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.
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.
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}}
Exact energies for the Hubbard dimer ($U=4t$) as functions of $\lambda$ on the real axis (\subref{subfig:FCI_real}) and in the complex plane (\subref{subfig:FCI_cplx}).
Only the interacting closed-shell singlets are plotted in the complex plane, becoming degenerate at the EP (black dot).
To illustrate the concepts discussed throughout this article, we consider the symmetric Hubbard dimer at half filling, \ie\ with two opposite-spin fermions.
Analytically solvable models are essential in theoretical chemistry and physics as their mathematical simplicity compared to realistic systems (e.g., atoms and molecules) allows new concepts and methods to be
easily tested while retaining the key physical phenomena.
To illustrate the formation of an EP, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t\rightarrow\lambda t$ to give the parameterised Hamiltonian $\hH(\lambda)$.
While the open-shell triplet ($E_{\text{T}}$) and singlet ($E_{\text{S}}$) are independent of $\lambda$, the closed-shell singlet ground state ($E_{-}$) and doubly-excited state ($E_{+}$) couple strongly to form an avoided crossing at $\lambda=0$ (see Fig.~\ref{subfig:FCI_real}).
At non-zero values of $U$ and $t$, these closed-shell singlets can only become degenerate at a pair of complex conjugate points in the complex $\lambda$ plane
Crucially, the energy surface becomes non-analytic at $\lambda_{\text{EP}}$ and a square-root singularity forms with two branch cuts running along the imaginary axis from $\lambda_{\text{EP}}$ to $\pm\i\infty$ (see Fig.~\ref{subfig:FCI_cplx}).
On the real $\lambda$ axis, these EPs lead to the singlet avoided crossing at $\lambda=\Re(\lambda_{\text{EP}})$.
The ``shape'' of this avoided crossing is related to the magnitude of $\Im(\lambda_{\text{EP}})$, with smaller values giving a ``sharper'' interaction.
Remarkably, the existence of these square-root singularities means that following a complex contour around an EP in the complex $\lambda$ plane will interconvert the closed-shell ground and excited states (see Fig.~\ref{subfig:FCI_cplx}).
As a result, completely encircling an EP leads to the interconversion of the two interacting states, while a second complete rotation returns the two states to their original energies.
Additionally, the wave functions pick up a geometric phase in the process, and four complete loops are required to recover their starting forms.\cite{MoiseyevBook}
In the Hartree--Fock (HF) approximation, the many-electron wave function is approximated as a single Slater determinant $\Psi^{\text{HF}}(\vb{x}_1,\ldots,\vb{x}_N)$, where $\vb{x}=(\sigma,\vb{r})$ is a composite vector gathering spin and spatial coordinates.
This Slater determinant is defined as an antisymmetric combination of $\Ne$ (real-valued) occupied one-electron spin-orbitals $\phi_p(\vb{x})$, which are, by definition, eigenfunctions of the one-electron Fock operator
From hereon, $i$ and $j$ denote occupied orbitals, $a$ and $b$ denote unoccupied (or virtual) orbitals, while $p$, $q$, $r$, and $s$ denote arbitrary orbitals.
%\titou{In a recent paper, \cite{Burton_2019} using holomorphic Hartree--Fock (h-HF) \cite{Hiscock_2014,Burton_2018,Burton_2016} as an analytic continuation of conventional HF theory, we have demonstrated, on a simple model, that one can interconvert states of different symmetries and natures by following well-defined contours in the complex $\lambda$-plane, where $\lambda$ is the strength of the electron-electron interaction (see Fig.~\ref{fig:iAC}).
%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} one can ``morph'' a ground-state wave function into an excited-state wave function via a stationary path of HF solutions. \cite{Seidl_2018}
%In such a way, we could obtain a doubly-excited state wave function starting from the ground state wave function, a process which is not as easy as one might think. \cite{Gilbert_2008,Thom_2008,Shea_2018}
%One of the fundamental discovery we made was that Coulson-Fischer points (where multiple symmetry-broken solutions coalesce) play a central role and can be classified as \textit{quasi}-exceptional points, as the wave functions do not become self-orthogonal.
%The findings reported in Ref.~\onlinecite{Burton_2019} represent the very first study of non-Hermitian quantum mechanics for the exploration of multiple solutions at the HF level.
%It perfectly illustrates the deeper topology of electronic states revealed using a complex-scaled electron-electron interaction.
%Through the introduction of non-Hermiticity, we have provided a more general framework in which the complex and diverse characteristics of multiple solutions can be explored and understood.}
% HGAB: I don't think this parapgrah tells us anything we haven't discussed before
%A convenient way to investigate the convergence properties of the MP series is to analytically continue the coupling parameter $\lambda$ into the complex variable.
%By doing so, the Hamiltonian and the energy become complex-valued functions of $\lambda$,
%and the energy becomes a multivalued function on $K$ Riemann sheets (where $K$ is the number of basis functions).
%As mentioned above, by searching the singularities of the function $E(\lambda)$, one can get information on the convergence properties of the MP series.
%These singularities of the energy function are exactly the EPs connecting the electronic states as mentioned in Sec.~\ref{sec:intro}.
%The direct computation of the terms of the series is quite manageable up to fourth order in perturbation, while the fifth and sixth order in perturbation can still be obtained but at a rather high cost. \cite{JensenBook}
%In order to better understand the behaviour of the MP series and how it is connected to the singularity structure, we have to access high-order terms.
%For small systems, one can access the whole terms of the series using full configuration interaction (FCI).
%If the Hamiltonian $H(\lambda)$ is diagonalized in the FCI space, one gets the exact energies (in this finite Hilbert space) and the Taylor expansion with respect to $\lambda$ allows to access the MP perturbation series at any order.
i) the Epstein-Nesbet (EN) partitioning which consists in taking the diagonal elements of $\hH$ as the zeroth-order Hamiltonian. \cite{Nesbet_1955,Epstein_1926}
ii) the weak correlation partitioning in which the one-electron part is consider as the unperturbed Hamiltonian $\hH^{(0)}$ and the two-electron part is the perturbation operator $\hV$, and
iii) the strong coupling partitioning where the two operators are inverted compared to the weak correlation partitioning. \cite{Seidl_2018}
%When one relies on MP perturbation theory (and more generally on any perturbative partitioning), it would be reasonable to ask for a systematic improvement of the energy with respect to the perturbative order, \ie, one would expect that the more terms of the perturbative series one can compute, the closer the result from the exact energy.
%In other words, one would like a monotonic convergence of the MP series. Assuming this, the only limiting process to get the exact correlation energy (in a finite basis set) would be our ability to compute the terms of this perturbation series.
%Unfortunately this is not as easy as one might think because i) the terms of the perturbative series become rapidly computationally cumbersome, and ii) erratic behaviour of the perturbative coefficients are not uncommon.
%For example, in the late 80's, Gill and Radom reported deceptive and slow convergences in stretched systems \cite{Gill_1986,Gill_1988} (see also Refs.~\onlinecite{Handy_1985,Lepetit_1988}).
%In Ref.~\onlinecite{Gill_1986}, the authors showed that the RMP series is convergent, yet oscillatory which is far from being convenient if one is only able to compute the first few terms of the expansion.
%For example, in the case of the barrier to homolytic fission of \ce{He2^2+} in a minimal basis set, RMP5 is worse than RMP4 (see Fig.~2 in Ref.~\onlinecite{Gill_1986}).
%On the other hand, the UMP series is monotonically convergent after UMP5 but very slowly .
%Thus, one cannot practically use it for systems where only the first terms are computationally accessible.
%When a bond is stretched, in most cases the exact wave function becomes more and more of multi-reference nature.
%Yet, the HF wave function is restricted to be a single Slater determinant.
%It is then inappropriate to model (even qualitatively) stretched systems.
%Nevertheless, as explained in Sec.~\ref{sec:HF} and illustrated in Sec.~\ref{sec:HF_hubbard}, the HF wave function can undergo symmetry breaking to lower its energy by sacrificing one of the symmetry of the exact wave function in the process (see also the pedagogical example of \ce{H2} in Ref.~\onlinecite{SzaboBook}).
%One could then potentially claim that the RMP series exhibits deceptive convergence properties as the RHF Slater determinant is a poor approximation of the exact wave function for stretched system.
%However, as illustrated above, even in the unrestricted formalism which clearly represents a better description of a stretched system, the UMP series does not have the smooth and rapidly convergent behaviour that one would wish for.
%The reasons behind this ambiguous behaviour are further explained below.
%In the unrestricted framework the singlet ground state wave function is allowed to mix with triplet wave functions, leading to the so-called spin contamination issue. Gill \textit{et al.}~highlighted the link between slow convergence of the UMP series and spin contamination for \ce{H2} in a minimal basis. \cite{Gill_1988}
%Handy and coworkers reported the same behaviour of the UMP series (oscillatory and slowly monotonically convergent) in stretched \ce{H2O} and \ce{NH2}. \cite{Handy_1985} Lepetit \textit{et al.}~analysed the difference between the MP and EN partitioning for the UHF reference. \cite{Lepetit_1988}
%They concluded that the slow convergence of the UMP series is due to i) the fact that the MP denominator (see Eq.~\ref{eq:EMP2}) tends to a constant instead of vanishing, and ii) the lack of the singly-excited configurations (which only appears at fourth order) that strongly couple to the doubly-excited configurations.
%We believe that this divergent behaviour might therefore be attributed to the need for the single excitations to focus on correcting the structure of the reference orbitals rather than capturing the correlation energy.
%Cremer and He analysed 29 atomic and molecular systems at the FCI level \cite{Cremer_1996} and grouped them in two classes: i) the \textit{class A} systems where one observes a monotonic convergence \titou{of the RMP series?} to the FCI energy, and ii) the \textit{class B} systems for which convergence is erratic after initial oscillations.
%Their system set contains stretched molecules as well as molecules at their equilibrium geometry for various basis sets.
%They highlighted that \cite{Cremer_1996}
%\textit{``Class A systems are characterised by electronic structures with well-separated electron pairs while class B systems are characterized by electronic structures with electron clustering in one or more regions.''}
%Moreover, they analysed the contribution of the triple (T) excitations to the MP4, MP5 and MP6 energies next to the single, double and quadruple (SDQ) excitations contribution.
%They showed that class A systems have very little contribution from the triple excitations and that most of the correlation energy is due to pair correlation.
%On the other hand, class B systems have an important contribution from the triple excitations which alternates in sign resulting in an oscillation of the total correlation energy.
%This observation on the contribution to the MP$n$ energy corroborates the electronic structure discussed above.
%As one can only compute the first terms of the MP series, a smart way of getting more accurate results is to use extrapolation formula, \ie, estimating the limit of the series with only few terms.
%Cremer and He proved that using specific extrapolation formulas of the MP series for class A and class B systems improves the precision of the results compared to the formula used without resorting to classes. \cite{Cremer_1996}
%The mean absolute deviation taking the FCI correlation energies as reference is $0.3$ millihartree with the class-specific formula whereas the deviation increases to 12 millihartree using the general formula.
%Even if there were still shaded areas in their analysis and that their classification was incomplete, the work of Ref.~\onlinecite{Cremer_1996} clearly evidenced that understanding the origin of the different modes of convergence could potentially lead to a more rationalised use of MP perturbation theory and, hence, to more accurate correlation energy estimates.
Convergence of the RMP series as a function of the perturbation order $n$ for the Hubbard dimer at $U/t =3.5$ (where $r_c > 1$) 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 $\lambda$.
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 $\lambda$.
%The convergence of the UMP as a function of the ratio $U/t$ is shown in Fig.~\ref{subfig:UMP_cvg} for two specific values: the first ($U = 3t$) is well within the RMP convergence region, while the second ($U = 7t$) falls outside.
%Note that in the case of UMP, there are now two pairs of EPs as the open-shell singlet now couples strongly with both the ground and doubly-excited states.
%This has the clear tendency to move away from the origin the EP dictating the convergence of the ground-state energy, while deteriorating the convergence properties of the excited-state energy.
%For $U = 3t$ (see Fig.~\ref{subfig:UMP_3}), the ground-state energy is remarkably flat since the UHF energy is already a pretty good estimate of the exact energy thanks to the symmetry-breaking process.
%Most of the UMP expansion is actually correcting the spin-contamination in the wave function.
%For $U = 7t$ (see Fig.~\ref{subfig:UMP_7}), we are well towards the strong correlation regime, where we see that the UMP series is slowly convergent while RMP diverges.
%We see a single EP on the ground-state surface which falls just outside (maybe on?) the radius of convergence.
%An EP this close to the radius of convergence gives an increasingly slow convergence of the UMP series, but it will converge eventually as observed in Fig.~\ref{subfig:UMP_cvg}.
%\textit{``Class A systems are characterised by electronic structures with well-separated electron pairs while class B systems are characterized by electronic structures with electron clustering in one or more regions.''}
%Moreover, they analysed the contribution of the triple (T) excitations to the MP4, MP5 and MP6 energies next to the single, double and quadruple (SDQ) excitations contribution.
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.
%This observation on the contribution to the MP$n$ energy corroborates the electronic structure discussed above.
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}
%As one can only compute the first terms of the MP series, a smart way of getting more accurate results is to use extrapolation formula, \ie, estimating the limit of the series with only few terms.
%Cremer and He proved that using specific extrapolation formulas of the MP series for class A and class B systems improves the precision of the results compared to the formula used without resorting to classes. \cite{Cremer_1996}
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 Section~\ref{sec:Resummation}, we consider more advanced extrapolation routines that take account of EPs in the complex $\lambda$-plane.
%The mean absolute deviation taking the FCI correlation energies as reference is $0.3$ millihartree with the class-specific formula whereas the deviation increases to 12 millihartree using the general formula.
%Even if there were still shaded areas in their analysis and that their classification was incomplete, the work of Ref.~\onlinecite{Cremer_1996} clearly evidenced that understanding the origin of the different modes of convergence could potentially lead to a more rationalised use of MP perturbation theory and, hence, to more accurate correlation energy estimates.
\textit{``In the limit of large order, the series coefficients become equivalent to the Taylor series coefficients of the singularity closest to the origin. ''}
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.
%Other features characterising the convergence behaviour of a perturbation method are its rate of convergence, its length of recurring period, and its sign pattern.
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.
The mean-field potential $v^{\text{HF}}$ essentially represents a negatively charged field with the spatial extent
controlled by the extent of the HF orbitals, usually located close to the nuclei.
When $\lambda$ is negative, the mean-field potential becomes increasingly repulsive, while the explicit two-electron
Coulomb interaction becomes attractive.
There is therefore a critical value $\lc < 0$ where it becomes energetically favourable for the electrons
to dissociate and form a bound cluster at an infinite separation from the nuclei.\cite{Stillinger_2000}
This autoionisation effect is closely related to the critial point for electron binding in two-electron
atoms (see Ref.~\onlinecite{Baker_1971}).
Furthermore, a similar set of critical points exists along the positive real axis, corresponding to single-electron ionisation
processes.\cite{Sergeev_2005}
}
% CLASSIFICATIONS BY GOODSOON AND SERGEEV
\hugh{%
To further develop the link between the critical point and types of MP convergence, Sergeev and Goodson investigated
the relationship with the location of the dominant singularity that controls the radius of convergence.\cite{Goodson_2004}.
They demonstrated that the dominant singularity in class A corresponds to a dominant EP with a positive real component,
with the magnitude of the imaginary component controlling the oscillations in the signs of the MP
term.\cite{Goodson_2000a,Goodson200b}
In contrast, class B systems correspond to a dominant singularity on the negative real $\lambda$ axis representing
the MP critical point.
The divergence of class B systems, which contain closely spaced electrons (\eg, \ce{F-}), can then be understood as the
HF potential $v^{\text{HF}}$ is relatively localised and the autoionization is favoured at negative
$\lambda$ values closer to the origin.
With these insights, they regrouped the systems into new classes: i) $\alpha$ singularities which have ``large'' imaginary parts,
and ii) $\beta$ singularities which have very small imaginary parts.\cite{Goodson_2004,Sergeev_2006}
}
% RELATIONSHIP TO BASIS SET SIZE
\hugh{%
The existence of the MP critical point can also explain why the divergence observed by Olsen \etal\ in the \ce{Ne} atom
and \ce{HF} molecule occured when diffuse basis functions were included.\cite{Olsen_1996}
Clearly diffuse basis functions are required for the electrons to dissociate from the nuclei, and indeed using
only compact basis functions causes the critical point to disappear.
While a finite basis can only predict complex-conjugate branch point singularities, the critical point is modelled
by a cluster of sharp avoided crossings between the ground state and high-lying excited states.\cite{Sergeev_2005}
Alternatively, Sergeev \etal\ demonstrated that the inclusion of a ``ghost'' atom also
allows the formation of the critical point as the electrons form a bound cluster occuping the ghost atom orbitals.\cite{Sergeev_2005}
This effect explains the origin of the divergence in the \ce{HF} molecule as the \ce{F} valence electrons jump to \ce{H}
for a sufficiently negative $\lambda$.\cite{Sergeev_2005}
Furthermore, the two-state model of Olsen \etal{}\cite{Olsen_2000} was simply too minimal to understand the complexity of
divergences caused by the MP critical point.
}
% BASIS SET DEPENDENCE (INCLUDE?)
%Finally, it was shown that $\beta$ singularities are very sensitive to changes in the basis set but not to the bond stretching.
%On the contrary, $\alpha$ singularities are relatively insensitive to the basis sets but very sensitive to bond stretching.
%According to Goodson, \cite{Goodson_2004} the singularity structure of stretched molecules is difficult because there is more than one significant singularity.
%This is consistent with the observation of Olsen and coworkers \cite{Olsen_2000} on the \ce{HF} molecule at equilibrium and stretched geometries.
%To the best of our knowledge, the effect of bond stretching on singularities, its link with spin contamination and symmetry breaking of the wave function has not been as well understood as the ionization phenomenon and its link with diffuse functions.
%In the 2000's, Sergeev and Goodson \cite{Sergeev_2005, Sergeev_2006} analysed this problem from a more mathematical point of view by looking at the whole singularity structure where Olsen and collaborators were trying to find the dominant singularity causing the divergence. \cite{Olsen_1996,Olsen_2000,Olsen_2019}
%Singularities of $\alpha$-type are related to large avoided crossing between the ground and low-lying excited states, whereas $\beta$ singularities come from a sharp avoided crossing between the ground state and a highly diffuse state.
%They succeeded to explain the divergence of the series caused by $\beta$ singularities following previous work by Stillinger. \cite{Stillinger_2000}
%This reasoning is done on the exact Hamiltonian and energy, \ie, the Hamiltonian in the complete Hilbert space, this is the exact energy which exhibits this singularity on the negative real axis.
%However, in a finite basis set which does not span the complete Hilbert space, one can prove that, for a Hermitian Hamiltonian, the singularities of $E(\lambda)$ occurs in complex conjugate pairs with non-zero imaginary parts.
%Sergeev and Goodson proved, \cite{Sergeev_2005} as predicted by Stillinger, \cite{Stillinger_2000} that in a finite basis set the critical point on the real axis is modelled by a cluster of sharp avoided crossings with diffuse functions, equivalently by a cluster of $\beta$ singularities in the negative half plane.
%This explains that Olsen \textit{et al.}, because they used a simple two-state model, only observed the first singularity of this cluster of singularities causing the divergence. \cite{Olsen_2000}
% RELATIONSHIP TO QUANTUM PHASE TRANSITION
\hugh{%
When a Hamiltonian is parametrised by a variable such as $\lambda$, the existence of abrupt changes in the
eigenstates as a function of $\lambda$ indicate the presence of a zero-temperature quantum phase transition (QPT).%
%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.
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.
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}
(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.
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}
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 $P^2(\lambda)-4 Q(\lambda) R(\lambda)$.
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_2000}, 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 $\lambda_\text{EP}=\pm i 4t/U$.
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}).
\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.
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 $\lambda < r_c$), 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}
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}
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.
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.
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.
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).
