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@ -166,7 +166,7 @@ Each of these points is further illustrated with the ubiquitous Hubbard dimer at
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% SPIKE THE READER % SPIKE THE READER
\hugh{Perturbation theory isn't usually considered in the complex plane. Perturbation theory isn't usually considered in the complex plane.
Normally it is applied using real numbers as one of very few availabe tools for Normally it is applied using real numbers as one of very few availabe tools for
describing realistic quantum systems where exact solutions of the Schr\"odinger equation are impossible.\cite{Dirac_1929} describing realistic quantum systems where exact solutions of the Schr\"odinger equation are impossible.\cite{Dirac_1929}
In particular, time-independent Rayleigh--Schr\"odinger perturbation theory\cite{RayleighBook,Schrodinger_1926} In particular, time-independent Rayleigh--Schr\"odinger perturbation theory\cite{RayleighBook,Schrodinger_1926}
@ -174,7 +174,6 @@ has emerged as an instrument of choice among the vast array of methods developed
\cite{SzaboBook,JensenBook,CramerBook,HelgakerBook,ParrBook,FetterBook,ReiningBook} \cite{SzaboBook,JensenBook,CramerBook,HelgakerBook,ParrBook,FetterBook,ReiningBook}
However, the properties of perturbation theory in the complex plane However, the properties of perturbation theory in the complex plane
are essential for understanding the quality of perturbative approximations on the real axis. are essential for understanding the quality of perturbative approximations on the real axis.
}
% Good old Schroedinger % Good old Schroedinger
%The electronic Schr\"odinger equation, %The electronic Schr\"odinger equation,
@ -196,19 +195,19 @@ are essential for understanding the quality of perturbative approximations on th
% Moller-Plesset % Moller-Plesset
%\hugh{Accurately predicting the electronic energy is the primary focus of electronic structure theory, in %\hugh{Accurately predicting the electronic energy is the primary focus of electronic structure theory, in
%principle providing a fundamental understanding of chemical structure, bonding, and reactivity. %principle providing a fundamental understanding of chemical structure, bonding, and reactivity.
\hugh{In electronic structure theory,} the workhorse of time-independent perturbation theory is M\o{}ller--Plesset (MP) %perturbation In electronic structure theory, the workhorse of time-independent perturbation theory is M\o{}ller--Plesset (MP) %perturbation
theory,\cite{Moller_1934} which remains one of the most popular methods for computing the electron theory,\cite{Moller_1934} which remains one of the most popular methods for computing the electron
correlation energy.\cite{Wigner_1934,Lowdin_1958} correlation energy.\cite{Wigner_1934,Lowdin_1958}
%\trashHB{an old yet important concept, first introduced by Wigner \cite{Wigner_1934} and later defined by L\"owdin. \cite{Lowdin_1958}} %\trashHB{an old yet important concept, first introduced by Wigner \cite{Wigner_1934} and later defined by L\"owdin. \cite{Lowdin_1958}}
This approach estimates the exact electronic energy by constructing a perturbative correction on top This approach estimates the exact electronic energy by constructing a perturbative correction on top
of a mean-field Hartree--Fock (HF) approximation.\cite{SzaboBook} of a mean-field Hartree--Fock (HF) approximation.\cite{SzaboBook}
The popularity of MP theory stems from its black-box nature, \hugh{size-extensivity,} and relatively low computational scaling, The popularity of MP theory stems from its black-box nature, size-extensivity, and relatively low computational scaling,
making it easily applied in a broad range of molecular research.\cite{HelgakerBook} making it easily applied in a broad range of molecular research.\cite{HelgakerBook}
However, it is now widely recognised that the series of MP approximations (defined for a given perturbation However, it is now widely recognised that the series of MP approximations (defined for a given perturbation
order $n$ as MP$n$) can show erratic, slow, or divergent behaviour that limit its systematic improvability.% order $n$ as MP$n$) can show erratic, slow, or divergent behaviour that limit its systematic improvability.%
\cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988} \cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988}
As a result, practical applications typically employ only the lowest-order MP2 approach, while As a result, practical applications typically employ only the lowest-order MP2 approach, while
the successive MP3, MP4, and MP5 (and higher \hugh{order}) terms are generally not considered to offer enough improvement the successive MP3, MP4, and MP5 (and higher order) terms are generally not considered to offer enough improvement
to justify their increased cost. to justify their increased cost.
Turning the MP approximations into a convergent and Turning the MP approximations into a convergent and
systematically improvable series largely remains an open challenge. systematically improvable series largely remains an open challenge.
@ -269,16 +268,15 @@ In doing so, we will demonstrate how understanding the MP energy in the complex
be harnessed to significantly improve estimates of the exact energy using only the lowest-order terms be harnessed to significantly improve estimates of the exact energy using only the lowest-order terms
in the MP series. in the MP series.
\trashHB{The present review is organised as follows.} In Sec.~\ref{sec:EPs}, we introduce the key concepts such as Rayleigh-Schr\"odinger perturbation theory and the mean-field HF approximation, and discuss their non-Hermitian analytic continuation into the complex plane.
In Sec.~\ref{sec:EPs}, we introduce the key concepts such as Rayleigh-Schr\"odinger perturbation theory and the mean-field HF approximation, and discuss their \hugh{non-Hermitian} analytic continuation into the complex plane.
Section \ref{sec:MP} presents MP perturbation theory and we report a comprehensive historical overview of the research that Section \ref{sec:MP} presents MP perturbation theory and we report a comprehensive historical overview of the research that
has been performed on the physics of MP singularities. has been performed on the physics of MP singularities.
In Sec.~\ref{sec:Resummation}, we discuss several resummation techniques for improving the accuracy In Sec.~\ref{sec:Resummation}, we discuss several resummation techniques for improving the accuracy
of low-order MP approximations, including Pad\'e and quadratic approximants. of low-order MP approximations, including Pad\'e and quadratic approximants.
Finally, we draw our conclusions in Sec.~\ref{sec:ccl} \hugh{and highlight our perspective on directions for Finally, we draw our conclusions in Sec.~\ref{sec:ccl} and highlight our perspective on directions for
future research}. future research.
Throughout this review, we present illustrative and pedagogical examples based on the ubiquitous Throughout this review, we present illustrative and pedagogical examples based on the ubiquitous
Hubbard dimer, \hugh{reinforcing} the amazing versatility of this powerful simplistic model. Hubbard dimer, reinforcing the amazing versatility of this powerful simplistic model.
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\section{Exceptional Points in Electronic Structure} \section{Exceptional Points in Electronic Structure}