single excitation to the
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doubly-excited second excitation (Fig.~\ref{fig:UMP}).
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This new ground-state EP always appears outside the unit cylinder and guarantees convergence of the ground-state energy.
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@ -1002,8 +1002,8 @@ class A systems generally include well-separated and weakly correlated electron
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are characterised by dense electron clustering in one or more spatial regions.\cite{Cremer_1996}
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In class A systems, they showed that the majority of the correlation energy arises from pair correlation,
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with little contribution from triple excitations.
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On the other hand, triple excitations have an important contribution in class B systems, including providing
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orbital relaxation \titou{to doubly-excited states}, and these contributions lead to oscillations of the total correlation energy.
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On the other hand, triple excitations have an important contribution in class B systems, including
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orbital relaxation to doubly-excited configurations, and these contributions lead to oscillations of the total correlation energy.
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Using these classifications, Cremer and He then introduced simple extrapolation formulas for estimating the
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exact correlation energy $\Delta E$ using terms up to MP6\cite{Cremer_1996}
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@ -1102,7 +1102,7 @@ The three remaining Hermitian archetypes seem to be rarely observed in MP pertur
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In contrast, the non-Hermitian coupled cluster perturbation theory,%
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\cite{Pawlowski_2019a,Pawlowski_2019b,Pawlowski_2019c,Pawlowski_2019d,Pawlowski_2019e} exhibits a range of archetypes
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including the interspersed zigzag, triadic, ripple, geometric, and zigzag-geometric forms.
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This analysis highlights the importance of the primary critical point in controlling the high-order convergence,
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This analysis highlights the importance of the primary singularity in controlling the high-order convergence,
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regardless of whether this point is inside or outside the complex unit circle. \cite{Handy_1985,Olsen_2000}
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%=======================================
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@ -1165,7 +1165,7 @@ While a finite basis can only predict complex-conjugate branch point singulariti
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by a cluster of sharp avoided crossings between the ground state and high-lying excited states.\cite{Sergeev_2005}
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Alternatively, Sergeev \etal\ demonstrated that the inclusion of a ``ghost'' atom also
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allows the formation of the critical point as the electrons form a bound cluster occupying the ghost atom orbitals.\cite{Sergeev_2005}
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This effect explains the origin of the divergence in the \ce{HF} molecule as the fluorine valence electrons jump to \titou{the} hydrogen at
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This effect explains the origin of the divergence in the \ce{HF} molecule as the fluorine valence electrons jump to the hydrogen at
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a sufficiently negative $\lambda$ value.\cite{Sergeev_2005}
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Furthermore, the two-state model of Olsen and collaborators \cite{Olsen_2000} was simply too minimal to understand the complexity of
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divergences caused by the MP critical point.
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@ -1175,7 +1175,7 @@ When a Hamiltonian is parametrised by a variable such as $\lambda$, the existenc
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eigenstates as a function of $\lambda$ indicate the presence of a zero-temperature quantum phase transition (QPT).%
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\cite{Heiss_1988,Heiss_2002,Borisov_2015,Sindelka_2017,CarrBook,Vojta_2003,SachdevBook,GilmoreBook}
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Meanwhile, as an avoided crossing becomes increasingly sharp, the corresponding EPs move increasingly close to the real axis, eventually forming a critical point.
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The existence of an EP \emph{on} the real axis is therefore diagnostic of a QPT.\cite{Cejnar_2005, Cejnar_2007a}
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\hugh{The existence of an EP \emph{on} the real axis is therefore diagnostic of a QPT.\cite{Cejnar_2005, Cejnar_2007a}}
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Since the MP critical point corresponds to a singularity on the real $\lambda$ axis, it can immediately be
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recognised as a QPT with respect to varying the perturbation parameter $\lambda$.
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However, a conventional QPT can only occur in the thermodynamic limit, which here is analogous to the complete
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@ -1374,7 +1374,7 @@ radius of convergence (see Fig.~\ref{fig:RadConv}).
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\includegraphics[height=0.23\textheight]{fig9a}
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\includegraphics[height=0.23\textheight]{fig9b}
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\caption{\label{fig:PadeRMP}
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RMP ground-state energy as a function of $\lambda$ \titou{in the Hubbard dimer} obtained using various \titou{truncated Taylor series and approximants}
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RMP ground-state energy as a function of $\lambda$ in the Hubbard dimer obtained using various truncated Taylor series and approximants
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at $U/t = 3.5$ (left) and $U/t = 4.5$ (right).}
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\end{figure*}
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%%%%%%%%%%%%%%%%%
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@ -1427,7 +1427,7 @@ Despite this limitation, the successive diagonal Pad\'e approximants (\ie, $d_A
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often define a convergent perturbation series in cases where the Taylor series expansion diverges.
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\begin{table}[b]
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\caption{RMP ground-state energy estimate at $\lambda = 1$ \titou{of the Hubbard dimer} provided by various truncated Taylor
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\caption{RMP ground-state energy estimate at $\lambda = 1$ of the Hubbard dimer provided by various truncated Taylor
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series and Pad\'e approximants at $U/t = 3.5$ and $4.5$.
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We also report the distance of the closest pole to the origin $\abs{\lc}$ provided by the diagonal Pad\'e approximants.
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\label{tab:PadeRMP}}
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@ -1473,7 +1473,7 @@ a convergent series.
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\begin{figure}[t]
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\includegraphics[width=\linewidth]{fig10}
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\caption{\label{fig:QuadUMP}
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UMP energies \titou{in the Hubbard dimer} as a function of $\lambda$ obtained using various \titou{approximants} at $U/t = 3$.}
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UMP energies in the Hubbard dimer as a function of $\lambda$ obtained using various approximants at $U/t = 3$.}
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\end{figure}
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%%%%%%%%%%%%%%%%%
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@ -1500,7 +1500,7 @@ function $E(\lambda)$ via a generalised version of the square-root singularity
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expression \cite{Mayer_1985,Goodson_2011,Goodson_2019}
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\begin{equation}
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\label{eq:QuadApp}
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\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)} ],
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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)} ],
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\end{equation}
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with the polynomials
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\begin{align}
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@ -1542,7 +1542,7 @@ The remedy for this problem involves applying a suitable transformation of the c
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\begin{table}[b]
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\caption{Estimate for the distance of the closest singularity (pole or branch point) to the origin $\abs{\lc}$
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in the UMP energy function \titou{of the Hubbard dimer} provided by various \titou{truncated Taylor series and approximants} at $U/t = 3$ and $7$.
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in the UMP energy function of the Hubbard dimer provided by various truncated Taylor series and approximants at $U/t = 3$ and $7$.
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The truncation degree of the Taylor expansion $n$ of $E(\lambda)$ and the number of branch
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points $n_\text{bp} = \max(2d_p,d_q+d_r)$ generated by the quadratic approximants are also reported.
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\label{tab:QuadUMP}}
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@ -1597,7 +1597,7 @@ The remedy for this problem involves applying a suitable transformation of the c
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\end{subfigure}
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\caption{%
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Comparison of the [3/2,2] and [3/0,4] quadratic approximants with the exact UMP energy surface in the complex $\lambda$
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plane \titou{in the Hubbard dimer} with $U/t = 3$.
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plane in the Hubbard dimer with $U/t = 3$.
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Both quadratic approximants correspond to the same truncation degree of the Taylor series and model the branch points
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using a radicand polynomial of the same order.
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However, the [3/2,2] approximant introduces poles into the surface that limits it accuracy, while the [3/0,4] approximant
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@ -1644,7 +1644,7 @@ energy using low-order perturbation expansions.
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\begin{table}[h]
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\caption{
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Estimate and associated error of the exact UMP energy \titou{of the Hubbard dimer} at $U/t = 7$ for
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Estimate and associated error of the exact UMP energy of the Hubbard dimer at $U/t = 7$ for
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various approximants using up to ten terms in the Taylor expansion.
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\label{tab:UMP_order10}}
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\begin{ruledtabular}
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@ -1714,7 +1714,7 @@ terms of a perturbation series, even if it diverges.
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\begin{table}[th]
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\caption{
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Acceleration of the diagonal Pad\'e approximant sequence for the RMP energy
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\titou{of the Hubbard dimer at $U/t = 3.5$ and $4.5$} using the Shanks transformation.
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of the Hubbard dimer at $U/t = 3.5$ and $4.5$ using the Shanks transformation.
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\label{tab:RMP_shank}}
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\begin{ruledtabular}
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\begin{tabular}{lcccc}
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@ -1757,9 +1757,8 @@ the cost of larger denominators is an overall slower rate of convergence.
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\includegraphics[width=\linewidth]{fig12}
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\caption{%
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Comparison of the scaled RMP10 Taylor expansion with the exact RMP energy as a function
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of $\lambda$ for the \trash{symmetric} Hubbard dimer at $U/t = 4.5$.
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of $\lambda$ for the Hubbard dimer at $U/t = 4.5$.
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The two functions correspond closely within the radius of convergence.
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\titou{T2: are we keeping this?}
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}
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\label{fig:rmp_anal_cont}
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\end{figure}
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@ -1794,7 +1793,7 @@ the contour.
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Once the contour values of $E(\lambda')$ are converged, Cauchy's integral formula Eq.~\eqref{eq:Cauchy} can
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be invoked to compute the value at $E(\lambda=1)$ and obtain a final estimate of the exact energy.
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The authors illustrate this protocol for the dissociation curve of \ce{LiH} and the stretched water
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molecule \trash{to obtain} \titou{and obtained?} encouragingly accurate results.\cite{Mihalka_2019}
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molecule and obtained encouragingly accurate results.\cite{Mihalka_2019}
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%%%%%%%%%%%%%%%%%%%%
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\section{Concluding Remarks}
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