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@ -165,38 +165,53 @@ Each of these points is further illustrated with the ubiquitous Hubbard dimer at
\label{sec:intro} \label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%
% SPIKE THE READER
\hugh{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
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}
has emerged as an instrument of choice among the vast array of methods developed for this purpose.%
\cite{SzaboBook,JensenBook,CramerBook,HelgakerBook,ParrBook,FetterBook,ReiningBook}
However, the properties of perturbation theory in the complex plane
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,
\begin{equation} %\begin{equation}
\hH \Psi = E \Psi, % \hH \Psi = E \Psi,
\end{equation} %\end{equation}
is the starting point for a fundamental understanding of the behaviour of electrons and, thence, of chemical structure, bonding and reactivity. %is the starting point for a fundamental understanding of the behaviour of electrons and, thence, of chemical structure, bonding and reactivity.
However, as famously stated by Dirac: \cite{Dirac_1929} %However, as famously stated by Dirac: \cite{Dirac_1929}
\begin{quote} %\begin{quote}
\textit{``The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. % \textit{``The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.
It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.''} % It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.''}
\end{quote} %\end{quote}
Indeed, as anticipated by Dirac, accurately predicting the energetics of electronic states from first principles has been one of the grand challenges faced by theoretical chemists and physicists since the dawn of quantum mechanics. %Indeed, as anticipated by Dirac, accurately predicting the energetics of electronic states from first principles has been one of the grand challenges faced by theoretical chemists and physicists since the dawn of quantum mechanics.
% RSPT % RSPT
Together with the variational principle, perturbation theory is one of the very few essential tool for describing realistic quantum systems for which it is impossible to find the exact solution of the Schr\"odinger equation. %Together with the variational principle, perturbation theory is one of the very few essential tool for describing realistic quantum systems for which it is impossible to find the exact solution of the Schr\"odinger equation.
In particular, time-independent Rayleigh--Schr\"odinger perturbation theory \cite{RayleighBook,Schrodinger_1926} has cemented itself as an instrument of choice in the armada of theoretical and computational methods that have been developed for this purpose. \cite{SzaboBook,JensenBook,CramerBook,HelgakerBook,ParrBook,FetterBook,ReiningBook} %In particular, time-independent Rayleigh--Schr\"odinger perturbation theory \cite{RayleighBook,Schrodinger_1926} has cemented itself as an instrument of choice in the armada of theoretical and computational methods that have been developed for this purpose. \cite{SzaboBook,JensenBook,CramerBook,HelgakerBook,ParrBook,FetterBook,ReiningBook}
% Moller-Plesset % Moller-Plesset
The workhorse of time-independent perturbation theory is most certainly M\o{}ller--Plesset (MP) perturbation %\hugh{Accurately predicting the electronic energy is the primary focus of electronic structure theory, in
theory \cite{Moller_1934} which remains one of the most popular methods for computing the electron correlation energy, %principle providing a fundamental understanding of chemical structure, bonding, and reactivity.
an old yet important concept, first introduced by Wigner \cite{Wigner_1934} and later defined by L\"owdin. \cite{Lowdin_1958} \hugh{In electronic structure theory,} the workhorse of time-independent perturbation theory is M\o{}ller--Plesset (MP) %perturbation
In this approach, the exact electronic energy is estimated by constructing a perturbative correction on top theory,\cite{Moller_1934} which remains one of the most popular methods for computing the electron
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}}
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 and relatively low computational scaling, The popularity of MP theory stems from its black-box nature, \hugh{size-extensivity,} and relatively low computational scaling,
making it easily applied in a broad range of molecular research. making it easily applied in a broad range of molecular research.\cite{HelgakerBook}
However, it is now widely understood 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) terms are generally not considered to offer enough improvement the successive MP3, MP4, and MP5 (and higher \hugh{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 systematically improvable series largely remains an open challenge. Turning the MP approximations into a convergent and
systematically improvable series largely remains an open challenge.
% COMPLEX PLANE % COMPLEX PLANE
Our conventional view of electronic structure theory is centred around the Hermitian notion of quantised energy levels, Our conventional view of electronic structure theory is centred around the Hermitian notion of quantised energy levels,
@ -207,7 +222,7 @@ However, an entirely different perspective on quantisation can be found by analy
quantum mechanics into the complex domain. quantum mechanics into the complex domain.
In this inherently non-Hermitian framework, the energy levels emerge as individual \textit{sheets} of a complex In this inherently non-Hermitian framework, the energy levels emerge as individual \textit{sheets} of a complex
multi-valued function and can be connected as one continuous \textit{Riemann surface}.\cite{BenderPTBook} multi-valued function and can be connected as one continuous \textit{Riemann surface}.\cite{BenderPTBook}
This connection is possible because orderability of real numbers is lost when energies are extended to the This connection is possible because the orderability of real numbers is lost when energies are extended to the
complex domain. complex domain.
As a result, our quantised view of conventional quantum mechanics only arises from As a result, our quantised view of conventional quantum mechanics only arises from
restricting our domain to Hermitian approximations. restricting our domain to Hermitian approximations.
@ -254,12 +269,16 @@ 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.
The present review is organised as follows. \trashHB{The present review is organised as follows.}
In Sec.~\ref{sec:EPs}, we introduce key concepts, such as Rayleigh-Schr\"odinger perturbation theory and the mean-field HF approximation, and discuss their 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} deals with MP perturbation theory and we report an exhaustive historical overview of the various research activities that have been performed on the physics of singularities. Section \ref{sec:MP} presents MP perturbation theory and we report a comprehensive historical overview of the research that
We discuss several resummation techniques (such as Pad\'e and quadratic approximants) in Sec.~\ref{sec:Resummation}. has been performed on the physics of MP singularities.
Finally, we draw our conclusions in Sec.~\ref{sec:ccl}. In Sec.~\ref{sec:Resummation}, we discuss several resummation techniques for improving the accuracy
For most of the concepts presented in this review, we report illustrative and pedagogical examples based on the ubiquitous Hubbard dimer, showing the amazing versatility of this simple yet powerful model system. 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
future research}.
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.
%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%
\section{Exceptional Points in Electronic Structure} \section{Exceptional Points in Electronic Structure}
@ -1776,7 +1795,7 @@ the cost of larger denominators is an overall slower rate of convergence.
\includegraphics[width=\linewidth]{fig12} \includegraphics[width=\linewidth]{fig12}
\caption{% \caption{%
Comparison of the scaled RMP10 Taylor expansion with the exact RMP energy as a function Comparison of the scaled RMP10 Taylor expansion with the exact RMP energy as a function
of $\lambda$ for the symmetric Hubbard dimer at $U/t = 4$. of $\lambda$ for the symmetric Hubbard dimer at $U/t = 4.5$.
The two functions correspond closely within the radius of convergence. The two functions correspond closely within the radius of convergence.
} }
\label{fig:rmp_anal_cont} \label{fig:rmp_anal_cont}

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Notebooks/HughHubbarrdDimer.nb
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