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\begin{abstract}
We explore the extension of quantum chemistry in the complex plane and its link with perturbation theory.
We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory.
We observe that the physics of a quantum system is intimately connected to the position of energy singularities in the complex plane, known as exceptional 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, 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.
We also discuss several resummation techniques (such as Pad\'e and quadratic approximants) able to improve the overall accuracy of the M{\o}ller--Plesset perturbative series in both convergent and divergent cases.
Each of these points is further illustrated with the ubiquitous Hubbard dimer at half filling which proves to be a versatile model system in order to understand the subtle concepts of the analytic continuation of perturbation theory into the complex plane.
After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree--Fock approximation and Rayleigh-Schr\"odinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities.
In particular, we highlight the seminal work of several research groups on the convergence behaviour of perturbative series obtained within M{\o}ller--Plesset perturbation theory, and its links with quantum phase transitions.
We also discuss several resummation techniques (such as Pad\'e and quadratic approximants) that can improve the overall accuracy of the M{\o}ller--Plesset perturbative series in both convergent and divergent cases.
Each of these points is pedagogically illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.
\end{abstract}
\maketitle