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Phys.}\ }\textbf {\bibinfo {volume} {144}},\ + \bibinfo {pages} {094112} (\bibinfo {year} {2016})}\BibitemShut {NoStop}% +\end{thebibliography}% diff --git a/arXiv/EPAWTFT.tex b/arXiv/EPAWTFT.tex new file mode 100644 index 0000000..bedc7cd --- /dev/null +++ b/arXiv/EPAWTFT.tex @@ -0,0 +1,1918 @@ +\documentclass[aps,prb,reprint,showkeys,superscriptaddress]{revtex4-1} +\usepackage{subcaption} +\usepackage{bm,graphicx,tabularx,array,booktabs,dcolumn,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,siunitx} +\usepackage[version=4]{mhchem} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{txfonts} + +\usepackage[normalem]{ulem} +\definecolor{hughgreen}{RGB}{0, 128, 0} +\newcommand{\titou}[1]{\textcolor{red}{#1}} +\newcommand{\hugh}[1]{\textcolor{hughgreen}{#1}} +\newcommand{\hughDraft}[1]{\textcolor{orange}{#1}} +\newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}} +\newcommand{\trashHB}[1]{\textcolor{hughgreen}{\sout{#1}}} +\newcommand{\antoine}[1]{\textcolor{cyan}{#1}} +\newcommand{\trashantoine}[1]{\textcolor{cyan}{\sout{#1}}} + +\usepackage[ + colorlinks=true, + citecolor=blue, + linkcolor=blue, + filecolor=blue, + urlcolor=blue, + breaklinks=true + ]{hyperref} +\urlstyle{same} + +\newcommand{\ctab}{\multicolumn{1}{c}{---}} +\newcommand{\latin}[1]{#1} +%\newcommand{\latin}[1]{\textit{#1}} +\newcommand{\ie}{\latin{i.e.}} +\newcommand{\eg}{\latin{e.g.}} +\newcommand{\etal}{\textit{et al.}} + +\newcommand{\mc}{\multicolumn} +\newcommand{\fnm}{\footnotemark} +\newcommand{\fnt}{\footnotetext} +\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\mr}{\multirow} + +% operators +\newcommand{\bH}{\mathbf{H}} +\newcommand{\bV}{\mathbf{V}} +\newcommand{\bh}{\mathbf{h}} +\newcommand{\bQ}{\mathbf{Q}} +\newcommand{\bSig}{\mathbf{\Sigma}} +\newcommand{\br}{\mathbf{r}} +\newcommand{\bp}{\mathbf{p}} +\newcommand{\cP}{\mathcal{P}} +\newcommand{\cS}{\mathcal{S}} +\newcommand{\cT}{\mathcal{T}} +\newcommand{\cC}{\mathcal{C}} +\newcommand{\PT}{\mathcal{PT}} + +\newcommand{\EPT}{E_{\PT}} +\newcommand{\laPT}{\lambda_{\PT}} + +\newcommand{\EEP}{E_\text{EP}} +\newcommand{\laEP}{\lambda_\text{EP}} + + +\newcommand{\Ne}{N} % Number of electrons +\newcommand{\Nn}{M} % Number of nuclei +\newcommand{\hI}{\Hat{I}} +\newcommand{\hH}{\Hat{H}} +\newcommand{\hS}{\Hat{S}} +\newcommand{\hT}{\Hat{T}} +\newcommand{\hW}{\Hat{W}} +\newcommand{\hV}{\Hat{V}} +\newcommand{\hc}[2]{\Hat{c}_{#1}^{#2}} +\newcommand{\hn}[1]{\Hat{n}_{#1}} +\newcommand{\n}[1]{n_{#1}} +\newcommand{\Dv}{\Delta v} + +\newcommand{\ra}{\rightarrow} +\newcommand{\up}{\uparrow} +\newcommand{\dw}{\downarrow} + +% Center tabularx columns +\newcolumntype{Y}{>{\centering\arraybackslash}X} + +% HF rotation angles +\newcommand{\ta}{\theta^{\,\alpha}} +\newcommand{\tb}{\theta^{\,\beta}} +\newcommand{\ts}{\theta^{\sigma}} + +% Some constants +\renewcommand{\i}{\mathrm{i}} % Imaginary unit +\newcommand{\e}{\mathrm{e}} % Euler number +\newcommand{\rc}{r_{\text{c}}} +\newcommand{\lc}{\lambda_{\text{c}}} +\newcommand{\lp}{\lambda_{\text{p}}} +\newcommand{\lep}{\lambda_{\text{EP}}} + +% Some energies +\newcommand{\Emp}{E_{\text{MP}}} + +% Blackboard bold +\newcommand{\bbR}{\mathbb{R}} +\newcommand{\bbC}{\mathbb{C}} + +% Making life easier +\newcommand{\Lup}{\mathcal{L}^{\uparrow}} +\newcommand{\Ldown}{\mathcal{L}^{\downarrow}} +\newcommand{\Lsi}{\mathcal{L}^{\sigma}} +\newcommand{\Rup}{\mathcal{R}^{\uparrow}} +\newcommand{\Rdown}{\mathcal{R}^{\downarrow}} +\newcommand{\Rsi}{\mathcal{R}^{\sigma}} +\newcommand{\vhf}{\Hat{v}_{\text{HF}}} +\newcommand{\whf}{\Psi_{\text{HF}}} + + +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France.} +\newcommand{\UOX}{Physical and Theoretical Chemical Laboratory, Department of Chemistry, University of Oxford, Oxford, OX1 3QZ, U.K.} +\begin{document} + +\title{Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them} + +\author{Antoine \surname{Marie}} +\affiliation{\LCPQ} +\author{Hugh G.~A.~\surname{Burton}} +\email[Corresponding author: ]{hugh.burton@chem.ox.ac.uk} +\affiliation{\UOX} +\author{Pierre-Fran\c{c}ois \surname{Loos}} +\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr} +\affiliation{\LCPQ} + + +\begin{abstract} +We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. +We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. +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. +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. +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. +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. +\end{abstract} + +\keywords{perturbation theory, complex plane, exceptional point, divergent series, resummation} + +\maketitle + +%%%%%%%%%%%%%%%%%%%%%%% +\section{Introduction} +\label{sec:intro} +%%%%%%%%%%%%%%%%%%%%%%% + +% SPIKE THE READER +Perturbation theory isn't usually considered in the complex plane. +Normally it is applied using real numbers as one of very few available tools for +describing realistic quantum systems. +In particular, time-independent Rayleigh--Schr\"odinger perturbation theory\cite{RayleighBook,Schrodinger_1926} +has emerged as an instrument of choice among the vast array of methods developed for this purpose.% +\cite{SzaboBook,JensenBook,CramerBook,HelgakerBook,ParrBook,FetterBook,ReiningBook} +However, the properties of perturbation theory in the complex plane +are essential for understanding the quality of perturbative approximations on the real axis. + +% Moller-Plesset +In electronic structure theory, the workhorse of time-independent perturbation theory is M\o{}ller--Plesset (MP) +theory,\cite{Moller_1934} which remains one of the most popular methods for computing the electron +correlation energy.\cite{Wigner_1934,Lowdin_1958} +This approach estimates the exact electronic energy by constructing a perturbative correction on top +of a mean-field Hartree--Fock (HF) approximation.\cite{SzaboBook} +The popularity of MP theory stems from its black-box nature, size-extensivity, and relatively low computational scaling, +making it easily applied in a broad range of molecular research.\cite{HelgakerBook} +However, it is now widely recognised that the series of MP approximations (defined for a given perturbation +order $n$ as MP$n$) can show erratic, slow, or divergent behaviour that limit its systematic improvability.% +\cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988} +As a result, practical applications typically employ only the lowest-order MP2 approach, while +the successive MP3, MP4, and MP5 (and higher order) terms are generally not considered to offer enough improvement +to justify their increased cost. +Turning the MP approximations into a convergent and +systematically improvable series largely remains an open challenge. + +% COMPLEX PLANE +Our conventional view of electronic structure theory is centred around the Hermitian notion of quantised energy levels, +where the different electronic states of a molecule are discrete and energetically ordered. +The lowest energy state defines the ground electronic state, while higher energy states +represent electronic excited states. +However, an entirely different perspective on quantisation can be found by analytically continuing +quantum mechanics into the complex domain. +In this inherently non-Hermitian framework, the energy levels emerge as individual \textit{sheets} of a complex +multi-valued function and can be connected as one continuous \textit{Riemann surface}.\cite{BenderPTBook} +This connection is possible because the orderability of real numbers is lost when energies are extended to the +complex domain. +As a result, our quantised view of conventional quantum mechanics only arises from +restricting our domain to Hermitian approximations. + +% NON-HERMITIAN HAMILTONIANS +Non-Hermitian Hamiltonians already have a long history in quantum chemistry and have been extensively used to +describe metastable resonance phenomena.\cite{MoiseyevBook} +Through the methods of complex-scaling\cite{Moiseyev_1998} and complex absorbing +potentials,\cite{Riss_1993,Ernzerhof_2006,Benda_2018} outgoing resonances can be stabilised as square-integrable +wave functions with a complex energy that allows the resonance energy and lifetime to be computed. +We refer the interested reader to the excellent book by Moiseyev for a general overview. \cite{MoiseyevBook} + +% EXCEPTIONAL POINTS +The Riemann surface for the electronic energy $E(\lambda)$ with a coupling parameter $\lambda$ can be +constructed by analytically continuing the function into the complex $\lambda$ domain. +In the process, the ground and excited states become smoothly connected and form a continuous complex-valued +energy surface. +\textit{Exceptional points} (EPs) can exist on this energy surface, corresponding to branch point +singularities where two (or more) states become exactly degenerate.% +\cite{MoiseyevBook,Heiss_1988,Heiss_1990,Heiss_1999,Berry_2011,Heiss_2012,Heiss_2016,Benda_2018} +While EPs can be considered as the non-Hermitian analogues of conical intersections,\cite{Yarkony_1996} +the behaviour of their eigenvalues near a degeneracy could not be more different. +Incredibly, following the eigenvalues around an EP leads to the interconversion of the degenerate states, +and multiple loops around the EP are required to recover the initial energy.\cite{MoiseyevBook,Heiss_2016,Benda_2018} +In contrast, encircling a conical intersection leaves the states unchanged. +Furthermore, while the eigenvectors remain orthogonal at a conical intersection, the eigenvectors at an EP +become identical and result in a \textit{self-orthogonal} state. \cite{MoiseyevBook} +An EP effectively creates a ``portal'' between ground and excited-states in the complex plane.% +\cite{Burton_2019,Burton_2019a} +This transition between states has been experimentally observed in electronics, +microwaves, mechanics, acoustics, atomic systems and optics.\cite{Bittner_2012,Chong_2011,Chtchelkatchev_2012,Doppler_2016,Guo_2009,Hang_2013,Liertzer_2012,Longhi_2010,Peng_2014, Peng_2014a,Regensburger_2012,Ruter_2010,Schindler_2011,Szameit_2011,Zhao_2010,Zheng_2013,Choi_2018,El-Ganainy_2018} + +% MP THEORY IN THE COMPLEX PLANE +The MP energy correction can be considered as a function of the perturbation parameter $\lambda$. +When the domain of $\lambda$ is extended to the complex plane, EPs can also occur in the MP energy. +Although these EPs are generally complex-valued, +their positions are intimately related to the +convergence of the perturbation expansion on the real axis.% +\cite{BenderBook,Olsen_1996,Olsen_2000,Olsen_2019,Mihalka_2017a,Mihalka_2017b,Mihalka_2019} +Furthermore, the existence of an avoided crossing on the real axis is indicative of a nearby EP +in the complex plane. +Our aim in this article is to provide a comprehensive review of the fundamental relationship between EPs +and the convergence properties of the MP series. +In doing so, we will demonstrate how understanding the MP energy in the complex plane can +be harnessed to significantly improve estimates of the exact energy using only the lowest-order terms +in the MP series. + +In Sec.~\ref{sec:EPs}, we introduce the key concepts such as Rayleigh--Schr\"odinger perturbation theory and the mean-field HF approximation, and discuss their non-Hermitian analytic continuation into the complex plane. +Section \ref{sec:MP} presents MP perturbation theory and we report a comprehensive historical overview of the research that +has been performed on the physics of MP singularities. +In Sec.~\ref{sec:Resummation}, we discuss several resummation techniques for improving the accuracy +of low-order MP approximations, including Pad\'e and quadratic approximants. +Finally, we draw our conclusions in Sec.~\ref{sec:ccl} and highlight our perspective on directions for +future research. +Throughout this review, we present illustrative and pedagogical examples based on the ubiquitous +Hubbard dimer, reinforcing the amazing versatility of this powerful simplistic model. + +%%%%%%%%%%%%%%%%%%%%%%% +\section{Exceptional Points in Electronic Structure} +\label{sec:EPs} +%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Time-Independent Schr\"odinger Equation} +\label{sec:TDSE} +%%%%%%%%%%%%%%%%%%%%%%% +Within the Born-Oppenheimer approximation, the exact molecular Hamiltonian with $\Ne$ electrons and +$\Nn$ (clamped) nuclei is defined for a given nuclear framework as +\begin{equation}\label{eq:ExactHamiltonian} + \hH(\vb{R}) = + - \frac{1}{2} \sum_{i}^{\Ne} \grad_i^2 + - \sum_{i}^{\Ne} \sum_{A}^{\Nn} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}} + + \sum_{i2t$, the closed-shell orbital restriction prevents RHF from +modelling the correct physics with the two electrons on opposite sites. + +%%% FIG 3 (?) %%% +% Analytic Continuation of HF +%%%%%%%%%%%%%%%%% +\begin{figure*}[t] + \begin{subfigure}{0.49\textwidth} + \includegraphics[height=0.65\textwidth,trim={0pt 0pt 0pt -35pt},clip]{fig3a} + \subcaption{\label{subfig:UHF_cplx_angle}} + \end{subfigure} + \begin{subfigure}{0.49\textwidth} + \includegraphics[height=0.65\textwidth]{fig3b} + \subcaption{\label{subfig:UHF_cplx_energy}} + \end{subfigure} + \caption{% + (\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta_{\text{UHF}}$ for $\lambda \in \bbC$ in the Hubbard dimer for $U/t = 2$. + Symmetry-broken solutions correspond to individual sheets and become equivalent at + the \textit{quasi}-EP $\lambda_{\text{c}}$ (black dot). + The RHF solution is independent of $\lambda$, giving the constant plane at $\pi/2$. + (\subref{subfig:UHF_cplx_energy}) The corresponding HF energy surfaces show a non-analytic + point at the \textit{quasi}-EP. + \label{fig:HF_cplx}} +\end{figure*} +%%%%%%%%%%%%%%%%% + +As the on-site repulsion is increased from 0, the HF approximation reaches a critical value at $U=2t$ where a symmetry-broken +UHF solution appears with a lower energy than the RHF one. +Note that the RHF wave function remains a genuine solution of the HF equations for $U \ge 2t$, but corresponds to a saddle point +of the HF energy rather than a minimum. +This critical point is analogous to the infamous Coulson--Fischer point identified in the hydrogen dimer.\cite{Coulson_1949} +For $U \ge 2t$, the optimal orbital rotation angles for the UHF orbitals become +\begin{subequations} +\begin{align} + \ta_\text{UHF} & = \arctan (-\frac{2t}{\sqrt{U^2 - 4t^2}}), + \label{eq:ta_uhf} + \\ + \tb_\text{UHF} & = \arctan (+\frac{2t}{\sqrt{U^2 - 4t^2}}), + \label{eq:tb_uhf} +\end{align} +\end{subequations} +with the corresponding UHF ground-state energy (Fig.~\ref{fig:HF_real}) +\begin{equation} + E_\text{UHF} \equiv E_\text{HF}(\ta_\text{UHF}, \tb_\text{UHF}) = - \frac{2t^2}{U}. +\end{equation} +Time-reversal symmetry dictates that this UHF wave function must be degenerate with its spin-flipped counterpart, obtained +by swapping $\ta_{\text{UHF}}$ and $\tb_{\text{UHF}}$ in Eqs.~\eqref{eq:ta_uhf} and \eqref{eq:tb_uhf}. +This type of symmetry breaking is also called a spin-density wave in the physics community as the system +``oscillates'' between the two symmetry-broken configurations. \cite{GiulianiBook} +Symmetry breaking can also occur in RHF theory when a charge-density wave is formed from an oscillation +between the two closed-shell configurations with both electrons localised on one site or the other.\cite{StuberPaldus,Fukutome_1981} + +%===============================================% +\subsection{Self-Consistency as a Perturbation} +%===============================================% + +% INTRODUCE PARAMETRISED FOCK HAMILTONIAN +The inherent non-linearity in the Fock eigenvalue problem arises from self-consistency +in the HF approximation, and is usually solved through an iterative approach.\cite{Roothaan_1951,Hall_1951} +Alternatively, the non-linear terms arising from the Coulomb and exchange operators can +be considered as a perturbation from the core Hamiltonian \eqref{eq:Hcore} by introducing the +transformation $U \to \lambda\, U$, giving the parametrised Fock operator +\begin{equation} + \Hat{f}(\vb{x} ; \lambda) = \Hat{h}(\vb{x}) + \lambda\, \vhf(\vb{x}). +\end{equation} +The orbitals in the reference problem $\lambda=0$ correspond to the symmetry-pure eigenfunctions of the one-electron core +Hamiltonian, while self-consistent solutions at $\lambda = 1$ represent the orbitals of the true HF solution. + +% INTRODUCE COMPLEX ANALYTIC-CONTINUATION +For real $\lambda$, the self-consistent HF energies at given (real) $U$ and $t$ values +in the Hubbard dimer directly mirror the energies shown in Fig.~\ref{fig:HF_real}, +with coalesence points at +\begin{equation} + \lambda_{\text{c}} = \pm \frac{2t}{U}. + \label{eq:scaled_fock} +\end{equation} +In contrast, when $\lambda$ becomes complex, the HF equations become non-Hermitian and +each HF solution can be analytically continued for all $\lambda$ values using +the holomorphic HF approach.\cite{Hiscock_2014,Burton_2016,Burton_2018} +Remarkably, the coalescence point in this analytic continuation emerges as a +\textit{quasi}-EP on the real $\lambda$ axis (Fig.~\ref{fig:HF_cplx}), where +the different HF solutions become equivalent but not self-orthogonal.\cite{Burton_2019} +By analogy with perturbation theory, the regime where this \textit{quasi}-EP occurs +within $\lambda_{\text{c}} \le 1$ can be interpreted as an indication that +the symmetry-pure reference orbitals no longer provide a qualitatively +accurate representation for the true HF ground state at $\lambda = 1$. +For example, in the Hubbard dimer with $U > 2t$, one finds $\lambda_{\text{c}} < 1$ and the symmetry-pure orbitals +do not provide a good representation of the HF ground state. +In contrast, $U < 2t$ yields $\lambda_{\text{c}} > 1$ and corresponds to +the regime where the HF ground state is correctly represented by symmetry-pure orbitals. + +% COMPLEX ADIABATIC CONNECTION +We have recently shown that the complex scaled Fock operator \eqref{eq:scaled_fock} +also allows states of different symmetries to be interconverted by following a well-defined +contour in the complex $\lambda$-plane.\cite{Burton_2019} +In particular, by slowly varying $\lambda$ in a similar (yet different) manner +to an adiabatic connection in density-functional theory,\cite{Langreth_1975,Gunnarsson_1976,Zhang_2004} +a ground-state wave function can be ``morphed'' into an excited-state wave function +via a stationary path of HF solutions. +This novel approach to identifying excited-state wave functions demonstrates the fundamental +role of \textit{quasi}-EPs in determining the behaviour of the HF approximation. +Furthermore, the complex-scaled Fock operator can be used routinely to construct analytic +continuations of HF solutions beyond the points where real HF solutions +coalesce and vanish.\cite{Burton_2019b} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{M{\o}ller--Plesset Perturbation Theory in the Complex Plane} +\label{sec:MP} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%=====================================================% +\subsection{Background Theory} +%=====================================================% + +In electronic structure, the HF Hamiltonian \eqref{eq:HFHamiltonian} is often used as the zeroth-order Hamiltonian +to define M\o{}ller--Plesset (MP) perturbation theory.\cite{Moller_1934} +This approach can recover a large proportion of the electron correlation energy,\cite{Lowdin_1955a,Lowdin_1955b,Lowdin_1955c} +and provides the foundation for numerous post-HF approximations. +With the MP partitioning, the parametrised perturbation Hamiltonian becomes +\begin{multline}\label{eq:MPHamiltonian} + \hH(\lambda) = + \sum_{i}^{N} \qty[ - \frac{\grad_i^2}{2} - \sum_{A}^{M} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}} ] + \\ + + (1-\lambda) \sum_{i}^{N} \vhf(\vb{x}_i) + + \lambda\sum_{i 1$) and $4.5$ (where $r_c < 1$). + The Riemann surfaces associated with the exact energies of the RMP Hamiltonian \eqref{eq:H_RMP} are also represented for these two values of $U/t$ as functions of complex $\lambda$. + \label{fig:RMP}} +\end{figure*} + +The behaviour of the RMP and UMP series observed in \ce{H2} can also be illustrated by considering +the analytic Hubbard dimer with a complex-valued perturbation strength. +In this system, the stretching of the \ce{H\bond{-}H} bond is directly mirrored by an increase in the ratio $U/t$. +Using the ground-state RHF reference orbitals leads to the parametrised RMP Hamiltonian +\begin{widetext} +\begin{equation} +\label{eq:H_RMP} +\bH_\text{RMP}\qty(\lambda) = + \begin{pmatrix} + -2t + U - \lambda U/2 & 0 & 0 & \lambda U/2 \\ + 0 & U - \lambda U/2 & \lambda U/2 & 0 \\ + 0 & \lambda U/2 & U - \lambda U/2 & 0 \\ + \lambda U/2 & 0 & 0 & 2t + U - \lambda U/2 \\ + \end{pmatrix}, +\end{equation} +\end{widetext} +which yields the ground-state energy +\begin{equation} + \label{eq:E0MP} + E_{-}(\lambda) = U - \frac{\lambda U}{2} - \frac{1}{2} \sqrt{(4t)^2 + \lambda ^2 U^2}. +\end{equation} +From this expression, the EPs can be identified as $\lep = \pm \i 4t / U$, +giving the radius of convergence +\begin{equation} + \rc = \abs{\frac{4t}{U}}. +\end{equation} +Remarkably, these EPs are identical to the exact EPs discussed in Sec.~\ref{sec:example}. +The Taylor expansion of the RMP energy can then be evaluated to obtain the $k$th-order MP correction +\begin{equation} + E_\text{RMP}^{(k)} = U \delta_{0,k} - \frac{1}{2} \frac{U^k}{(4t)^{k-1}} \mqty( 1/2 \\ k/2). +\end{equation} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% RADIUS OF CONVERGENCE PLOTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +The RMP series is convergent for $U = 3.5\,t$ with $\rc > 1$, as illustrated for the individual terms at each +perturbation order in Fig.~\ref{subfig:RMP_cvg}. +In contrast, for $U = 4.5t$ one finds $\rc < 1$, and the RMP series becomes divergent. +The corresponding Riemann surfaces for $U = 3.5\,t$ and $4.5\,t$ are shown in Figs.~\ref{subfig:RMP_3.5} and +\ref{subfig:RMP_4.5}, respectively, with the single EP at $\lep$ (black dot) and the radius of convergence indicated +by the vertical cylinder of unit radius. +For the divergent case, $\lep$ lies inside this cylinder of convergence, while in the convergent case $\lep$ lies +outside this cylinder. +In both cases, the EP connects the ground state with the doubly-excited state, and thus the convergence behaviour +for the two states using the ground-state RHF orbitals is identical. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% RADIUS OF CONVERGENCE PLOTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{figure}[htb] + \includegraphics[width=\linewidth]{fig5} + \caption{ + Radius of convergence $r_c$ for the RMP ground state (red), the UMP ground state (blue), and the UMP excited state (orange) + series of the Hubbard dimer as functions of the ratio $U/t$. + \label{fig:RadConv}} +\end{figure} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +%%% FIG 3 %%% +\begin{figure*} + \begin{subfigure}{0.32\textwidth} + \includegraphics[height=0.75\textwidth]{fig6a} + \subcaption{\label{subfig:UMP_3} $U/t = 3$} + \end{subfigure} + % + \begin{subfigure}{0.32\textwidth} + \includegraphics[height=0.75\textwidth]{fig6b} + \subcaption{\label{subfig:UMP_cvg}} + \end{subfigure} + % + \begin{subfigure}{0.32\textwidth} + \includegraphics[height=0.75\textwidth]{fig6c} + \subcaption{\label{subfig:UMP_7} $U/t = 7$} + \end{subfigure} \caption{ + Convergence of the UMP series as a function of the perturbation order $n$ for the Hubbard dimer at $U/t = 3$ and $7$. + The Riemann surfaces associated with the exact energies of the UMP Hamiltonian \eqref{eq:H_UMP} are also represented for these two values of $U/t$ as functions of $\lambda$. + \label{fig:UMP}} +\end{figure*} + +The behaviour of the UMP series is more subtle than the RMP series as the spin-contamination in the wave function +introduces additional coupling between the singly- and doubly-excited configurations. +Using the ground-state UHF reference orbitals in the Hubbard dimer yields the parametrised UMP Hamiltonian +\begin{widetext} +\begin{equation} +\label{eq:H_UMP} +\bH_\text{UMP}\qty(\lambda) = + \begin{pmatrix} + -2t^2 \lambda/U & 0 & 0 & 2t^2 \lambda/U \\ + 0 & U - 2t^2 \lambda/U & 2t^2\lambda/U & 2t \sqrt{U^2 - (2t)^2} \lambda/U \\ + 0 & 2t^2\lambda/U & U - 2t^2 \lambda/U & -2t \sqrt{U^2 - (2t)^2} \lambda/U \\ + 2t^2 \lambda/U & 2t \sqrt{U^2 - (2t)^2} \lambda/U & -2t \sqrt{U^2 - (2t)^2} \lambda/U & 2U(1-\lambda) + 6t^2\lambda/U \\ + \end{pmatrix}. +\end{equation} +\end{widetext} +While a closed-form expression for the ground-state energy exists, it is cumbersome and we eschew reporting it. +Instead, the radius of convergence of the UMP series can be obtained numerically as a function of $U/t$, as shown +in Fig.~\ref{fig:RadConv}. +These numerical values reveal that the UMP ground-state series has $\rc > 1$ for all $U/t$ and always converges. +However, in the strong correlation limit (large $U/t$), this radius of convergence tends to unity, indicating that +the convergence of the corresponding UMP series becomes increasingly slow. +Furthermore, the doubly-excited state using the ground-state UHF orbitals has $\rc < 1$ for almost any value +of $U/t$, reaching the limiting value of $1/2$ for $U/t \to \infty$. Hence, the +excited-state UMP series will always diverge. + +% DISCUSSION OF UMP RIEMANN SURFACES +The convergence behaviour can be further elucidated by considering the full structure of the UMP energies +in the complex $\lambda$-plane (see Figs.~\ref{subfig:UMP_3} and \ref{subfig:UMP_7}). +These Riemann surfaces are illustrated for $U = 3t$ and $7t$ alongside the perturbation terms at each order +in Fig.~\ref{subfig:UMP_cvg}. +At $U = 3t$, the RMP series is convergent, while RMP becomes divergent for $U=7t$. +The ground-state UMP expansion is convergent in both cases, although the rate of convergence is significantly slower +for larger $U/t$ as the radius of convergence becomes increasingly close to one (Fig.~\ref{fig:RadConv}). + +% EFFECT OF SYMMETRY BREAKING +As the UHF orbitals break the spin symmetry, new coupling terms emerge between the electronic states that +cause fundamental changes to the structure of EPs in the complex $\lambda$-plane. +For example, while the RMP energy shows only one EP between the ground and +doubly-excited states (Fig.~\ref{fig:RMP}), the UMP energy has two pairs of complex-conjugate EPs: one connecting the ground state with the +singly-excited open-shell singlet, and the other connecting this single excitation to the +doubly-excited second excitation (Fig.~\ref{fig:UMP}). +This new ground-state EP always appears outside the unit cylinder and guarantees convergence of the ground-state energy. +However, the excited-state EP is moved within the unit cylinder and causes the +convergence of the excited-state UMP series to deteriorate. +Our interpretation of this effect is that the symmetry-broken orbital optimisation has redistributed the strong +coupling between the ground- and doubly-excited states into weaker couplings between all states, and has thus +sacrificed convergence of the excited-state series so that the ground-state convergence can be maximised. + +Since the UHF ground state already provides a good approximation to the exact energy, the ground-state sheet of +the UMP energy is relatively flat and the corresponding EP in the Hubbard dimer always lies outside the unit cylinder. +The slow convergence observed in stretched \ce{H2}\cite{Gill_1988} can then be seen as this EP +moves increasingly close to the unit cylinder at large $U/t$ and $\rc$ approaches one (from above). +Furthermore, the majority of the UMP expansion in this regime is concerned with removing spin-contamination from the wave +function rather than improving the energy. +It is well-known that the spin-projection needed to remove spin-contamination can require non-linear combinations +of highly-excited determinants,\cite{Lowdin_1955c} and thus it is not surprising that this process proceeds +very slowly as the perturbation order is increased. + +%==========================================% +\subsection{Classifying Types of Convergence} +%==========================================% + +% CREMER AND HE +As computational implementations of higher-order MP terms improved, the systematic investigation +of convergence behaviour in a broader class of molecules became possible. +Cremer and He introduced an efficient MP6 approach and used it to analyse the RMP convergence of +29 atomic and molecular systems.\cite{Cremer_1996} +They established two general classes: ``class A'' systems that exhibit monotonic convergence; +and ``class B'' systems for which convergence is erratic after initial oscillations. +By analysing the different cluster contributions to the MP energy terms, they proposed that +class A systems generally include well-separated and weakly correlated electron pairs, while class B systems +are characterised by dense electron clustering in one or more spatial regions.\cite{Cremer_1996} +In class A systems, they showed that the majority of the correlation energy arises from pair correlation, +with little contribution from triple excitations. +On the other hand, triple excitations have an important contribution in class B systems, including +orbital relaxation to doubly-excited configurations, and these contributions lead to oscillations of the total correlation energy. + +Using these classifications, Cremer and He then introduced simple extrapolation formulas for estimating the +exact correlation energy $\Delta E$ using terms up to MP6\cite{Cremer_1996} +\begin{subequations} +\begin{align} + \label{eq:CrHeA} +\Delta E_{\text{A}} + &= \Emp^{(2)} + \Emp^{(3)} + \Emp^{(4)} + + \frac{\Emp^{(5)}}{1 - (\Emp^{(6)} / \Emp^{(5)})}, + \\ + \label{eq:CrHeB} +\Delta E_{\text{B}} + &= \Emp^{(2)} + \Emp^{(3)} + \qty(\Emp^{(4)} + \Emp^{(5)}) \exp(\Emp^{(6)} / \Emp^{(5)}). +\end{align} +\end{subequations} +These class-specific formulas reduced the mean absolute error from the FCI correlation energy by a +factor of four compared to previous class-independent extrapolations, +highlighting how one can leverage a deeper understanding of MP convergence to improve estimates of +the correlation energy at lower computational costs. +In Sec.~\ref{sec:Resummation}, we consider more advanced extrapolation routines that take account of EPs in the complex $\lambda$-plane. + +In the late 90's, Olsen \etal\ discovered an even more concerning behaviour of the MP series. \cite{Olsen_1996} +They showed that the series could be divergent even in systems that were considered to be well understood, +such as \ce{Ne} or the \ce{HF} molecule. \cite{Olsen_1996, Christiansen_1996} +Cremer and He had already studied these two systems and classified them as \textit{class B} systems.\cite{Cremer_1996} +However, Olsen and co-workers performed their analysis in larger basis sets containing diffuse functions, +finding that the corresponding MP series becomes divergent at (very) high order. +The discovery of this divergent behaviour is particularly worrying as large basis sets +are required to get meaningful and accurate energies.\cite{Loos_2019d,Giner_2019} +Furthermore, diffuse functions are particularly important for anions and/or Rydberg excited states, where the wave function +is inherently more diffuse than the ground state.\cite{Loos_2018a,Loos_2020a} + +Olsen \etal\ investigated the causes of these divergences and the different types of convergence by +analysing the relation between the dominant singularity (\ie, the closest singularity to the origin) +and the convergence behaviour of the series.\cite{Olsen_2000} +Their analysis is based on Darboux's theorem: \cite{Goodson_2011} +\begin{quote} +\textit{``In the limit of large order, the series coefficients become equivalent to + the Taylor series coefficients of the singularity closest to the origin. ''} +\end{quote} +Following this theory, a singularity in the unit circle is designated as an intruder state, +with a front-door (or back-door) intruder state if the real part of the singularity is positive (or negative). + +Using their observations in Ref.~\onlinecite{Olsen_1996}, Olsen and collaborators proposed +a simple method that performs a scan of the real axis to detect the avoided crossing responsible +for the dominant singularities in the complex plane. \cite{Olsen_2000} +By modelling this avoided crossing using a two-state Hamiltonian, one can obtain an approximation for +the dominant singularities as the EPs of the two-state matrix +\begin{equation} + \label{eq:Olsen_2x2} + \underbrace{\mqty(\alpha & \delta \\ \delta & \beta )}_{\bH} + = \underbrace{\mqty(\alpha + \alpha_{\text{s}} & 0 \\ 0 & \beta + \beta_{\text{s}} )}_{\bH^{(0)}} + + \underbrace{\mqty( -\alpha_{\text{s}} & \delta \\ \delta & - \beta_{\text{s}})}_{\bV}, +\end{equation} +where the diagonal matrix is the unperturbed Hamiltonian matrix $\bH^{(0)}$ with level shifts +$\alpha_{\text{s}}$ and $\beta_{\text{s}}$, and $\bV$ represents the perturbation. + +The authors first considered molecules with low-lying doubly-excited states with the same spatial +and spin symmetry as the ground state. \cite{Olsen_2000} +In these systems, the exact wave function has a non-negligible contribution from the doubly-excited states, +and thus the low-lying excited states are likely to become intruder states. +For \ce{CH_2} in a diffuse, yet rather small basis set, the series is convergent at least up to the 50th order, and +the dominant singularity lies close (but outside) the unit circle, causing slow convergence of the series. +These intruder-state effects are analogous to the EP that dictates the convergence behaviour of +the RMP series for the Hubbard dimer (Fig.~\ref{fig:RMP}). +Furthermore, the authors demonstrated that the divergence for \ce{Ne} is due to a back-door intruder state +that arise when the ground state undergoes sharp avoided crossings with highly diffuse excited states. +This divergence is related to a more fundamental critical point in the MP energy surface that we will +discuss in Sec.~\ref{sec:MP_critical_point}. + +Finally, Ref.~\onlinecite{Olsen_1996} proved that the extrapolation formulas of Cremer and He \cite{Cremer_1996} +[see Eqs.~\eqref{eq:CrHeA} and \eqref{eq:CrHeB}] are not mathematically motivated when considering the complex +singularities causing the divergence, and therefore cannot be applied for all systems. +For example, the \ce{HF} molecule contains both back-door intruder states and low-lying doubly-excited states that +result in alternating terms up to 10th order. +The series becomes monotonically convergent at higher orders since +the two pairs of singularities are approximately the same distance from the origin. + +More recently, this two-state model has been extended to non-symmetric Hamiltonians as\cite{Olsen_2019} +\begin{equation} + \underbrace{\mqty(\alpha & \delta_1 \\ \delta_2 & \beta)}_{\bH} = \underbrace{\mqty(\alpha & 0 \\ 0 & \beta + \gamma )}_{\bH^{(0)}} + \underbrace{\mqty( 0 & \delta_2 \\ \delta_1 & - \gamma)}_{\bV}. +\end{equation} +This extension allows various choices of perturbation to be analysed, including coupled cluster +perturbation expansions \cite{Pawlowski_2019a,Pawlowski_2019b,Pawlowski_2019c,Pawlowski_2019d,Pawlowski_2019e} +and other non-Hermitian perturbation methods. +Note that new forms of perturbation expansions only occur when the sign of $\delta_1$ and $\delta_2$ differ. +Using this non-Hermitian two-state model, the convergence of a perturbation series can be characterised +according to a so-called ``archetype'' that defines the overall ``shape'' of the energy convergence.\cite{Olsen_2019} +For Hermitian Hamiltonians, these archetypes can be subdivided into five classes +(zigzag, interspersed zigzag, triadic, ripples, and geometric), +while two additional archetypes (zigzag-geometric and convex-geometric) are observed in non-Hermitian Hamiltonians. +% +The geometric archetype appears to be the most common for MP expansions,\cite{Olsen_2019} but the +ripples archetype corresponds to some of the early examples of MP convergence. \cite{Handy_1985,Lepetit_1988,Leininger_2000} +The three remaining Hermitian archetypes seem to be rarely observed in MP perturbation theory. +In contrast, the non-Hermitian coupled cluster perturbation theory,% +\cite{Pawlowski_2019a,Pawlowski_2019b,Pawlowski_2019c,Pawlowski_2019d,Pawlowski_2019e} exhibits a range of archetypes +including the interspersed zigzag, triadic, ripple, geometric, and zigzag-geometric forms. +This analysis highlights the importance of the primary singularity in controlling the high-order convergence, +regardless of whether this point is inside or outside the complex unit circle. \cite{Handy_1985,Olsen_2000} + +%======================================= +\subsection{M{\o}ller--Plesset Critical Point} +\label{sec:MP_critical_point} +%======================================= + +% STILLINGER INTRODUCES THE CRITICAL POINT +In the early 2000's, Stillinger reconsidered the mathematical origin behind the divergent series with odd-even +sign alternation.\cite{Stillinger_2000} +This type of convergence behaviour corresponds to Cremer and He's class B systems with closely spaced +electron pairs and includes \ce{Ne}, \ce{HF}, \ce{F-}, and \ce{H2O}.\cite{Cremer_1996} +Stillinger proposed that these series diverge due to a dominant singularity +on the negative real $\lambda$ axis, corresponding to a multielectron autoionisation threshold.\cite{Stillinger_2000} +To understand Stillinger's argument, consider the parametrised MP Hamiltonian in the form +\begin{multline} +\label{eq:HamiltonianStillinger} + \hH(\lambda) = + \sum_{i}^{\Ne} \Bigg[ + \overbrace{-\frac{1}{2}\grad_i^2 + - \sum_{A}^{\Nn} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}}}^{\text{independent of $\lambda$}} + \\ + + \underbrace{(1-\lambda)\vhf(\vb{x}_i)}_{\text{repulsive for $\lambda < 1$}} + + \underbrace{\lambda\sum_{i 0$. +The reference Slater determinant for a doubly-occupied atom can be represented using RHF +orbitals [see Eq.~\eqref{eq:RHF_orbs}] with $\ta_{\text{RHF}} = \tb_{\text{RHF}} = 0$, +which corresponds to strictly localising the two electrons on the left site. +%and energy +%\begin{equation} +% E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon). +%\end{equation} +With this representation, the parametrised asymmetric RMP Hamiltonian becomes +\begin{widetext} +\begin{equation} +\label{eq:H_asym} +\bH_\text{asym}\qty(\lambda) = +\begin{pmatrix} + 2(U-\epsilon) - \lambda U & -\lambda t & -\lambda t & 0 \\ + -\lambda t & (U-\epsilon) - \lambda U & 0 & -\lambda t \\ + -\lambda t & 0 & (U-\epsilon) -\lambda U & -\lambda t \\ + 0 & -\lambda t & -\lambda t & \lambda U \\ +\end{pmatrix}. +\end{equation} +\end{widetext} + +% DERIVING BEHAVIOUR OF THE CRITICAL SITE +For the ghost site to perfectly represent ionised electrons, the hopping term between the two sites must vanish (\ie, $t=0$). +This limit corresponds to the dissociative regime in the asymmetric Hubbard dimer as discussed in Ref.~\onlinecite{Carrascal_2018}, +and the RMP energies become +\begin{subequations} +\begin{align} + E_{-} &= 2(U - \epsilon) - \lambda U, + \\ + E_{\text{S}} &= (U - \epsilon) - \lambda U, + \\ + E_{+} &= U \lambda, +\end{align} +\end{subequations} +as shown in Fig.~\ref{subfig:rmp_cp} (dashed lines). +The RMP critical point then corresponds to the intersection $E_{-} = E_{+}$, giving the critical $\lambda$ value +\begin{equation} + \lc = 1 - \frac{\epsilon}{U}. +\end{equation} +Clearly the radius of convergence $\rc = \abs{\lc}$ is controlled directly by the ratio $\epsilon / U$, +with a convergent RMP series occurring for $\epsilon > 2 U$. +The on-site repulsion $U$ controls the strength of the HF potential localised around the ``atomic site'', with a +stronger repulsion encouraging the electrons to be ionised at a less negative value of $\lambda$. +Large $U$ can be physically interpreted as strong electron repulsion effects in electron dense molecules. +In contrast, smaller $\epsilon$ gives a weaker attraction to the atomic site, +representing strong screening of the nuclear attraction by core and valence electrons, +and again a less negative $\lambda$ is required for ionisation to occur. +Both of these factors are common in atoms on the right-hand side of the periodic table, \eg, \ce{F}, +\ce{O}, \ce{Ne}. +Molecules containing these atoms are therefore often class $\beta$ systems with +a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_2006} + +% EXACT VERSUS APPROXIMATE +The critical point in the exact case $t=0$ lies on the negative real $\lambda$ axis (Fig.~\ref{subfig:rmp_cp}: dashed lines), +mirroring the behaviour of a quantum phase transition.\cite{Kais_2006} +However, in practical calculations performed with a finite basis set, the critical point is modelled as a cluster +of branch points close to the real axis. +The use of a finite basis can be modelled in the asymmetric dimer by making the second site a less +idealised destination for the ionised electrons with a non-zero (yet small) hopping term $t$. +Taking the value $t=0.1$ (Fig.~\ref{subfig:rmp_cp}: solid lines), the critical point becomes a +sharp avoided crossing with a complex-conjugate pair of EPs close to the real axis (Fig.~\ref{subfig:rmp_cp_surf}). +In the limit $t \to 0$, these EPs approach the real axis (Fig.~\ref{subfig:rmp_ep_to_cp}), +mirroring Sergeev's discussion on finite basis +set representations of the MP critical point.\cite{Sergeev_2006} + +%------------------------------------------------------------------% +% Figure on the UMP critical point +%------------------------------------------------------------------% +\begin{figure*}[t] + \begin{subfigure}{0.32\textwidth} + \includegraphics[height=0.75\textwidth,trim={0pt 5pt -10pt 15pt},clip]{fig8a} + \subcaption{\label{subfig:ump_cp}} + \end{subfigure} + % + \begin{subfigure}{0.32\textwidth} + \includegraphics[height=0.75\textwidth]{fig8b} + \subcaption{\label{subfig:ump_cp_surf}} + \end{subfigure} + % + \begin{subfigure}{0.32\textwidth} + \includegraphics[height=0.75\textwidth]{fig8c} + \subcaption{\label{subfig:ump_ep_to_cp}} + \end{subfigure} +% \includegraphics[height=0.65\textwidth,trim={0pt 5pt 0pt 15pt}, clip]{ump_critical_point} +\caption{% + The UMP ground-state EP in the symmetric Hubbard dimer becomes a critical point in the strong correlation limit (\ie, large $U/t$). + (\subref{subfig:ump_cp}) As $U/t$ increases, the avoided crossing on the real $\lambda$ axis + becomes increasingly sharp. + (\subref{subfig:ump_cp_surf}) Complex energy surfaces for $U = 5t$. + (\subref{subfig:ump_ep_to_cp}) Convergence of the EPs at $\lep$ onto the real axis for $U/t \to \infty$. + %mirrors the formation of the RMP critical point and other QPTs in the complete basis set limit. +\label{fig:UMP_cp}} + +\end{figure*} +%------------------------------------------------------------------% + +% RELATIONSHIP BETWEEN QPT AND UMP +Returning to the symmetric Hubbard dimer, we showed in Sec.~\ref{sec:spin_cont} that the slow +convergence of the strongly correlated UMP series +was due to a complex-conjugate pair of EPs just outside the radius of convergence. +These EPs have positive real components and small imaginary components (see Fig.~\ref{fig:UMP}), suggesting a potential +connection to MP critical points and QPTs (see Sec.~\ref{sec:MP_critical_point}). +For $\lambda>1$, the HF potential becomes an attractive component in Stillinger's +Hamiltonian displayed in Eq.~\eqref{eq:HamiltonianStillinger}, while the explicit electron-electron interaction +becomes increasingly repulsive. +Closed-shell critical points along the positive real $\lambda$ axis then represent +points where the two-electron repulsion overcomes the attractive HF potential +and a single electron dissociates from the molecule (see Ref.~\onlinecite{Sergeev_2006}). + +In contrast, symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons. +Consider one of the two reference UHF solutions where the spin-up and spin-down electrons are localised on the left and right sites respectively. +The spin-up HF potential will then be a repulsive interaction from the spin-down electron +density that is centred around the right site (and vice-versa). +As $\lambda$ becomes greater than 1 and the HF potentials become attractive, there will be a sudden +driving force for the electrons to swap sites. +This swapping process can also be represented as a double excitation, and thus an avoided crossing will occur +for $\lambda \geq 1$ (Fig.~\ref{subfig:ump_cp}). +While this appears to be an avoided crossing between the ground and first-excited state, +the presence of an earlier excited-state avoided crossing means that the first-excited state qualitatively +represents the reference double excitation for $\lambda > 1/2$. + +% SHARPNESS AND QPT +The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$. +For small $U/t$, the HF potentials will be weak and the electrons will delocalise over the two sites, +both in the UHF reference and the exact wave function. +This delocalisation dampens the electron swapping process and leads to a ``shallow'' avoided crossing +that corresponds to EPs with non-zero imaginary components (solid lines in Fig.~\ref{subfig:ump_cp}). +As $U/t$ becomes larger, the HF potentials become stronger and the on-site repulsion dominates the hopping +term to make electron delocalisation less favourable. +In other words, the electrons localise on individual sites to form a Wigner crystal. +These effects create a stronger driving force for the electrons to swap sites until eventually this swapping +occurs exactly at $\lambda = 1$. +In this limit, the ground-state EPs approach the real axis (Fig.~\ref{subfig:ump_ep_to_cp}) and the avoided +crossing creates a gradient discontinuity in the ground-state energy (dashed lines in Fig.~\ref{subfig:ump_cp}). +We therefore find that, in the strong correlation limit, the symmetry-broken ground-state EP becomes +a new type of MP critical point and represents a QPT as the perturbation parameter $\lambda$ is varied. +Furthermore, this argument explains why the dominant UMP singularity lies so close, but always outside, the +radius of convergence (see Fig.~\ref{fig:RadConv}). + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Resummation Methods} +\label{sec:Resummation} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%% +\begin{figure*} + \includegraphics[height=0.23\textheight]{fig9a} + \includegraphics[height=0.23\textheight]{fig9b} + \caption{\label{fig:PadeRMP} + RMP ground-state energy as a function of $\lambda$ in the Hubbard dimer obtained using various truncated Taylor series and approximants + at $U/t = 3.5$ (left) and $U/t = 4.5$ (right).} +\end{figure*} +%%%%%%%%%%%%%%%%% + + +%As frequently claimed by Carl Bender, +It is frequently stated that +\textit{``the most stupid thing to do with a series is to sum it.''} +Nonetheless, quantum chemists are basically doing this on a daily basis. +As we have seen throughout this review, the MP series can often show erratic, +slow, or divergent behaviour. +In these cases, estimating the correlation energy by simply summing successive +low-order terms is almost guaranteed to fail. +Here, we discuss alternative tools that can be used to sum slowly convergent or divergent series. +These so-called ``resummation'' techniques form a vast field of research and thus we will +provide details for only the most relevant methods. +We refer the interested reader to more specialised reviews for additional information.% +\cite{Goodson_2011,Goodson_2019} + + +%==========================================% +\subsection{Pad\'e Approximant} +%==========================================% + +The failure of a Taylor series for correctly modelling the MP energy function $E(\lambda)$ +arises because one is trying to model a complicated function containing multiple branches, branch points, and +singularities using a simple polynomial of finite order. +A truncated Taylor series can only predict a single sheet and does not have enough +flexibility to adequately describe functions such as the MP energy. +Alternatively, the description of complex energy functions can be significantly improved +by introducing Pad\'e approximants, \cite{Pade_1892} and related techniques. \cite{BakerBook,BenderBook} + +A Pad\'e approximant can be considered as the best approximation of a function by a +rational function of given order. +More specifically, a $[d_A/d_B]$ Pad\'e approximant is defined as +\begin{equation} + \label{eq:PadeApp} + E_{[d_A/d_B]}(\lambda) = \frac{A(\lambda)}{B(\lambda)} + = \frac{\sum_{k=0}^{d_A} a_k\, \lambda^k}{1 + \sum_{k=1}^{d_B} b_k\, \lambda^k}, +\end{equation} +where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting terms for each power of $\lambda$. +Pad\'e approximants are extremely useful in many areas of physics and +chemistry\cite{Loos_2013,Pavlyukh_2017,Tarantino_2019,Gluzman_2020} as they can model poles, +which appear at the roots of $B(\lambda)$. +However, they are unable to model functions with square-root branch points +(which are ubiquitous in the singularity structure of perturbative methods) +and more complicated functional forms appearing at critical points +(where the nature of the solution undergoes a sudden transition). +Despite this limitation, the successive diagonal Pad\'e approximants (\ie, $d_A = d_B $) +often define a convergent perturbation series in cases where the Taylor series expansion diverges. + +\begin{table}[b] + \caption{RMP ground-state energy estimate at $\lambda = 1$ of the Hubbard dimer provided by various truncated Taylor + 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. + \label{tab:PadeRMP}} + \begin{ruledtabular} + \begin{tabular}{lccccc} + & & \mc{2}{c}{$\abs{\lc}$} & \mc{2}{c}{$E_{-}(\lambda = 1)$} \\ + \cline{3-4} \cline{5-6} + Method & Degree & $U/t = 3.5$ & $U/t = 4.5$ & $U/t = 3.5$ & $U/t = 4.5$ \\ + \hline + Taylor & 2 & & & $-1.01563$ & $-1.01563$ \\ + & 3 & & & $-1.01563$ & $-1.01563$ \\ + & 4 & & & $-0.86908$ & $-0.61517$ \\ + & 5 & & & $-0.86908$ & $-0.61517$ \\ + & 6 & & & $-0.92518$ & $-0.86858$ \\ + \hline + Pad\'e & [1/1] & $2.29$ & $1.78$ & $-1.61111$ & $-2.64286$ \\ + & [2/2] & $2.29$ & $1.78$ & $-0.82124$ & $-0.48446$ \\ + & [3/3] & $1.73$ & $1.34$ & $-0.91995$ & $-0.81929$ \\ + & [4/4] & $1.47$ & $1.14$ & $-0.90579$ & $-0.74866$ \\ + & [5/5] & $1.35$ & $1.05$ & $-0.90778$ & $-0.76277$ \\ + \hline + Exact & & $1.14$ & $0.89$ & $-0.90754$ & $-0.76040$ \\ + \end{tabular} + \end{ruledtabular} +\end{table} + +Figure~\ref{fig:PadeRMP} illustrates the improvement provided by diagonal Pad\'e +approximants compared to the usual Taylor expansion in cases where the RMP series of +the Hubbard dimer converges ($U/t = 3.5$) and diverges ($U/t = 4.5$). +More quantitatively, Table \ref{tab:PadeRMP} gathers estimates of the RMP ground-state +energy at $\lambda = 1$ provided by various truncated Taylor series and Pad\'e +approximants for these two values of the ratio $U/t$. +While the truncated Taylor series converges laboriously to the exact energy as the truncation +degree increases at $U/t = 3.5$, the Pad\'e approximants yield much more accurate results. +Furthermore, the distance of the closest pole to the origin $\abs{\lc}$ in the Pad\'e approximants +indicate that they provide a relatively good approximation to the position of the +true branch point singularity in the RMP energy. +For $U/t = 4.5$, the Taylor series expansion performs worse and eventually diverges, +while the Pad\'e approximants still offer relatively accurate energies and recovers +a convergent series. + +%%%%%%%%%%%%%%%%% +\begin{figure}[t] + \includegraphics[width=\linewidth]{fig10} + \caption{\label{fig:QuadUMP} + UMP energies in the Hubbard dimer as a function of $\lambda$ obtained using various approximants at $U/t = 3$.} +\end{figure} +%%%%%%%%%%%%%%%%% + +We can expect the UMP energy function to be much more challenging +to model properly as it contains three connected branches +(see Figs.~\ref{subfig:UMP_3} and \ref{subfig:UMP_7}). +Figure~\ref{fig:QuadUMP} and Table~\ref{tab:QuadUMP} indicate that this is indeed the case. +In particular, Fig.~\ref{fig:QuadUMP} illustrates that the Pad\'e approximants are trying to model +the square root branch point that lies close to $\lambda = 1$ by placing a pole on the real axis +(\eg, [3/3]) or with a very small imaginary component (\eg, [4/4]). +The proximity of these poles to the physical point $\lambda = 1$ means that any error in the Pad\'e +functional form becomes magnified in the estimate of the exact energy, as seen for the low-order +approximants in Table~\ref{tab:QuadUMP}. +However, with sufficiently high degree polynomials, one obtains +accurate estimates for the position of the closest singularity and the ground-state energy at $\lambda = 1$, +even in cases where the convergence of the UMP series is incredibly slow +(see Fig.~\ref{subfig:UMP_cvg}). + +%==========================================% +\subsection{Quadratic Approximant} +%==========================================% +Quadratic approximants are designed to model the singularity structure of the energy +function $E(\lambda)$ via a generalised version of the square-root singularity +expression \cite{Mayer_1985,Goodson_2011,Goodson_2019} +\begin{equation} + \label{eq:QuadApp} + 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} +with the polynomials +\begin{align} + \label{eq:PQR} + P(\lambda) & = \sum_{k=0}^{d_P} p_k \lambda^k, + & + Q(\lambda) & = \sum_{k=0}^{d_Q} q_k \lambda^k, + & + R(\lambda) & = \sum_{k=0}^{d_R} r_k \lambda^k, +\end{align} +defined such that $d_P + d_Q + d_R = n - 1$, and $n$ is the truncation order of the Taylor series of $E(\lambda)$. +Recasting Eq.~\eqref{eq:QuadApp} as a second-order expression in $E(\lambda)$, \ie, +\begin{equation} + Q(\lambda) E^2(\lambda) - P(\lambda) E(\lambda) + R(\lambda) \sim \order*{\lambda^{n+1}}, +\end{equation} +and substituting $E(\lambda$) by its $n$th-order expansion and the polynomials by +their respective expressions \eqref{eq:PQR} yields $n+1$ linear equations for the coefficients +$p_k$, $q_k$, and $r_k$ (where we are free to assume that $q_0 = 1$). +A quadratic approximant, characterised by the label $[d_P/d_Q,d_R]$, generates, by construction, +$n_\text{bp} = \max(2d_p,d_q+d_r)$ branch points at the roots of the polynomial +$P^2(\lambda) - 4 Q(\lambda) R(\lambda)$ and $d_q$ poles at the roots of $Q(\lambda)$. + +Generally, the diagonal sequence of quadratic approximant, +\ie, $[0/0,0]$, $[1/0,0]$, $[1/0,1]$, $[1/1,1]$, $[2/1,1]$, +is of particular interest as the order of the corresponding Taylor series increases on each step. +However, while a quadratic approximant can reproduce multiple branch points, it can only describe +a total of two branches. +This constraint can hamper the faithful description of more complicated singularity structures such as the MP energy surface. +Despite this limitation, Ref.~\onlinecite{Goodson_2000a} demonstrates that quadratic approximants +provide convergent results in the most divergent cases considered by Olsen and +collaborators\cite{Christiansen_1996,Olsen_1996} +and Leininger \etal \cite{Leininger_2000} + +As a note of caution, Ref.~\onlinecite{Goodson_2019} suggests that low-order +quadratic approximants can struggle to correctly model the singularity structure when +the energy function has poles in both the positive and negative half-planes. +In such a scenario, the quadratic approximant will tend to place its branch points in-between, potentially introducing singularities quite close to the origin. +The remedy for this problem involves applying a suitable transformation of the complex plane (such as a bilinear conformal mapping) which leaves the points at $\lambda = 0$ and $\lambda = 1$ unchanged. \cite{Feenberg_1956} + +\begin{table}[b] + \caption{Estimate for the distance of the closest singularity (pole or branch point) to the origin $\abs{\lc}$ + in the UMP energy function of the Hubbard dimer provided by various 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 + points $n_\text{bp} = \max(2d_p,d_q+d_r)$ generated by the quadratic approximants are also reported. + \label{tab:QuadUMP}} + \begin{ruledtabular} + \begin{tabular}{lccccccc} + & & & & \mc{2}{c}{$\abs{\lc}$} & \mc{2}{c}{$E_{-}(\lambda = 1)$} \\ + \cline{5-6}\cline{7-8} + \mc{2}{c}{Method} & $n$ & $n_\text{bp}$ & $U/t = 3$ & $U/t = 7$ & $U/t = 3$ & $U/t = 7$ \\ + \hline + Taylor & & 2 & & & & $-0.74074$ & $-0.29155$ \\ + & & 3 & & & & $-0.78189$ & $-0.29690$ \\ + & & 4 & & & & $-0.82213$ & $-0.30225$ \\ + & & 5 & & & & $-0.85769$ & $-0.30758$ \\ + & & 6 & & & & $-0.88882$ & $-0.31289$ \\ + \hline + Pad\'e & [1/1] & 2 & & $9.000$ & $49.00$ & $-0.75000$ & $-0.29167$ \\ + & [2/2] & 4 & & $0.974$ & $1.003$ & $\hphantom{-}0.75000$ & $-17.9375$ \\ + & [3/3] & 6 & & $1.141$ & $1.004$ & $-1.10896$ & $-1.49856$ \\ + & [4/4] & 8 & & $1.068$ & $1.003$ & $-0.85396$ & $-0.33596$ \\ + & [5/5] & 10 & & $1.122$ & $1.004$ & $-0.97254$ & $-0.35513$ \\ + \hline + Quadratic & [2/1,2] & 6 & 4 & $1.086$ & $1.003$ & $-1.01009$ & $-0.53472$ \\ + & [2/2,2] & 7 & 4 & $1.082$ & $1.003$ & $-1.00553$ & $-0.53463$ \\ + & [3/2,2] & 8 & 6 & $1.082$ & $1.001$ & $-1.00568$ & $-0.52473$ \\ + & [3/2,3] & 9 & 6 & $1.071$ & $1.002$ & $-0.99973$ & $-0.53102$ \\ + & [3/3,3] & 10 & 6 & $1.071$ & $1.002$ & $-0.99966$ & $-0.53103$ \\[0.5ex] + (pole-free) & [3/0,2] & 6 & 6 & $1.059$ & $1.003$ & $-1.13712$ & $-0.57199$ \\ + & [3/0,3] & 7 & 6 & $1.073$ & $1.002$ & $-1.00335$ & $-0.53113$ \\ + & [3/0,4] & 8 & 6 & $1.071$ & $1.002$ & $-1.00074$ & $-0.53116$ \\ + & [3/0,5] & 9 & 6 & $1.070$ & $1.002$ & $-1.00042$ & $-0.53114$ \\ + & [3/0,6] & 10 & 6 & $1.070$ & $1.002$ & $-1.00039$ & $-0.53113$ \\ + \hline + Exact & & & & $1.069$ & $1.002$ & $-1.00000$ & $-0.53113$ \\ + \end{tabular} + \end{ruledtabular} +\end{table} + +\begin{figure*} + \begin{subfigure}{0.32\textwidth} + \includegraphics[height=0.85\textwidth]{fig11a} + \subcaption{\label{subfig:322quad} [3/2,2] Quadratic} + \end{subfigure} + % + \begin{subfigure}{0.32\textwidth} + \includegraphics[height=0.85\textwidth]{fig11b} + \subcaption{\label{subfig:exact} Exact} + \end{subfigure} + % + \begin{subfigure}{0.32\textwidth} + \includegraphics[height=0.85\textwidth]{fig11c} + \subcaption{\label{subfig:304quad} [3/0,4] Quadratic} + \end{subfigure} +\caption{% +Comparison of the [3/2,2] and [3/0,4] quadratic approximants with the exact UMP energy surface in the complex $\lambda$ +plane 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 +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 +is free of poles.} +\label{fig:nopole_quad} +\end{figure*} + +For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximants +are quite poor approximations, but the $[1/0,1]$ version perfectly models the RMP energy +function by predicting a single pair of EPs at $\lambda_\text{EP} = \pm \i 4t/U$. +This is expected from the form of the RMP energy [see Eq.~\eqref{eq:E0MP}], which matches +the ideal target for quadratic approximants. +Furthermore, the greater flexibility of the diagonal quadratic approximants provides a significantly +improved model of the UMP energy in comparison to the Pad\'e approximants or Taylor series. +In particular, these quadratic approximants provide an effective model for the avoided crossings +(Fig.~\ref{fig:QuadUMP}) and an improved estimate for the distance of the +closest branch point to the origin. +Table~\ref{tab:QuadUMP} shows that they provide remarkably accurate +estimates of the ground-state energy at $\lambda = 1$. + +While the diagonal quadratic approximants provide significantly improved estimates of the +ground-state energy, we can use our knowledge of the UMP singularity structure to develop +even more accurate results. +We have seen in previous sections that the UMP energy surface +contains only square-root branch cuts that approach the real axis in the limit $U/t \to \infty$. +Since there are no true poles on this surface, we can obtain more accurate quadratic approximants by +taking $d_q = 0$ and increasing $d_r$ to retain equivalent accuracy in the square-root term [see Eq.\eqref{eq:QuadApp}]. +Figure~\ref{fig:nopole_quad} illustrates this improvement for the pole-free [3/0,4] quadratic +approximant compared to the [3/2,2] approximant with the same truncation degree in the Taylor +expansion. +Clearly, modelling the square-root branch point using $d_q = 2$ has the negative effect of +introducing spurious poles in the energy, while focussing purely on the branch point with $d_q = 0$ +leads to a significantly improved model. +Table~\ref{tab:QuadUMP} shows that these pole-free quadratic approximants +provide a rapidly convergent series with essentially exact energies at low order. + + +Finally, to emphasise the improvement that can be gained by using either Pad\'e, diagonal quadratic, +or pole-free quadratic approximants, we collect the energy and error obtained using only the first 10 terms of the UMP +Taylor series in Table~\ref{tab:UMP_order10}. +The accuracy of these approximants reinforces how our understanding of the MP +energy surface in the complex plane can be leveraged to significantly improve estimates of the exact +energy using low-order perturbation expansions. + +\begin{table}[h] + \caption{ + Estimate and associated error of the exact UMP energy of the Hubbard dimer at $U/t = 7$ for + various approximants using up to ten terms in the Taylor expansion. + \label{tab:UMP_order10}} + \begin{ruledtabular} + \begin{tabular}{lccc} + \mc{2}{c}{Method} & $E_{-}(\lambda = 1)$ & \% Abs.\ Error \\ + \hline + Taylor & 10 & $-0.33338$ & $37.150$ \\ + Pad\'e & [5/5] & $-0.35513$ & $33.140$ \\ + Quadratic (diagonal) & [3/3,3] & $-0.53103$ & $\hphantom{0}0.019$ \\ + Quadratic (pole-free)& [3/0,6] & $-0.53113$ & $\hphantom{0}0.005$ \\ + \hline + Exact & & $-0.53113$ & \\ + \end{tabular} + \end{ruledtabular} +\end{table} + +%==========================================% +\subsection{Shanks Transformation} +\label{sec:Shanks} +%==========================================% + +While the Pad\'e and quadratic approximants can yield a convergent series representation +in cases where the standard MP series diverges, there is no guarantee that the rate of convergence +will be fast enough for low-order approximations to be useful. +However, these low-order partial sums or approximants often contain a remarkable amount of information +that can be used to extract further information about the exact result. +The Shanks transformation presents one approach for extracting this information +and accelerating the rate of convergence of a sequence.\cite{Shanks_1955,BenderBook} + +Consider the partial sums +$S_n = \sum_{k=0}^{n} s_k$ +defined from the truncated summation of an infinite series +$S = \sum_{k=0}^{\infty} s_k$. +If the series converges, then the partial sums will tend to the exact result +\begin{equation} + \lim_{n \to \infty} S_n = S. +\end{equation} +The Shanks transformation attempts to generate increasingly accurate estimates of this +limit by defining a new series as +\begin{equation} + T(S_n) = \frac{S_{n+1} S_{n-1} - S_{n}^2}{S_{n+1} - 2 S_{n} + S_{n-1}}. +\end{equation} +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. +However, it is only designed to accelerate converging partial sums with +the approximate form $S_n \approx S + \alpha\,\beta^n$. +Furthermore, while this transformation can accelerate the convergence of a series, +there is no guarantee that this acceleration will be fast enough to significantly +improve the accuracy of low-order approximations. + +To the best of our knowledge, the Shanks transformation has never previously been applied +to accelerate the convergence of the MP series. +We have therefore applied it to the convergent Taylor series, Pad\'e approximants, and quadratic +approximants for RMP and UMP in the symmetric Hubbard dimer. +The UMP approximants converge too slowly for the Shanks transformation +to provide any improvement, even in the case where the quadratic approximants are already +very accurate. +In contrast, acceleration of the diagonal Pad\'e approximants for the RMP cases +can significantly improve the estimate of the energy using low-order perturbation terms, +as shown in Table~\ref{tab:RMP_shank}. +Even though the RMP series diverges at $U/t = 4.5$, the combination +of diagonal Pad\'e approximants with the Shanks transformation reduces the absolute error in +the best energy estimate to 0.002\,\% using only the first 10 terms in the Taylor series. +This remarkable result indicates just how much information is contained in the first few +terms of a perturbation series, even if it diverges. + +\begin{table}[th] + \caption{ + Acceleration of the diagonal Pad\'e approximant sequence for the RMP energy + of the Hubbard dimer at $U/t = 3.5$ and $4.5$ using the Shanks transformation. + \label{tab:RMP_shank}} + \begin{ruledtabular} + \begin{tabular}{lcccc} + & & & \mc{2}{c}{$E_{-}(\lambda = 1)$} \\ + \cline{4-5} + Method & Degree & Series Term & $U/t = 3.5$ & $U/t = 4.5$ \\ + \hline + Pad\'e & [1/1] & $S_1$ & $-1.61111$ & $-2.64286$ \\ + & [2/2] & $S_2$ & $-0.82124$ & $-0.48446$ \\ + & [3/3] & $S_3$ & $-0.91995$ & $-0.81929$ \\ + & [4/4] & $S_4$ & $-0.90579$ & $-0.74866$ \\ + & [5/5] & $S_5$ & $-0.90778$ & $-0.76277$ \\ + \hline + Shanks & & $T(S_2)$ & $-0.90898$ & $-0.77432$ \\ + & & $T(S_3)$ & $-0.90757$ & $-0.76096$ \\ + & & $T(S_4)$ & $-0.90753$ & $-0.76042$ \\ + \hline + Exact & & & $-0.90754$ & $-0.76040$ \\ + \end{tabular} + \end{ruledtabular} +\end{table} + +%==========================================% +\subsection{Analytic Continuation} +%==========================================% + +Recently, Mih\'alka \etal\ have studied the effect of different partitionings, such as MP or EN theory, on the position of +branch points and the convergence properties of Rayleigh--Schr\"odinger perturbation theory\cite{Mihalka_2017b} (see also +Ref.~\onlinecite{Surjan_2000}). +Taking the equilibrium and stretched water structures as an example, they estimated the radius of convergence using quadratic +Pad\'e approximants. +The EN partitioning provided worse convergence properties than the MP partitioning, which is believed to be +because the EN denominators are generally smaller than the MP denominators. +To remedy the situation, they showed that introducing a suitably chosen level shift parameter can turn a +divergent series into a convergent one by increasing the magnitude of these denominators.\cite{Mihalka_2017b} +However, like the UMP series in stretched \ce{H2},\cite{Lepetit_1988} +the cost of larger denominators is an overall slower rate of convergence. + +\begin{figure} + \includegraphics[width=\linewidth]{fig12} + \caption{% + Comparison of the scaled RMP10 Taylor expansion with the exact RMP energy as a function + of $\lambda$ for the Hubbard dimer at $U/t = 4.5$. + The two functions correspond closely within the radius of convergence. + } + \label{fig:rmp_anal_cont} +\end{figure} + +In a later study by the same group, they used analytic continuation techniques +to resum a divergent MP series such as a stretched water molecule.\cite{Mihalka_2017a} +Any MP series truncated at a given order $n$ can be used to define the scaled function +\begin{equation} + E_{\text{MP}n}(\lambda) = \sum_{k=0}^{n} \lambda^{k} E_\text{MP}^{(k)}. +\end{equation} +Reliable estimates of the energy can be obtained for values of $\lambda$ where the MP series is rapidly +convergent (\ie, for $\abs{\lambda} < \rc$), as shown in Fig.~\ref{fig:rmp_anal_cont} for the RMP10 series +of the symmetric Hubbard dimer with $U/t = 4.5$. +These values can then be analytically continued using a polynomial- or Pad\'e-based fit to obtain an +estimate of the exact energy at $\lambda = 1$. +However, choosing the functional form for the best fit remains a difficult and subtle challenge. + +This technique was first generalised using complex scaling parameters to construct an analytic +continuation by solving the Laplace equations.\cite{Surjan_2018} +It was then further improved by introducing Cauchy's integral formula\cite{Mihalka_2019} +\begin{equation} + \label{eq:Cauchy} + E(\lambda) = \frac{1}{2\pi \i} \oint_{\mathcal{C}} \frac{E(\lambda')}{\lambda' - \lambda}, +\end{equation} +which states that the value of the energy can be computed at $\lambda$ inside the complex +contour $\mathcal{C}$ using only the values along the same contour. +Starting from a set of points in a ``trusted'' region where the MP series is convergent, their approach +self-consistently refines estimates of the $E(\lambda')$ values on a contour that includes the physical point +$\lambda = 1$. +The shape of this contour is arbitrary, but there must be no branch points or other singularities inside +the contour. +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. +The authors illustrate this protocol for the dissociation curve of \ce{LiH} and the stretched water +molecule and obtained encouragingly accurate results.\cite{Mihalka_2019} + +%%%%%%%%%%%%%%%%%%%% +\section{Concluding Remarks} +\label{sec:ccl} +%%%%%%%%%%%%%%%%%%%% + +% INTRO TO CONC. +To accurately model chemical systems, one must choose a computational protocol from an ever growing +collection of theoretical methods. +Until the Schr\"odinger equation is solved exactly, this choice must make a compromise on the accuracy +of certain properties depending on the system that is being studied. +It is therefore essential that we understand the strengths and weaknesses of different methods, +and why one might fail in cases where others work beautifully. +In this review, we have seen that the success and failure of perturbation-based methods are +directly connected to the position of exceptional point singularities in the complex plane. + +% HISTORICAL OVERVIEW +We began by presenting the fundamental concepts behind non-Hermitian extensions of quantum chemistry into the complex plane, +including the Hartree--Fock approximation and Rayleigh--Schr\"odinger perturbation theory. +We then provided a comprehensive review of the various research that has been performed +around the physics of complex singularities in perturbation theory, with a particular focus on M{\o}ller--Plesset theory. +Seminal contributions from various research groups have revealed highly oscillatory, +slowly convergent, or catastrophically divergent behaviour of the restricted and/or unrestricted MP perturbation series.% +\cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988} +In particular, the spin-symmetry-broken unrestricted MP series is notorious +for giving incredibly slow convergence.\cite{Gill_1986,Nobes_1987,Gill_1988a,Gill_1988} +All these behaviours can be rationalised and explained by the position of exceptional points +and other singularities that arise when perturbation theory is extended across the complex plane. + +% CLASSIFICATIONS +The classifications of different convergence types developed by Cremer and He,\cite{Cremer_1996} +Olsen \etal,\cite{Christiansen_1996,Olsen_1996,Olsen_2000,Olsen_2019} +or Sergeev and Goodson\cite{Goodson_2000a,Goodson_2000b,Goodson_2004,Sergeev_2005,Sergeev_2006} are particularly +worth highlighting. +In Cremer and He's original classification, ``class A'' systems exhibit monotonic convergence and generally +correspond to weakly correlated electron pairs, while ``class B'' systems show erratic convergence after initial +oscillations and generally contain spatially dense electron clusters.\cite{Cremer_1996} +Further insights were provided by Olsen and coworkers +who employed a two-state model to understand the various convergence behaviours of Hermitian and non-Hermitian +perturbation series.\cite{Christiansen_1996,Olsen_1996,Olsen_2000,Olsen_2019} +The careful analysis from Sergeev and Goodson later refined these classes depending on the position of the +singularity closest to the origin, giving $\alpha$ singularities which have large imaginary component, +and $\beta$ singularities which have a very small imaginary component.% +\cite{Goodson_2000a,Goodson_2000b,Goodson_2004,Sergeev_2005,Sergeev_2006} +Remarkably, the position of $\beta$ singularities close to the real axis can be justified as a critical +point where one (or more) electron is ionised from the molecule, creating a quantum phase transition.\cite{Stillinger_2000} +We have shown that the slow convergence of symmetry-broken MP approximations can also be driven by a $\beta$ +singularity and is closely related to these quantum phase transitions. + +% RESUMMATION +We have also discussed several resummation techniques that can be used to improve energy estimates +for both convergent and divergent series, including Pad\'e and quadratic approximants. +Furthermore, we have provided the first illustration of how the Shanks transformation can accelerate +convergence of MP approximants to improve the accuracy of low-order approximations. +Using these resummation and acceleration methods to turn low-order truncated MP series into convergent and +systematically improvable series can dramatically improve the accuracy and applicability of these perturbative methods. +However, the application of these approaches requires the evaluation of higher-order MP coefficients +(\eg, MP3, MP4, MP5, etc) that are generally expensive to compute in practice. +There is therefore a strong demand for computationally efficient approaches to evaluate general terms in the MP +series, and the development of stochastic,\cite{Thom_2007,Neuhauser_2012,Willow_2012,Takeshita_2017,Li_2019} +or linear-scaling approximations\cite{Rauhut_1998,Schutz_1999} +may prove fruitful avenues in this direction. + +% ORBITAL OPTIMISATION EXCITED STATES +The present review has only considered the convergence of the MP series using the RHF or UHF +reference orbitals. +However, numerous recent studies have shown that the use of orbitals optimised in the presence of the MP2 +correction\cite{Bozkaya_2011,Neese_2009,Lee_2018} or Kohn--Sham density-functional theory (DFT) orbitals +can significantly improve the accuracy of the MP3 correction,\cite{Bertels_2019,Rettig_2020} +particularly in the presence of symmetry-breaking. +Beyond intuitive heuristics, it is not clear why these alternative orbitals provide such accurate results, +and a detailed investigation of their MP energy function in the complex plane is therefore bound to provide +fascinating insights. +Furthermore, the convergence properties of the excited-state MP series using orbital-optimised higher energy +HF solutions\cite{Gilbert_2008,Barca_2014,Barca_2018a,Barca_2018b} remains entirely unexplored.\cite{Lee_2019,CarterFenk_2020} + +% HUBBARD +Finally, the physical concepts and mathematical tools presented in this manuscript have been illustrated +on the symmetric (or asymmetric in one occasion) Hubbard dimer at half-filling. +Although extremely simple, these illustrations highlight the incredible versatility of the Hubbard model +for understanding the subtle features of perturbation theory in the complex plane, alongisde other examples +such as Kohn-Sham DFT, \cite{Carrascal_2015,Cohen_2016} linear-response theory,\cite{Carrascal_2018} +many-body perturbation theory,\cite{Romaniello_2009,Romaniello_2012,DiSabatino_2015,Hirata_2015,Tarantino_2017,Olevano_2019} +ensemble DFT, \cite{Deur_2017,Deur_2018,Senjean_2018,Sagredo_2018,Fromager_2020} thermal DFT,\cite{Smith_2016,Smith_2018} +wave function methods,\cite{Stein_2014,Henderson_2015,Shepherd_2016} and many more. +In particular, we have shown that the Hubbard dimer contains sufficient flexibility to describe +the effects of symmetry breaking, the MP critical point, and resummation techniques, in contrast to the more +minimalistic models considered previously. +We therefore propose that the Hubbard dimer provides the ideal arena for further developing our fundamental understanding +and applications of perturbation theory. + +% DIRECTIONS +Perturbation theory isn't usually considered in the complex plane. +But when it is, a lot can be learnt about the performance of perturbation theory on the real axis. +These insights can allow incredibly accurate results to be obtained using only the lowest-order terms in a perturbation series. +Yet perturbation theory represents only one method for approximating the exact energy, and few other methods +have been considered through similar complex non-Hermitian extensions. +There is therefore much still to be discovered about the existence and consequences of exceptional points +throughout electronic structure theory. + +%%%%%%%%%%%%%%%%%%%%%%%% +\begin{acknowledgements} +%%%%%%%%%%%%%%%%%%%%%%%% +This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481). +HGAB gratefully acknowledges New College, Oxford for funding through the Astor Junior Research Fellowship. +%%%%%%%%%%%%%%%%%%%%%% +\end{acknowledgements} +%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%% +\bibliography{EPAWTFT} +%%%%%%%%%%%%%%%%%%%%%% + +\end{document} diff --git a/arXiv/fig10.pdf b/arXiv/fig10.pdf new file mode 100644 index 0000000..da8c123 Binary files /dev/null and b/arXiv/fig10.pdf differ diff --git a/arXiv/fig11a.pdf b/arXiv/fig11a.pdf new file mode 100644 index 0000000..cfa889a Binary files /dev/null and b/arXiv/fig11a.pdf differ diff --git a/arXiv/fig11b.pdf b/arXiv/fig11b.pdf new file mode 100644 index 0000000..e8f55f6 Binary files /dev/null and b/arXiv/fig11b.pdf 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