remove easy corrections
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@ -85,7 +85,7 @@
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@article{Rauhut_1998,
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author = {G. Rauhut, P. Pulay and Hans-Joachim Werner},
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doi = {10.1002/(SICI)1096-987X(199808)19:11<1241::AID-JCC4>3.0.CO;2-K},
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journal = {J. Comp. Chem.},
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journal = {J. Comput. Chem.},
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pages = {1241},
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title = {Integral transformation with low‐order scaling for large local second‐order {M\oller--Plesset} calculations},
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volume = {19},
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@ -129,19 +129,16 @@
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\begin{abstract}
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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 \hugh{complex-valued} energy singularities
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\trashHB{in the complex plane}, known as exceptional points.
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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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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 \trashHB{pedagogically} 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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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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\keywords{perturbation theory, complex plane, exceptional point, divergent series, resummation}
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\maketitle
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%\raggedbottom
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%\tableofcontents
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%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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@ -177,7 +174,7 @@ systematically improvable series largely remains an open challenge.
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% COMPLEX PLANE
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Our conventional view of electronic structure theory is centred around the Hermitian notion of quantised energy levels,
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where the different electronic states of a \hugh{molecule} are discrete and energetically ordered.
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where the different electronic states of a molecule are discrete and energetically ordered.
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The lowest energy state defines the ground electronic state, while higher energy states
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represent electronic excited states.
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However, an entirely different perspective on quantisation can be found by analytically continuing
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@ -220,7 +217,7 @@ microwaves, mechanics, acoustics, atomic systems and optics.\cite{Bittner_2012,C
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% MP THEORY IN THE COMPLEX PLANE
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The MP energy correction can be considered as a function of the perturbation parameter $\lambda$.
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When the domain of $\lambda$ is extended to the complex plane, EPs can also occur in the MP energy.
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Although these EPs \hugh{are generally complex-valued} \trashHB{exist in the complex plane},
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Although these EPs are generally complex-valued,
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their positions are intimately related to the
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convergence of the perturbation expansion on the real axis.%
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\cite{BenderBook,Olsen_1996,Olsen_2000,Olsen_2019,Mihalka_2017a,Mihalka_2017b,Mihalka_2019}
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@ -292,11 +289,11 @@ unless otherwise stated, atomic units will be used throughout.
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\begin{figure*}[t]
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\begin{subfigure}{0.49\textwidth}
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\includegraphics[height=0.65\textwidth]{fig1a}
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\subcaption{\titou{Real axis} \label{subfig:FCI_real}}
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\subcaption{Real axis \label{subfig:FCI_real}}
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\end{subfigure}
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\begin{subfigure}{0.49\textwidth}
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\includegraphics[height=0.65\textwidth]{fig1b}
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\subcaption{\titou{Complex plane} \label{subfig:FCI_cplx}}
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\subcaption{Complex plane \label{subfig:FCI_cplx}}
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\end{subfigure}
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\caption{%
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Exact energies for the Hubbard dimer ($U=4t$) as functions of $\lambda$ on the real axis (\subref{subfig:FCI_real}) and in the complex plane (\subref{subfig:FCI_cplx}).
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@ -334,7 +331,7 @@ and the electrons localise on opposite sites to minimise their Coulomb repulsion
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This phenomenon is often referred to as Wigner crystallisation. \cite{Wigner_1934}
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To illustrate the formation of an EP, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t \to \lambda t$ to give the parameterised Hamiltonian $\hH(\lambda)$.
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When $\lambda$ is real, the Hamiltonian \trashHB{\eqref{eq:H_FCI}} is Hermitian with the distinct (real-valued) (eigen)energies
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When $\lambda$ is real, the Hamiltonian is Hermitian with the distinct (real-valued) (eigen)energies
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\begin{subequations}
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\begin{align}
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E_{\mp} &= \frac{1}{2} \qty(U \mp \sqrt{ (4 \lambda t)^2 + U^2 } ),
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@ -434,7 +431,7 @@ of the reference and perturbation Hamiltonians in Eq.~\eqref{eq:SchrEq-PT}.\cite
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%
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% LAMBDA IN THE COMPLEX PLANE
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From complex analysis, \cite{BenderBook} the radius of convergence for the energy can be obtained by looking for the
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\hugh{non-analytic} singularities of $E(\lambda)$ in the complex $\lambda$ plane.
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non-analytic singularities of $E(\lambda)$ in the complex $\lambda$ plane.
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This property arises from the following theorem: \cite{Goodson_2011}
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\begin{quote}
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\it
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@ -477,7 +474,7 @@ ultimately determines the convergence properties of the perturbation series.
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%===========================================%
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% SUMMARY OF HF
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In the \trash{Hartree--Fock (HF)} \titou{HF} approximation, the many-electron wave function is approximated as a single Slater determinant $\whf(\vb{x}_1,\ldots,\vb{x}_\Ne)$, where $\vb{x} = (\sigma,\vb{r})$ is a composite vector gathering spin and spatial coordinates.
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In the HF approximation, the many-electron wave function is approximated as a single Slater determinant $\whf(\vb{x}_1,\ldots,\vb{x}_\Ne)$, where $\vb{x} = (\sigma,\vb{r})$ is a composite vector gathering spin and spatial coordinates.
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This Slater determinant is defined as an antisymmetric combination of $\Ne$ (real-valued) occupied one-electron spin-orbitals $\phi_p(\vb{x})$, which are, by definition, eigenfunctions of the one-electron Fock operator
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\begin{equation}\label{eq:FockOp}
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\Hat{f}(\vb{x}) \phi_p(\vb{x}) = \qty[ \Hat{h}(\vb{x}) + \vhf(\vb{x}) ] \phi_p(\vb{x}) = \epsilon_p \phi_p(\vb{x}).
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@ -525,12 +522,12 @@ From hereon, $i$ and $j$ denote occupied orbitals, $a$ and $b$ denote unoccupied
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% BRIEF FLAVOURS OF HF
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In the most flexible variant of real HF theory (generalised HF) the one-electron orbitals can be complex-valued
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and contain a mixture of spin-up and spin-down components.\cite{Mayer_1993,Jimenez-Hoyos_2011}
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However, the application of HF \titou{theory} with some level of constraint on the orbital structure is far more common.
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However, the application of HF theory with some level of constraint on the orbital structure is far more common.
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Forcing the spatial part of the orbitals to be the same for spin-up and spin-down electrons leads to restricted HF (RHF) method,
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while allowing different orbitals for different spins leads to the so-called unrestricted HF (UHF) approach.\cite{StuberPaldus}
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The advantage of the UHF approximation is its ability to correctly describe strongly correlated systems,
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such as antiferromagnetic phases\cite{Slater_1951} or the dissociation of the hydrogen dimer.\cite{Coulson_1949}
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However, by allowing different orbitals for different spins, the UHF \hugh{wave function} is no longer required to be an eigenfunction of
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However, by allowing different orbitals for different spins, the UHF wave function is no longer required to be an eigenfunction of
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the total spin operator $\hat{\mathcal{S}}^2$, leading to ``spin-contamination''.
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%================================================================%
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@ -679,9 +676,9 @@ 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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\hugh{Furthermore, the complex-scaled Fock operator can be used routinely construct analytic
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Furthermore, the complex-scaled Fock operator can be used routinely \titou{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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coalesce and vanish.\cite{Burton_2019b}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{M{\o}ller--Plesset Perturbation Theory in the Complex Plane}
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@ -716,7 +713,7 @@ E_{\text{MP1}} = E_{\text{MP}}^{(0)} + E_{\text{MP}}^{(1)} = E_\text{HF}.
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\end{equation}
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The second-order MP2 energy correction is given by
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\begin{equation}\label{eq:EMP2}
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\hugh{E_{\text{MP}}^{(2)}} = \frac{1}{4} \sum_{ij} \sum_{ab} \frac{\abs{\mel{ij}{}{ab}}^2}{\epsilon_i + \epsilon_j - \epsilon_a - \epsilon_b},
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E_{\text{MP}}^{(2)} = \frac{1}{4} \sum_{ij} \sum_{ab} \frac{\abs{\mel{ij}{}{ab}}^2}{\epsilon_i + \epsilon_j - \epsilon_a - \epsilon_b},
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\end{equation}
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where $\mel{pq}{}{rs} = \braket{pq}{rs} - \braket{pq}{sr}$ are the anti-symmetrised two-electron integrals
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in the molecular spin-orbital basis\cite{Gill_1994}
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@ -779,7 +776,7 @@ The divergence of RMP expansions for stretched bonds can be easily understood fr
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Firstly, the exact wave function becomes increasingly multi-configurational as the bond is stretched, and the
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RHF wave function no longer provides a qualitatively correct reference for the perturbation expansion.
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Secondly, the energy gap between the bonding and antibonding orbitals associated with the stretch becomes
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increasingly small at larger bond lengths, leading to a divergence, for example, in the \trash{second-order MP} \titou{MP2} correction \eqref{eq:EMP2}.
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increasingly small at larger bond lengths, leading to a divergence, for example, in the MP2 correction \eqref{eq:EMP2}.
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In contrast, the origin of slow UMP convergence is less obvious as the reference UHF energy remains
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qualitatively correct at large bond lengths and the orbital degeneracy is avoided.
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Furthermore, this slow convergence can also be observed in molecules with a UHF ground state at the equilibrium
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@ -790,7 +787,7 @@ Using the UHF framework allows the singlet ground state wave function to mix wit
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leading to spin contamination where the wave function is no longer an eigenfunction of the $\Hat{\cS}^2$ operator.
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The link between slow UMP convergence and this spin-contamination was first systematically investigated
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by Gill \etal\ using the minimal basis \ce{H2} model.\cite{Gill_1988}
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In this work, the authors %compared titou{the UMP series with the exact RHF- and UHF-based FCI expansions (T2: I don't understand this)} and
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In this work, the authors
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identified that the slow UMP convergence arises from its failure to correctly predict the amplitude of the
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low-lying double excitation.
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This erroneous description of the double excitation amplitude has the same origin as the spin-contamination in the reference
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@ -887,7 +884,7 @@ The Taylor expansion of the RMP energy can then be evaluated to obtain the $k$th
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% RADIUS OF CONVERGENCE PLOTS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The RMP series is convergent for $U = 3.5\,t$ with $\rc > 1$, as illustrated for the individual terms at each
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\hugh{perturbation} order in Fig.~\ref{subfig:RMP_cvg}.
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perturbation order in Fig.~\ref{subfig:RMP_cvg}.
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In contrast, for $U = 4.5t$ one finds $\rc < 1$, and the RMP series becomes divergent.
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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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@ -968,8 +965,8 @@ for larger $U/t$ as the radius of convergence becomes increasingly close to one
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% EFFECT OF SYMMETRY BREAKING
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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 \trashHB{state} and
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\trashHB{the} 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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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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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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@ -997,7 +994,7 @@ very slowly as the perturbation order is increased.
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As computational implementations of higher-order MP terms improved, the systematic investigation
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of convergence behaviour in a broader class of molecules became possible.
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Cremer and He introduced an efficient MP6 approach and used it to analyse the RMP convergence of
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29 atomic and molecular systems \trashHB{with respect to the FCI energy}.\cite{Cremer_1996}
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29 atomic and molecular systems.\cite{Cremer_1996}
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They established two general classes: ``class A'' systems that exhibit monotonic convergence;
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and ``class B'' systems for which convergence is erratic after initial oscillations.
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By analysing the different cluster contributions to the MP energy terms, they proposed that
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@ -1078,7 +1075,7 @@ This divergence is related to a more fundamental critical point in the MP energy
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discuss in Sec.~\ref{sec:MP_critical_point}.
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Finally, Ref.~\onlinecite{Olsen_1996} proved that the extrapolation formulas of Cremer and He \cite{Cremer_1996}
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\titou{[see Eqs.~\eqref{eq:CrHeA} and \eqref{eq:CrHeB}]} are not mathematically motivated when considering the complex
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[see Eqs.~\eqref{eq:CrHeA} and \eqref{eq:CrHeB}] are not mathematically motivated when considering the complex
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singularities causing the divergence, and therefore cannot be applied for all systems.
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For example, the \ce{HF} molecule contains both back-door intruder states and low-lying doubly-excited states that
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result in alternating terms up to 10th order.
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@ -1385,7 +1382,7 @@ radius of convergence (see Fig.~\ref{fig:RadConv}).
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%As frequently claimed by Carl Bender,
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It is frequently stated that
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\textit{``the most stupid thing \hugh{to} \trashHB{that one can} do with a series is to sum it.''}
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\textit{``the most stupid thing to do with a series is to sum it.''}
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Nonetheless, quantum chemists are basically doing this on a daily basis.
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As we have seen throughout this review, the MP series can often show erratic,
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slow, or divergent behaviour.
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@ -1423,7 +1420,7 @@ Pad\'e approximants are extremely useful in many areas of physics and
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chemistry\cite{Loos_2013,Pavlyukh_2017,Tarantino_2019,Gluzman_2020} as they can model poles,
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which appear at the roots of $B(\lambda)$.
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However, they are unable to model functions with square-root branch points
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(which are ubiquitous in the singularity structure of \trashHB{a typical} perturbative \hugh{methods} \trashHB{treatment})
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(which are ubiquitous in the singularity structure of perturbative methods)
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and more complicated functional forms appearing at critical points
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(where the nature of the solution undergoes a sudden transition).
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Despite this limitation, the successive diagonal Pad\'e approximants (\ie, $d_A = d_B $)
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@ -1531,9 +1528,7 @@ Generally, the diagonal sequence of quadratic approximant,
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is of particular interest as the order of the corresponding Taylor series increases on each step.
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However, while a quadratic approximant can reproduce multiple branch points, it can only describe
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a total of two branches.
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%\titou{Since every branch points must therefore correspond to a degeneracy of the same two branches,}
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This constraint
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can hamper the faithful description of more complicated singularity structures such as the MP energy surface.
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This constraint can hamper the faithful description of more complicated singularity structures such as the MP energy surface.
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Despite this limitation, Ref.~\onlinecite{Goodson_2000a} demonstrates that quadratic approximants
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provide convergent results in the most divergent cases considered by Olsen and
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collaborators\cite{Christiansen_1996,Olsen_1996}
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@ -1821,7 +1816,7 @@ We began by presenting the fundamental concepts behind non-Hermitian extensions
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including the Hartree--Fock approximation and Rayleigh--Schr\"odinger perturbation theory.
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We then provided a comprehensive review of the various research that has been performed
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around the physics of complex singularities in perturbation theory, with a particular focus on M{\o}ller--Plesset theory.
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Seminal contributions from various research groups \trashHB{around the world} have revealed highly oscillatory,
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Seminal contributions from various research groups have revealed highly oscillatory,
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slowly convergent, or catastrophically divergent behaviour of the restricted and/or unrestricted MP perturbation series.%
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\cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988}
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In particular, the spin-symmetry-broken unrestricted MP series is notorious
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@ -1884,7 +1879,7 @@ for understanding the subtle features of perturbation theory in the complex plan
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such as Kohn-Sham DFT, \cite{Carrascal_2015,Cohen_2016} linear-response theory,\cite{Carrascal_2018}
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many-body perturbation theory,\cite{Romaniello_2009,Romaniello_2012,DiSabatino_2015,Hirata_2015,Tarantino_2017,Olevano_2019}
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ensemble DFT, \cite{Deur_2017,Deur_2018,Senjean_2018,Sagredo_2018,Fromager_2020} thermal DFT,\cite{Smith_2016,Smith_2018}
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\titou{wave function methods},\cite{Stein_2014,Henderson_2015,Shepherd_2016} and many more.
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wave function methods,\cite{Stein_2014,Henderson_2015,Shepherd_2016} and many more.
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In particular, we have shown that the Hubbard dimer contains sufficient flexibility to describe
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the effects of symmetry breaking, the MP critical point, and resummation techniques, in contrast to the more
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minimalistic models considered previously.
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