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add Antoine example and degreenified Hugh comments

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@ -195,36 +195,36 @@ We refer the interested reader to the excellent book of Moiseyev for a general o
\label{sec:TDSE} \label{sec:TDSE}
%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%
Within the Born-Oppenheimer approximation, the exact molecular Hamiltonian with $\Ne$ electrons and Within the Born-Oppenheimer approximation, the exact molecular Hamiltonian with $\Ne$ electrons and
$\Nn$ (clamped) nuclei is defined \hugh{for a given nuclear framework} as $\Nn$ (clamped) nuclei is defined for a given nuclear framework as
\begin{equation}\label{eq:ExactHamiltonian} \begin{equation}\label{eq:ExactHamiltonian}
\hugh{\hH(\vb{R})} = \hH(\vb{R}) =
- \frac{1}{2} \sum_{i}^{\Ne} \grad_i^2 - \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_{i}^{\Ne} \sum_{A}^{\Nn} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}}
+ \sum_{i<j}^{\Ne}\frac{1}{\abs{\vb{r}_i-\vb{r}_j}}, + \sum_{i<j}^{\Ne}\frac{1}{\abs{\vb{r}_i-\vb{r}_j}},
\end{equation} \end{equation}
where $\vb{r}_i$ defines the position of the $i$-th electron, $\vb{R}_{A}$ and $Z_{A}$ are the position where $\vb{r}_i$ defines the position of the $i$-th electron, $\vb{R}_{A}$ and $Z_{A}$ are the position
and charge of the $A$-th nucleus respectively, \hugh{and $\vb{R} = (\vb{R}_{1}, \dots, \vb{R}_{\Nn})$ is a and charge of the $A$-th nucleus respectively, and $\vb{R} = (\vb{R}_{1}, \dots, \vb{R}_{\Nn})$ is a
collective vector for the nuclear positions.} collective vector for the nuclear positions.
The first term represents the kinetic energy of the electrons, while The first term represents the kinetic energy of the electrons, while
the two following terms account for the electron-nucleus attraction and the electron-electron repulsion. the two following terms account for the electron-nucleus attraction and the electron-electron repulsion.
% EXACT SCHRODINGER EQUATION % EXACT SCHRODINGER EQUATION
The exact many-electron wave function \hugh{at a given nuclear geometry} $\Psi(\vb{R})$ corresponds The exact many-electron wave function at a given nuclear geometry $\Psi(\vb{R})$ corresponds
to the solution of the (time-independent) Schr\"{o}dinger equation to the solution of the (time-independent) Schr\"{o}dinger equation
\begin{equation} \begin{equation}
\hugh{\hH(\vb{R})\, \Psi(\vb{R}) = E(\vb{R})\, \Psi(\vb{R}),} \hH(\vb{R})\, \Psi(\vb{R}) = E(\vb{R})\, \Psi(\vb{R}),
\label{eq:SchrEq} \label{eq:SchrEq}
\end{equation} \end{equation}
with the eigenvalues $E(\vb{R})$ providing the exact energies. with the eigenvalues $E(\vb{R})$ providing the exact energies.
\hugh{The energy $E(\vb{R})$ can be considered as a ``one-to-many'' function since each input nuclear geometry The energy $E(\vb{R})$ can be considered as a ``one-to-many'' function since each input nuclear geometry
yields several eigenvalues corresponding to the ground and excited states of the exact spectrum.} yields several eigenvalues corresponding to the ground and excited states of the exact spectrum.
However, exact solutions to Eq.~\eqref{eq:SchrEq} are only possible in the simplest of systems, such as However, exact solutions to Eq.~\eqref{eq:SchrEq} are only possible in the simplest of systems, such as
the one-electron hydrogen atom and some specific two-electron systems with well-defined mathematical the one-electron hydrogen atom and some specific two-electron systems with well-defined mathematical
properties.\cite{Taut_1993,Loos_2009b,Loos_2010e,Loos_2012} properties.\cite{Taut_1993,Loos_2009b,Loos_2010e,Loos_2012}
\hugh{In practice, approximations to the exact Schr\"{o}dinger equation must be introduced, including In practice, approximations to the exact Schr\"{o}dinger equation must be introduced, including
the perturbation theories and Hartree--Fock approximation considered in this review the perturbation theories and Hartree--Fock approximation considered in this review
In what follows, we will drop the parametric dependence on the nuclear geometry and, In what follows, we will drop the parametric dependence on the nuclear geometry and,
unless otherwise stated, atomic units will be used throughout.} unless otherwise stated, atomic units will be used throughout.
%===================================% %===================================%
\subsection{Exceptional Points in the Hubbard Dimer} \subsection{Exceptional Points in the Hubbard Dimer}
@ -249,10 +249,8 @@ unless otherwise stated, atomic units will be used throughout.}
\end{figure*} \end{figure*}
To illustrate the concepts discussed throughout this article, we consider the symmetric Hubbard dimer at half filling, \ie\ with two opposite-spin fermions. To illustrate the concepts discussed throughout this article, we consider the symmetric Hubbard dimer at half filling, \ie\ with two opposite-spin fermions.
Analytic\trashHB{ally solvable} model systems are essential in theoretical chemistry and physics as their \hugh{mathematical} simplicity \trashHB{of the 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
mathematics} compared to realistic systems (e.g., atoms and molecules) allows new concepts and methods to be easily tested while retaining the key physical phenomena.
easily \trashHB{illustrated and} tested while retaining the key physical phenomena.
\hugh{(HGAB: This sentence felt too long to me. Feel free to re-instate words if you think they are neccessary)}
Using the (localised) site basis, the Hilbert space of the Hubbard dimer comprises the four configurations Using the (localised) site basis, the Hilbert space of the Hubbard dimer comprises the four configurations
\begin{align} \begin{align}
@ -274,12 +272,11 @@ where $t$ is the hopping parameter and $U$ is the on-site Coulomb repulsion.
We refer the interested reader to Refs.~\onlinecite{Carrascal_2015,Carrascal_2018} for more details about this system. We refer the interested reader to Refs.~\onlinecite{Carrascal_2015,Carrascal_2018} for more details about this system.
The parameter $U$ controls the strength of the electron correlation. The parameter $U$ controls the strength of the electron correlation.
In the weak correlation regime (small $U$), the kinetic energy dominates and the electrons are delocalised over both sites. In the weak correlation regime (small $U$), the kinetic energy dominates and the electrons are delocalised over both sites.
In the large-$U$ (or strong correlation) regime, the electron repulsion term \hugh{becomes dominant} \trashHB{drives the physics} In the large-$U$ (or strong correlation) regime, the electron repulsion term becomes dominant
and the electrons localise on opposite sites to minimise their Coulomb repulsion. and the electrons localise on opposite sites to minimise their Coulomb repulsion.
This phenomenon is often referred to as Wigner crystallisation. \cite{Wigner_1934} This phenomenon is often referred to as Wigner crystallisation. \cite{Wigner_1934}
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 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 $\hH(\lambda)$.
\hugh{to give the parameterised Hamiltonian $\hH(\lambda)$.}
When $\lambda$ is real, the Hamiltonian~\eqref{eq:H_FCI} is Hermitian with the distinct (real-valued) (eigen)energies When $\lambda$ is real, the Hamiltonian~\eqref{eq:H_FCI} is Hermitian with the distinct (real-valued) (eigen)energies
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
@ -321,7 +318,7 @@ As a result, completely encircling an EP leads to the interconversion of the two
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} 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}
% LOCATING EPS % LOCATING EPS
\hugh{To locate EPs in practice, one must simultaneously solve To locate EPs in practice, one must simultaneously solve
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:PolChar} \label{eq:PolChar}
@ -336,15 +333,15 @@ Equation \eqref{eq:PolChar} is the well-known secular equation providing the (ei
If the energy is also solution of Eq.~\eqref{eq:DPolChar}, then this energy value is at least two-fold degenerate. If the energy is also solution of Eq.~\eqref{eq:DPolChar}, then this energy value is at least two-fold degenerate.
These degeneracies can be conical intersections between two states with different symmetries These degeneracies can be conical intersections between two states with different symmetries
for real values of $\lambda$,\cite{Yarkony_1996} or EPs between two states with the for real values of $\lambda$,\cite{Yarkony_1996} or EPs between two states with the
same symmetry for complex values of $\lambda$.} same symmetry for complex values of $\lambda$.
%============================================================% %============================================================%
\subsection{Rayleigh-Schr\"odinger Perturbation Theory} \subsection{Rayleigh-Schr\"odinger Perturbation Theory}
%============================================================% %============================================================%
\hugh{One of the most common routes to approximately solving the Schr\"odinger equation One of the most common routes to approximately solving the Schr\"odinger equation
is to introduce a perturbative expansion of the exact energy.} is to introduce a perturbative expansion of the exact energy.
% SUMMARY OF RS-PT % SUMMARY OF RS-PT
Within Rayleigh-Schr\"odinger perturbation theory, the time-independent Schr\"odinger equation Within Rayleigh-Schr\"odinger perturbation theory, the time-independent Schr\"odinger equation
is recast as is recast as
@ -401,10 +398,12 @@ However, this series diverges for $x \ge 1$.
This divergence occurs because $f(x)$ has four singularities in the complex This divergence occurs because $f(x)$ has four singularities in the complex
($\e^{\i\pi/4}$, $\e^{-\i\pi/4}$, $\e^{\i3\pi/4}$, and $\e^{-\i3\pi/4}$) with a modulus equal to $1$, demonstrating ($\e^{\i\pi/4}$, $\e^{-\i\pi/4}$, $\e^{\i3\pi/4}$, and $\e^{-\i3\pi/4}$) with a modulus equal to $1$, demonstrating
that complex singularities are essential to fully understand the series convergence on the real axis.\cite{BenderBook} that complex singularities are essential to fully understand the series convergence on the real axis.\cite{BenderBook}
\titou{Include Antoine's example $\sum_{n=1}^\infty \lambda^n/n$ which is divergent at $\lambda = 1$ but convergent at $\lambda = -1$.}
The radius of convergence for the perturbation series Eq.~\eqref{eq:E_expansion} is therefore dictated by the magnitude \titou{$\abs{\lambda_0}$} of the The radius of convergence for the perturbation series Eq.~\eqref{eq:E_expansion} is therefore dictated by the magnitude $\abs{\lambda_c}$ of the
singularity in $E(\lambda)$ that is closest to the origin. singularity in $E(\lambda)$ that is closest to the origin.
Note that when $\lambda = \lambda_c$, one cannot \textit{a priori} predict if the series is convergent or not.
For example, the series $\sum_{k=1}^\infty \lambda^k/k$ diverges at $\lambda = 1$ but converges at $\lambda = -1$.
Like the exact system in Sec.~\ref{sec:example}, the perturbation energy $E(\lambda)$ represents Like the exact system in Sec.~\ref{sec:example}, the perturbation energy $E(\lambda)$ represents
a ``one-to-many'' function with the output elements representing an approximation to both the ground and excited states. a ``one-to-many'' function with the output elements representing an approximation to both the ground and excited states.
The most common singularities on $E(\lambda)$ therefore correspond to non-analytic EPs in the complex The most common singularities on $E(\lambda)$ therefore correspond to non-analytic EPs in the complex
@ -423,7 +422,7 @@ This Slater determinant is defined as an antisymmetric combination of $\Ne$ (rea
\begin{equation}\label{eq:FockOp} \begin{equation}\label{eq:FockOp}
\Hat{f}(\vb{x}) \phi_p(\vb{x}) = \qty[ \Hat{h}(\vb{x}) + \Hat{v}_\text{HF}(\vb{x}) ] \phi_p(\vb{x}) = \epsilon_p \phi_p(\vb{x}). \Hat{f}(\vb{x}) \phi_p(\vb{x}) = \qty[ \Hat{h}(\vb{x}) + \Hat{v}_\text{HF}(\vb{x}) ] \phi_p(\vb{x}) = \epsilon_p \phi_p(\vb{x}).
\end{equation} \end{equation}
Here the \hugh{(one-electron)} core Hamiltonian is Here the (one-electron) core Hamiltonian is
\begin{equation} \begin{equation}
\label{eq:Hcore} \label{eq:Hcore}
\Hat{h}(\vb{x}) = -\frac{\grad^2}{2} + \sum_{A}^{M} \frac{Z_A}{\abs{\vb{r}-\vb{R}_A}} \Hat{h}(\vb{x}) = -\frac{\grad^2}{2} + \sum_{A}^{M} \frac{Z_A}{\abs{\vb{r}-\vb{R}_A}}
@ -566,8 +565,8 @@ Time-reversal symmetry dictates that this UHF wave function must be degenerate w
by swapping $\ta^{\text{UHF}}$ and $\tb^{\text{UHF}}$ in Eqs.~\eqref{eq:ta_uhf} and \eqref{eq:tb_uhf}. 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 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} ``oscillates'' between the two symmetry-broken configurations. \cite{GiulianiBook}
\hugh{Symmetry breaking can also occur in RHF theory when a charge-density wave is formed from an oscillation 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}} 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} \subsection{Self-Consistency as a Perturbation} %OR {Complex adiabatic connection}
@ -580,7 +579,7 @@ Alternatively, the non-linear terms arising from the Coulomb and exchange operat
be considered as a perturbation from the core Hamiltonian \eqref{eq:Hcore} by introducing the be considered as a perturbation from the core Hamiltonian \eqref{eq:Hcore} by introducing the
transformation $U \rightarrow \lambda\, U$, giving the parametrised Fock operator transformation $U \rightarrow \lambda\, U$, giving the parametrised Fock operator
\begin{equation} \begin{equation}
\Hat{f}(\vb{x} \hugh{; \lambda}) = \Hat{h}(\vb{x}) + \lambda\, \Hat{v}_\text{HF}(\vb{x}). \Hat{f}(\vb{x} ; \lambda) = \Hat{h}(\vb{x}) + \lambda\, \Hat{v}_\text{HF}(\vb{x}).
\end{equation} \end{equation}
The orbitals in the reference problem $\lambda=0$ correspond to the symmetry-pure eigenfunctions of the one-electron core 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. Hamiltonian, while self-consistent solutions at $\lambda = 1$ represent the orbitals of the true HF solution.
@ -700,9 +699,9 @@ i) the Epstein-Nesbet (EN) partitioning which consists in taking the diagonal el
%Hence, the off-diagonal elements of $\hH$ are the perturbation operator, %Hence, the off-diagonal elements of $\hH$ are the perturbation operator,
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 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} iii) the strong coupling partitioning where the two operators are inverted compared to the weak correlation partitioning. \cite{Seidl_2018}
\hugh{While an in-depth comparison of these different approaches can offer insight into While an in-depth comparison of these different approaches can offer insight into
their relative strengths and weaknesses for various situations, we will restrict our current discussion their relative strengths and weaknesses for various situations, we will restrict our current discussion
to the convergence properties of the MP expansion.} to the convergence properties of the MP expansion.
%=====================================================% %=====================================================%
\subsection{M{\o}ller-Plesset Convergence in Molecular Systems} \subsection{M{\o}ller-Plesset Convergence in Molecular Systems}
@ -928,7 +927,6 @@ To do so, they analysed the relation between the dominant singularity (\ie, the
\begin{quote} \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. ''} \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} \end{quote}
\titou{T2: should we move this theorem earlier?}
Following the result of this theorem, the convergence patterns of the MP series can be explained by looking at the dominant singularity. Following the result of this theorem, the convergence patterns of the MP series can be explained by looking at the dominant singularity.
A singularity in the unit circle is designated as an intruder state, more precisely as a front-door (respectively back-door) intruder state if the real part of the singularity is positive (respectively negative). A singularity in the unit circle is designated as an intruder state, more precisely as a front-door (respectively back-door) intruder state if the real part of the singularity is positive (respectively negative).