\documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1} \usepackage{subcaption} \usepackage{bm,graphicx,tabularx,array,booktabs,dcolumn,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,siunitx,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{orange}{\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} \newcommand{\updot}{% \mathrel{\ooalign{\hfil$\vcenter{ \hbox{$\scriptscriptstyle\bullet$}}$\hfil\cr$\uparrow$\cr} }% } \newcommand{\dwdot}{% \mathrel{\ooalign{\hfil$\vcenter{ \hbox{$\scriptscriptstyle\bullet$}}$\hfil\cr$\downarrow$\cr} }% } \newcommand{\vac}{% \mathrel{\ooalign{\hfil$\vcenter{ \hbox{$\scriptscriptstyle\bullet$}}$\hfil\cr$ $\cr} }% } \newcommand{\uddot}{% \mathrel{\ooalign{\hfil$\vcenter{ \hbox{$\scriptscriptstyle\bullet$}}$\hfil\cr$\uparrow\downarrow$\cr} }% } % Center tabularx columns \newcolumntype{Y}{>{\centering\arraybackslash}X} % HF rotation angles \newcommand{\ta}{\theta_{\alpha}} \newcommand{\tb}{\theta_{\beta}} % 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{\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}{v_{\text{HF}}} \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France.} \newcommand{\UCAM}{Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, U.K.} \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{hugh.burton@chem.ox.ac.uk} \affiliation{\UOX} \author{Pierre-Fran\c{c}ois \surname{Loos}} \email{loos@irsamc.ups-tlse.fr} \affiliation{\LCPQ} \begin{abstract} In this review, we explore the 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 energy singularities in the complex plane, known as exceptionnal points. After a presentation of the fundamental notions of quantum chemistry in the complex plane, such as the mean-field Hartree--Fock approximation and Rayleigh-Schr\"odinger perturbation theory, and their illustration with the ubiquitous (symmetric) Hubbard dimer at half filling, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight the seminal work of several research groups on the convergence behaviour of perturbative series obtained within M{\o}ller--Plesset perturbation theory and its apparent link with quantum phase transitions. \end{abstract} \maketitle \raggedbottom \tableofcontents %%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \label{sec:intro} %%%%%%%%%%%%%%%%%%%%%%% Due to the ubiquitous influence of processes involving electronic states in physics, chemistry, and biology, their faithful description from first principles has been one of the grand challenges faced by theoretical chemists since the dawn of computational chemistry. Accurately predicting ground- and excited-state energies (hence excitation energies) is particularly valuable in this context, and it has concentrated most of the efforts within the community. An armada of theoretical and computational methods have been developed to this end, each of them being plagued by its own flaws. The fact that none of these methods is successful in every chemical scenario has encouraged chemists to carry on the development of new methodologies, their main goal being to get the most accurate energies (and properties) at the lowest possible computational cost in the most general context. One common feature of all these methods is that they rely on the notion of quantised energy levels of Hermitian quantum mechanics, in which the different electronic states of a molecule or an atom are energetically ordered, the lowest being the ground state while the higher ones are excited states. Within this quantised paradigm, electronic states look completely disconnected from one another. Many current methods study excited states using only ground-state information, creating a ground-state bias that leads to incorrect excitation energies. However, one can gain a different perspective on quantisation extending quantum chemistry into the complex domain. In a non-Hermitian complex picture, the energy levels are \textit{sheets} of a more complicated topological manifold called \textit{Riemann surface}, and they are smooth and continuous \textit{analytic continuation} of one another. In other words, our view of the quantised nature of conventional Hermitian quantum mechanics arises only from our limited perception of the more complex and profound structure of its non-Hermitian variant. \cite{MoiseyevBook,BenderPTBook} The realisation that ground and excited states both emerge from one single mathematical structure with equal importance suggests that excited-state energies can be computed from first principles in their own right. One could then exploit the structure of these Riemann surfaces to develop methods that directly target excited-state energies without needing ground-state information. \cite{Burton_2019,Burton_2019a} By analytically continuing the electronic energy $E(\lambda)$ in the complex domain (where $\lambda$ is a coupling parameter), the ground and excited states of a molecule can be smoothly connected. This connection is possible because by extending real numbers to the complex domain, the ordering property of real numbers is lost. Hence, electronic states can be interchanged away from the real axis since the concept of ground and excited states has been lost. Amazingly, this smooth and continuous transition from one state to another has recently been experimentally realised in physical settings such as 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} Exceptional points (EPs) are 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} They are the non-Hermitian analogs of conical intersections, \cite{Yarkony_1996} which are ubiquitous in non-adiabatic processes and play a key role in photochemical mechanisms. In the case of auto-ionising resonances, EPs have a role in deactivation processes similar to conical intersections in the decay of bound excited states. \cite{Benda_2018} Although Hermitian and non-Hermitian Hamiltonians are closely related, the behaviour of their eigenvalues near degeneracies is starkly different. For example, encircling non-Hermitian degeneracies at EPs leads to an interconversion of states, and two loops around the EP are necessary to recover the initial energy. \cite{MoiseyevBook,Heiss_2016,Benda_2018} Additionally, the wave function picks up a geometric phase (also known as Berry phase \cite{Berry_1984}) and four loops are required to recover the initial wave function. In contrast, encircling Hermitian degeneracies at conical intersections only introduces a geometric phase while leaving the states unchanged. More dramatically, whilst eigenvectors remain orthogonal at conical intersections, at non-Hermitian EPs the eigenvectors themselves become equivalent, resulting in a \textit{self-orthogonal} state. \cite{MoiseyevBook} More importantly here, although EPs usually lie off the real axis, these singular points are intimately related to the convergence properties of perturbative methods and avoided crossing on the real axis are indicative of singularities in the complex plane. \cite{BenderBook,Olsen_1996,Olsen_2000,Olsen_2019,Mihalka_2017a,Mihalka_2017b,Mihalka_2019} \titou{The use of non-Hermitian Hamiltonians in quantum chemistry has a long history; these Hamiltonians have been used extensively as a method for describing metastable resonance phenomena. \cite{MoiseyevBook} Through a complex-scaling of the electronic or atomic coordinates,\cite{Moiseyev_1998} or by introducing a complex absorbing potential,\cite{Riss_1993,Ernzerhof_2006,Benda_2018} outgoing resonance states are transformed into square-integrable wave functions that allow the energy and lifetime of the resonance to be computed. We refer the interested reader to the excellent book of Moiseyev for a general overview. \cite{MoiseyevBook}} \titou{Discussion around the different types of singularities in complex analysis. At a singular point, a function and/or its derivatives becomes infinite or undefined (hence non analytic). One very common type of singularities (belonging to the family of isolated singularities) are poles where the function behaves $1/(\lambda - \lambda_c)^n$ where $n \in \mathbb{N}^*$ is the order of the pole. Another class of singularities are branch points resulting from a multi-valued function such as a square root or a logarithm function and usually implying the presence of so-called branch cuts which are lines or curves where the function ``jumps'' from one value to another. Critical points are singularities which lie on the real axis and where the nature of the function undergoes a sudden transition. However, these do not clearly belong to a given class of singularities and they cannot be rigorously classified as they have more complicated functional forms. } \titou{T2: I THINK THAT IN GENERAL THE AXE LABELS ARE TOO SMALL.} %%%%%%%%%%%%%%%%%%%%%%% \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]{HF_cplx_angle} \subcaption{\label{subfig:UHF_cplx_angle}} \end{subfigure} \begin{subfigure}{0.49\textwidth} \includegraphics[height=0.65\textwidth]{HF_cplx_energy} \subcaption{\label{subfig:UHF_cplx_energy}} \end{subfigure} \caption{% (\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$ \titou{in the Hubbard dimer for $U/t = ??$}. 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 pair, 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} %OR {Complex adiabatic connection} %============================================================% % 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\, \Hat{v}_\text{HF}(\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 solutions 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} v^{\text{HF}}(\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 $\lambda$. \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 \trash{electron correlation} \titou{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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[htb] \includegraphics[width=\linewidth]{RadConv} \caption{ Radius of convergence $r_c$ for the RMP ground state (red), the UMP ground state (blue), and the UMP excited state (orange) series as functions of the ratio $U/t$. \label{fig:RadConv}} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%% The RMP series is convergent for $U = 3.5\,t$ with $\rc > 1$, as illustrated for the individual terms at each order of perturbation 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, the $\lep$ 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. %%% FIG 3 %%% \begin{figure*} \begin{subfigure}{0.32\textwidth} \includegraphics[height=0.75\textwidth]{fig3a} \subcaption{\label{subfig:UMP_3} $U/t = 3$} \end{subfigure} % \begin{subfigure}{0.32\textwidth} \includegraphics[height=0.75\textwidth]{fig3b} \subcaption{\label{subfig:UMP_cvg}} \end{subfigure} % \begin{subfigure}{0.32\textwidth} \includegraphics[height=0.75\textwidth]{fig3c} \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 \trash{molecular} \titou{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 state and the doubly-excited state (Fig.~\ref{fig:RMP}), the UMP energy has two 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} % Behaviour} % Further insights from a two-state model} %==========================================% % 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 with respect to the FCI energy.\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 providing orbital relaxation, 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} \Delta E_{\text{A}} &= \Emp^{(2)} + \Emp^{(3)} + \Emp^{(4)} + \frac{\Emp^{(5)}}{1 - (\Emp^{(6)} / \Emp^{(5)})}, \\[5pt] \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} are not mathematically motivated when considering the complex singularities causing the divergence, and therefore cannot be applied for all systems. For example, \ce{HF} 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 these 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 critical point 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)v^{\text{HF}}(\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 $\theta_{\alpha}^{\text{RHF}} = \theta_{\beta}^{\text{RHF}} = 0$, 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]{ump_cp} \subcaption{\label{subfig:ump_cp}} \end{subfigure} % \begin{subfigure}{0.32\textwidth} \includegraphics[height=0.75\textwidth]{ump_cp_surf} \subcaption{\label{subfig:ump_cp_surf}} \end{subfigure} % \begin{subfigure}{0.32\textwidth} \includegraphics[height=0.75\textwidth]{ump_ep_to_cp} \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 \titou{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 $\lambda > 1/2$ (see Fig.~\ref{subfig:ump_cp}). % 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 as $\lambda$ increases. 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}). %%==================================================== %\subsection{The physics of quantum phase transitions} %%==================================================== % %In the previous section, we saw that a careful analysis of the structure of the Hamiltonian allows us to predict the existence of a critical point. %In a finite basis set, this critical point is model by a cluster of $\beta$ singularities. %It is now well known that this phenomenon is a special case of a more general phenomenon. %Indeed, theoretical physicists proved that EPs close to the real axis are connected to \textit{quantum phase transitions} (QPTs). \cite{Heiss_1988,Heiss_2002,Borisov_2015,Sindelka_2017,CarrBook,Vojta_2003,SachdevBook,GilmoreBook} %In quantum mechanics, the Hamiltonian is almost always dependent of, at least, one parameter. %In some cases the variation of a parameter can lead to abrupt changes at a critical point. %These QPTs exist both for ground and excited states as shown by Cejnar and coworkers. \cite{Cejnar_2005,Cejnar_2007,Caprio_2008,Cejnar_2009,Sachdev_2011,Cejnar_2015,Cejnar_2016, Macek_2019,Cejnar_2020} %A ground-state QPT is characterised by the derivatives of the ground-state energy with respect to a non-thermal control parameter. \cite{Cejnar_2009, Sachdev_2011} %The transition is called discontinuous and of first order if the first derivative is discontinuous at the critical parameter value. %Otherwise, it is called continuous and of $m$th order (with $m \ge 2$) if the $m$th derivative is discontinuous. %A QPT can also be identify by the discontinuity of an appropriate order parameter (or one of its derivatives). % %The presence of an EP close to the real axis is characteristic of a sharp avoided crossing. %Yet, at such an avoided crossing, eigenstates change abruptly. %Although it is now well understood that EPs are closely related to QPTs, the link between the type of QPT (ground state or excited state, first or higher order) and EPs still need to be clarified. %One of the major obstacles that one faces in order to achieve this resides in the ability to compute the distribution of EPs. %The numerical assignment of an EP to two energies on the real axis is very difficult in large dimensions. %Hence, the design of specific methods are required to get information on the location of EPs. %Following this idea, Cejnar \textit{et al.}~developed a method based on a Coulomb analogy giving access to the density of EP close to the real axis. \cite{Cejnar_2005, Cejnar_2007} %More recently Stransky and coworkers proved that the distribution of EPs is characteristic of the QPT order. \cite{Stransky_2018} %In the thermodynamic limit, some of the EPs converge towards a critical point $\lambda_\text{c}$ on the real axis. %They showed that, within the interacting boson model, \cite{Lipkin_1965} EPs associated to first- and second-order QPT behave differently when the number of particles increases. %The position of these singularities converge towards the critical point on the real axis at different rates (exponentially and algebraically for the first and second orders, respectively) with respect to the number of particles. % %Moreover, Cejnar \textit{et al.}~studied the so-called shape-phase transitions of the interaction boson model from a QPT point of view. \cite{Cejnar_2000, Cejnar_2003, Cejnar_2007a, Cejnar_2009} %The phase of the ensemble of $s$ and $d$ bosons is characterised by a dynamical symmetry. %When a parameter is continuously modified the dynamical symmetry of the system can change at a critical value of this parameter, leading to a deformed phase. %They showed that at this critical value of the parameter, the system undergoes a QPT. %For example, without interaction the ground state is the spherical phase (a condensate of $s$ bosons) and when the interaction increases it leads to a deformed phase constituted of a mixture of $s$ and $d$ bosons states. %In particular, we see that the transition from the spherical phase to the axially symmetric one is analog to the symmetry breaking of the wave function of the hydrogen molecule when the bond is stretched. \cite{SzaboBook} %It seems like our understanding of the physics of spatial and/or spin symmetry breaking in HF theory can be enlightened by QPT theory. %Indeed, the second derivative of the HF ground-state energy is discontinuous at the point of spin symmetry-breaking which means that the system undergo a second-order QPT. % %Moreover, the $\beta$ singularities introduced by Sergeev and coworkers to describe the EPs modelling the formation of a bound cluster of electrons are actually a more general class of singularities. %The EPs close to the real axis (the so-called $\beta$ singularities) are connected to QPT because they result from a sharp avoided crossings at which the eigenstates change quickly. %However, the $\alpha$ singularities arise from large avoided crossings. %Thus, they cannot be connected to QPT. %The avoided crossings generating $\alpha$ singularities generally involve the ground state and low-lying doubly-excited states. %Those excited states have a non-negligible contribution to the exact FCI solution because they have (usually) the same spatial and spin symmetry as the ground state. %We believe that $\alpha$ singularities are connected to states with non-negligible contribution in the CI expansion thus to the dynamical part of the correlation energy, while $\beta$ singularities are linked to symmetry breaking and phase transitions of the wave function, \ie, to the multi-reference nature of the wave function thus to the static part of the correlation energy. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Resummation Methods} \label{sec:Resummation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% As frequently claimed by Carl Bender, \textit{``the most stupid thing that one can do with a series is to sum it.''} Nonetheless, quantum chemists are basically doing exactly this on a daily basis. Here, we discuss tools that can be used to sum divergent series. Resummation techniques is a vast field of research and, below, we provide details for a non-exhaustive list of these techniques. We refer the interested reader to more specialised reviews for additional information. \cite{Goodson_2011,Goodson_2019} %==========================================% \subsection{Pad\'e approximant} %==========================================% The inability of Taylor series to model properly the energy function $E(\lambda$) can be simply understood by the fact that one aims at modelling a complicated function with potentially poles and singularities by a simple polynomial of finite order. A truncated Taylor series just does not have enough flexibility to do the job properly. Nonetheless, the description of complex energy functions can be significantly improved thanks to Pad\'e approximant, \cite{Pade_1892} and related techniques. \cite{BakerBook,BenderBook} According to Wikipedia, \textit{``a Pad\'e approximant is 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}{\sum_{k=0}^{d_B} b_k \lambda^k} \end{equation} (with $b_0 = 1$), where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting terms according to 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 appears at the locations of 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 a typical perturbative treatment) and more complicated functional forms appearing at critical points (where the nature of the solution undergoes a sudden transition) for example. Figure \ref{fig:PadeRMP} illustrates the improvement brought by diagonal (\ie, $d_A = d_B$) Pad\'e approximants as 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 at $U/t = 3.5$ when one increases the truncation degree, the Pad\'e approximants yield much more accurate results with, additionally, a rather good estimate of the radius of convergence of the RMP series. For $U/t = 4.5$, the struggles of the truncated Taylor expansions are magnified and the Pad\'e approximants still provide quite accurate energies even outside the radius of convergence of the RMP series. \begin{table} \caption{RMP ground-state energy estimate at $\lambda = 1$ provided by various truncated Taylor series and Pad\'e approximants at $U/t = 3.5$ and $4.5$. We also report the estimate of the radius of convergence $r_c$ provided by the diagonal Pad\'e approximants. \label{tab:PadeRMP}} \begin{ruledtabular} \begin{tabular}{lccccc} & & \mc{2}{c}{$r_c$} & \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 & & & $-0.92518$ & $-0.86858$ \\ 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} %%%%%%%%%%%%%%%%% \begin{figure*} \includegraphics[height=0.23\textheight]{PadeRMP35} \includegraphics[height=0.23\textheight]{PadeRMP45} \caption{\label{fig:PadeRMP} RMP ground-state energy as a function of $\lambda$ obtained using various resummation techniques at $U/t = 3.5$ (left) and $U/t = 4.5$ (right).} \end{figure*} %%%%%%%%%%%%%%%%% %==========================================% \subsection{Quadratic approximant} %==========================================% In a nutshell, the idea behind quadratic approximant is 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(\lambda) = \frac{1}{2 Q(\lambda)} \qty[ P(\lambda) \pm \sqrt{P^2(\lambda) - 4 Q(\lambda) R(\lambda)} ] \end{equation} where \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} are polynomials, 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 $P^2(\lambda) - 4 Q(\lambda) R(\lambda)$. 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. Note that, by construction, a quadratic approximant has only two branches which hampers the faithful description of more complicated singularity structures. As shown in Ref.~\onlinecite{Goodson_2000a}, 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} For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximant are quite poor approximation, but its $[1/0,1]$ version already model perfectly the RMP energy function by predicting a single pair of EPs at $\lambda_\text{EP} = \pm i 4t/U$. This is expected knowing the form of the RMP energy [see Eq.~\eqref{eq:E0MP}] which perfectly suits the purpose of quadratic approximants. We can anticipate that the singularity structure of the UMP energy function is going to be much more challenging to model properly, and this is indeed the case as the UMP energy function contains three branches (see Figs.~\ref{subfig:UMP_3} and \ref{subfig:UMP_7}). However, by ramping up high enough the degree of the polynomials, one is able to get both, as shown in Fig.~\ref{fig:QuadUMP} and Table \ref{tab:QuadUMP}, accurate estimates of the radius of convergence of the UMP series and of the ground-state energy at $\lambda = 1$, even in cases where the convergence of the UMP series is painfully slow (see Fig.~\ref{subfig:UMP_cvg}). Figure \ref{fig:QuadUMP} evidences that the Pad\'e approximants are trying to model the square root singularity by placing a pole on the real axis (for [3/3]) or just off the real axis (for [4/4]). Thanks to greater flexibility, the quadratic approximants are able to model nicely the avoided crossing and the location of the singularities. Besides, they provide accurate estimates of the ground-state energy at $\lambda = 1$ (see Table \ref{tab:QuadUMP}). %%%%%%%%%%%%%%%%% \begin{figure} \includegraphics[width=\linewidth]{QuadUMP} \caption{\label{fig:QuadUMP} UMP energies as a function of $\lambda$ obtained using various resummation techniques at $U/t = 3$.} \end{figure} %%%%%%%%%%%%%%%%% \begin{table} \caption{Estimate of the radius of convergence $r_c$ of the UMP energy function provided by various resummation techniques 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}{$r_c$} & \mc{2}{c}{$E_{-}(\lambda)$} \\ \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 Pad\'e & [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 & & $1.122$ & $1.004$ & $-0.97254$ & $-0.35513$ \\ 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 & $1.071$ & $1.002$ & $-0.99966$ & $-0.53103$ \\ \hline Exact & & & & $1.069$ & $1.002$ & $-1.00000$ & $-0.53113$ \\ \end{tabular} \end{ruledtabular} \end{table} An interesting point raised in Ref.~\onlinecite{Goodson_2019} suggests that low-order quadratic approximants might struggle to model the correct singularity structure when the energy function has poles in both the positive and negative half-planes. In such a scenario, the quadratic approximant will have the tendency to place its branch points in-between, potentially introducing singularities quite close to the origin. A simple potential cure for this consists in applying a judicious transformation (like a bilinear conformal mapping) which does not affect the points at $\lambda = 0$ and $\lambda = 1$. \cite{Feenberg_1956} %==========================================% \subsection{Analytic continuation} %==========================================% Recently, Mih\'alka \textit{et al.} studied the partitioning effect on the convergence properties of Rayleigh-Schr\"odinger perturbation theory by considering the MP and the EN partitioning as well as an alternative partitioning \cite{Mihalka_2017a} (see also Ref.~\onlinecite{Surjan_2000}). Taking as an example (in particular) the water molecule at equilibrium and at stretched geometries, they could estimate the radius of convergence via a quadratic Pad\'e approximant and convert divergent perturbation expansions to convergent ones in some cases thanks to a judicious choice of the level shift parameter. In a subsequent study by the same group, \cite{Mihalka_2017b} they use analytic continuation techniques to resum divergent MP series \cite{Goodson_2011} taking again as an example the water molecule in a stretched geometry. In a nutshell, their idea consists in calculating the energy of the system for several values of $\lambda$ for which the MP series is rapidly convergent (\ie, for $\abs{\lambda} < r_c$), and to extrapolate the final energy to the physical system at $\lambda = 1$ via a polynomial- or Pad\'e-based fit. However, the choice of the functional form of the fit remains a subtle task. This technique was first generalised by using complex scaling parameters and applying analytic continuation by solving the Laplace equation, \cite{Surjan_2018} and then further improved thanks to Cauchy's integral formula \cite{Mihalka_2019} \begin{equation} \label{eq:Cauchy} \frac{1}{2\pi i} \oint_{\gamma} \frac{E(\lambda)}{\lambda - a} = E(a), \end{equation} which states that the value of the energy can be computed at $\lambda=a$ inside the complex contour $\gamma$ only by the knowledge of its values on the same contour. Their method consists in refining self-consistently the values of $E(\lambda)$ computed on a contour going through the physical point at $\lambda = 1$ and encloses points of the ``trusted'' region (where the MP series is convergent). The shape of this contour is arbitrary but no singularities are allowed inside the contour to ensure $E(\lambda)$ is analytic. When the values of $E(\lambda)$ on the so-called contour are converged, Cauchy's integrals formula \eqref{eq:Cauchy} is invoked to compute the values at $E(\lambda=1)$ which corresponds to the final estimate of the FCI energy. The authors illustrate this protocol on the dissociation curve of \ce{LiH} and the stretched water molecule showing encouraging results. \cite{Mihalka_2019} %%%%%%%%%%%%%%%%%%%% \section{Conclusion} %%%%%%%%%%%%%%%%%%%% In order to model accurately chemical systems, one must choose, in a ever growing zoo of methods, which computational protocol is adapted to the system of interest. This choice can be, moreover, motivated by the type of properties that one is interested in. That means that one must understand the strengths and weaknesses of each method, \ie, why one method might fail in some cases and work beautifully in others. We have seen that for methods relying on perturbation theory, their successes and failures are directly connected to the position of EPs in the complex plane. Exhaustive studies have been performed on the causes of failure of MP perturbation theory. First, it was understood that, for chemical systems for which the HF Slater determinant is a poor approximation to the exact wave function, MP perturbation theory fails too. Such systems can be, for example, molecules where the exact ground-state wave function is dominated by more than one configuration, \ie, multi-reference systems. More preoccupying cases were also reported. For instance, it has been shown that systems considered as well understood (\eg, \ce{Ne}) can exhibit divergent behaviour when the basis set is augmented with diffuse functions. Later, these erratic behaviours of the perturbation series were investigated and rationalised in terms of avoided crossings and singularities in the complex plane. It was shown that the singularities can be classified in two families. The first family includes $\alpha$ singularities resulting from a large avoided crossing between the ground state and a low-lying doubly-excited states. The $\beta$ singularities, which constitutes the second family, are artefacts generated by the incompleteness of the Hilbert space, and they are directly connected to an ionisation phenomenon occurring in the complete Hilbert space. These singularities are close to the real axis and connected with sharp avoided crossing between the ground state and a highly diffuse state. We have found that the $\beta$ singularities modelling the ionisation phenomenon described by Sergeev and Goodson are actually part of a more general class of singularities. Indeed, those singularities close to the real axis are connected to quantum phase transitions and symmetry breaking, and theoretical physics have demonstrated that the behaviour of the EPs depends of the type of transitions from which the EPs result (first or higher orders, ground state or excited state transitions). To conclude, this work shows that our understanding of the singularity structure of the energy is still incomplete but we hope that it opens new perspectives for the understanding of the physics of EPs in 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}