1908 lines
112 KiB
TeX
1908 lines
112 KiB
TeX
\documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1}
|
|
\usepackage{subcaption}
|
|
\usepackage{bm,graphicx,tabularx,array,booktabs,dcolumn,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,siunitx,mhchem}
|
|
\usepackage[utf8]{inputenc}
|
|
\usepackage[T1]{fontenc}
|
|
\usepackage{txfonts}
|
|
|
|
\usepackage[normalem]{ulem}
|
|
\definecolor{hughgreen}{RGB}{0, 128, 0}
|
|
\newcommand{\titou}[1]{\textcolor{red}{#1}}
|
|
\newcommand{\hugh}[1]{\textcolor{hughgreen}{#1}}
|
|
\newcommand{\hughDraft}[1]{\textcolor{orange}{#1}}
|
|
\newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}}
|
|
\newcommand{\trashHB}[1]{\textcolor{orange}{\sout{#1}}}
|
|
\newcommand{\antoine}[1]{\textcolor{cyan}{#1}}
|
|
|
|
\usepackage[
|
|
colorlinks=true,
|
|
citecolor=blue,
|
|
linkcolor=blue,
|
|
filecolor=blue,
|
|
urlcolor=blue,
|
|
breaklinks=true
|
|
]{hyperref}
|
|
\urlstyle{same}
|
|
|
|
\newcommand{\ctab}{\multicolumn{1}{c}{---}}
|
|
\newcommand{\latin}[1]{#1}
|
|
%\newcommand{\latin}[1]{\textit{#1}}
|
|
\newcommand{\ie}{\latin{i.e.}}
|
|
\newcommand{\eg}{\latin{e.g.}}
|
|
\newcommand{\etal}{\textit{et al.}}
|
|
|
|
\newcommand{\mc}{\multicolumn}
|
|
\newcommand{\fnm}{\footnotemark}
|
|
\newcommand{\fnt}{\footnotetext}
|
|
\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}}
|
|
\newcommand{\mr}{\multirow}
|
|
|
|
% operators
|
|
\newcommand{\bH}{\mathbf{H}}
|
|
\newcommand{\bV}{\mathbf{V}}
|
|
\newcommand{\bh}{\mathbf{h}}
|
|
\newcommand{\bQ}{\mathbf{Q}}
|
|
\newcommand{\bSig}{\mathbf{\Sigma}}
|
|
\newcommand{\br}{\mathbf{r}}
|
|
\newcommand{\bp}{\mathbf{p}}
|
|
\newcommand{\cP}{\mathcal{P}}
|
|
\newcommand{\cS}{\mathcal{S}}
|
|
\newcommand{\cT}{\mathcal{T}}
|
|
\newcommand{\cC}{\mathcal{C}}
|
|
\newcommand{\PT}{\mathcal{PT}}
|
|
|
|
\newcommand{\EPT}{E_{\PT}}
|
|
\newcommand{\laPT}{\lambda_{\PT}}
|
|
|
|
\newcommand{\EEP}{E_\text{EP}}
|
|
\newcommand{\laEP}{\lambda_\text{EP}}
|
|
|
|
|
|
\newcommand{\Ne}{N} % Number of electrons
|
|
\newcommand{\Nn}{M} % Number of nuclei
|
|
\newcommand{\hI}{\Hat{I}}
|
|
\newcommand{\hH}{\Hat{H}}
|
|
\newcommand{\hS}{\Hat{S}}
|
|
\newcommand{\hT}{\Hat{T}}
|
|
\newcommand{\hW}{\Hat{W}}
|
|
\newcommand{\hV}{\Hat{V}}
|
|
\newcommand{\hc}[2]{\Hat{c}_{#1}^{#2}}
|
|
\newcommand{\hn}[1]{\Hat{n}_{#1}}
|
|
\newcommand{\n}[1]{n_{#1}}
|
|
\newcommand{\Dv}{\Delta v}
|
|
|
|
\newcommand{\ra}{\rightarrow}
|
|
\newcommand{\up}{\uparrow}
|
|
\newcommand{\dw}{\downarrow}
|
|
|
|
% Center tabularx columns
|
|
\newcolumntype{Y}{>{\centering\arraybackslash}X}
|
|
|
|
% HF rotation angles
|
|
\newcommand{\ta}{\theta_{\alpha}}
|
|
\newcommand{\tb}{\theta_{\beta}}
|
|
|
|
% Some constants
|
|
\renewcommand{\i}{\mathrm{i}} % Imaginary unit
|
|
\newcommand{\e}{\mathrm{e}} % Euler number
|
|
\newcommand{\rc}{r_{\text{c}}}
|
|
\newcommand{\lc}{\lambda_{\text{c}}}
|
|
\newcommand{\lp}{\lambda_{\text{p}}}
|
|
\newcommand{\lep}{\lambda_{\text{EP}}}
|
|
|
|
% Some energies
|
|
\newcommand{\Emp}{E_{\text{MP}}}
|
|
|
|
% Blackboard bold
|
|
\newcommand{\bbR}{\mathbb{R}}
|
|
\newcommand{\bbC}{\mathbb{C}}
|
|
|
|
% Making life easier
|
|
\newcommand{\Lup}{\mathcal{L}^{\uparrow}}
|
|
\newcommand{\Ldown}{\mathcal{L}^{\downarrow}}
|
|
\newcommand{\Lsi}{\mathcal{L}^{\sigma}}
|
|
\newcommand{\Rup}{\mathcal{R}^{\uparrow}}
|
|
\newcommand{\Rdown}{\mathcal{R}^{\downarrow}}
|
|
\newcommand{\Rsi}{\mathcal{R}^{\sigma}}
|
|
\newcommand{\vhf}{v_{\text{HF}}}
|
|
|
|
|
|
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France.}
|
|
\newcommand{\UCAM}{Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, U.K.}
|
|
\newcommand{\UOX}{Physical and Theoretical Chemical Laboratory, Department of Chemistry, University of Oxford, Oxford, OX1 3QZ, U.K.}
|
|
\begin{document}
|
|
|
|
\title{Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them}
|
|
|
|
\author{Antoine \surname{Marie}}
|
|
\affiliation{\LCPQ}
|
|
\author{Hugh G.~A.~\surname{Burton}}
|
|
\email{hugh.burton@chem.ox.ac.uk}
|
|
\affiliation{\UOX}
|
|
\author{Pierre-Fran\c{c}ois \surname{Loos}}
|
|
\email{loos@irsamc.ups-tlse.fr}
|
|
\affiliation{\LCPQ}
|
|
|
|
|
|
\begin{abstract}
|
|
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 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
|
|
|
|
%\raggedbottom
|
|
%\tableofcontents
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Introduction}
|
|
\label{sec:intro}
|
|
%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
% SPIKE THE READER
|
|
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 \titou{to find?}.\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.
|
|
|
|
% Moller-Plesset
|
|
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
|
|
correlation energy.\cite{Wigner_1934,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}
|
|
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}
|
|
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.%
|
|
\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
|
|
the successive MP3, MP4, and MP5 (and higher order) terms are generally not considered to offer enough improvement
|
|
to justify their increased cost.
|
|
Turning the MP approximations into a convergent and
|
|
systematically improvable series largely remains an open challenge.
|
|
|
|
% COMPLEX PLANE
|
|
Our conventional view of electronic structure theory is centred around the Hermitian notion of quantised energy levels,
|
|
where the different electronic states of a molecular are discrete and energetically ordered.
|
|
The lowest energy state defines the ground electronic state, while higher energy states
|
|
represent electronic excited states.
|
|
However, an entirely different perspective on quantisation can be found by analytically continuing
|
|
quantum mechanics into the complex domain.
|
|
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}
|
|
This connection is possible because the orderability of real numbers is lost when energies are extended to the
|
|
complex domain.
|
|
As a result, our quantised view of conventional quantum mechanics only arises from
|
|
restricting our domain to Hermitian approximations.
|
|
|
|
% NON-HERMITIAN HAMILTONIANS
|
|
Non-Hermitian Hamiltonians already have a long history in quantum chemistry and have been extensively used to
|
|
describe metastable resonance phenomena.\cite{MoiseyevBook}
|
|
Through the methods of complex-scaling\cite{Moiseyev_1998} and complex absorbing
|
|
potentials,\cite{Riss_1993,Ernzerhof_2006,Benda_2018} outgoing resonances can be stabilised as square-integrable
|
|
wave functions with a complex energy that allows the resonance energy and lifetime to be computed.
|
|
We refer the interested reader to the excellent book by Moiseyev for a general overview. \cite{MoiseyevBook}
|
|
|
|
% EXCEPTIONAL POINTS
|
|
The Riemann surface for the electronic energy $E(\lambda)$ with a coupling parameter $\lambda$ can be
|
|
constructed by analytically continuing the function into the complex $\lambda$ domain.
|
|
In the process, the ground and excited states become smoothly connected and form a continuous complex-valued
|
|
energy surface.
|
|
\textit{Exceptional points} (EPs) can exist on this energy surface, corresponding to 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}
|
|
While EPs can be considered as the non-Hermitian analogues of conical intersections,\cite{Yarkony_1996}
|
|
the behaviour of their eigenvalues near a degeneracy could not be more different.
|
|
Incredibly, following the eigenvalues around an EP leads to the interconversion of the degenerate states,
|
|
and multiple loops around the EP are required to recover the initial energy.\cite{MoiseyevBook,Heiss_2016,Benda_2018}
|
|
In contrast, encircling a conical intersection leaves the states unchanged.
|
|
Furthermore, while the eigenvectors remain orthogonal at a conical intersection, the eigenvectors at an EP
|
|
become identical and result in a \textit{self-orthogonal} state. \cite{MoiseyevBook}
|
|
An EP effectively creates a ``portal'' between ground and excited-states in the complex plane.%
|
|
\cite{Burton_2019,Burton_2019a}
|
|
This transition between states has been experimentally observed in 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}
|
|
|
|
% MP THEORY IN THE COMPLEX PLANE
|
|
The MP energy correction can be considered as a function of the perturbation parameter $\lambda$.
|
|
When the domain of $\lambda$ is extended to the complex plane, EPs can also occur in the MP energy.
|
|
Although these EPs generally exist in the complex plane, their positions are intimately related to the
|
|
convergence of the perturbation expansion on the real axis.%
|
|
\cite{BenderBook,Olsen_1996,Olsen_2000,Olsen_2019,Mihalka_2017a,Mihalka_2017b,Mihalka_2019}
|
|
Furthermore, the existence of an avoided crossing on the real axis is indicative of a nearby EP
|
|
in the complex plane.
|
|
Our aim in this article is to provide a comprehensive review of the fundamental relationship between EPs
|
|
and the convergence properties of the MP series.
|
|
In doing so, we will demonstrate how understanding the MP energy in the complex plane can
|
|
be harnessed to significantly improve estimates of the exact energy using only the lowest-order terms
|
|
in the MP series.
|
|
|
|
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.
|
|
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.
|
|
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.
|
|
Finally, we draw our conclusions in Sec.~\ref{sec:ccl} and highlight our perspective on directions for
|
|
future research.
|
|
Throughout this review, we present illustrative and pedagogical examples based on the ubiquitous
|
|
Hubbard dimer, reinforcing the amazing versatility of this powerful simplistic model.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Exceptional Points in Electronic Structure}
|
|
\label{sec:EPs}
|
|
%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Time-Independent Schr\"odinger Equation}
|
|
\label{sec:TDSE}
|
|
%%%%%%%%%%%%%%%%%%%%%%%
|
|
Within the Born-Oppenheimer approximation, the exact molecular Hamiltonian with $\Ne$ electrons and
|
|
$\Nn$ (clamped) nuclei is defined for a given nuclear framework as
|
|
\begin{equation}\label{eq:ExactHamiltonian}
|
|
\hH(\vb{R}) =
|
|
- \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<j}^{\Ne}\frac{1}{\abs{\vb{r}_i-\vb{r}_j}},
|
|
\end{equation}
|
|
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, and $\vb{R} = (\vb{R}_{1}, \dots, \vb{R}_{\Nn})$ is a
|
|
collective vector for the nuclear positions.
|
|
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.
|
|
|
|
% EXACT SCHRODINGER EQUATION
|
|
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
|
|
\begin{equation}
|
|
\hH(\vb{R})\, \Psi(\vb{R}) = E(\vb{R})\, \Psi(\vb{R}),
|
|
\label{eq:SchrEq}
|
|
\end{equation}
|
|
with the eigenvalues $E(\vb{R})$ providing the exact energies.
|
|
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.
|
|
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
|
|
properties.\cite{Taut_1993,Loos_2009b,Loos_2010e,Loos_2012}
|
|
In practice, approximations to the exact Schr\"{o}dinger equation must be introduced, including
|
|
perturbation theories and the Hartree--Fock approximation considered in this review.
|
|
In what follows, we will drop the parametric dependence on the nuclear geometry and,
|
|
unless otherwise stated, atomic units will be used throughout.
|
|
|
|
%===================================%
|
|
\subsection{Exceptional Points in the Hubbard Dimer}
|
|
\label{sec:example}
|
|
%===================================%
|
|
|
|
%%% FIG 1 %%%
|
|
\begin{figure*}[t]
|
|
\begin{subfigure}{0.49\textwidth}
|
|
\includegraphics[height=0.65\textwidth]{fig1a}
|
|
\subcaption{\titou{Real axis} \label{subfig:FCI_real}}
|
|
\end{subfigure}
|
|
\begin{subfigure}{0.49\textwidth}
|
|
\includegraphics[height=0.65\textwidth]{fig1b}
|
|
\subcaption{\titou{Complex plane} \label{subfig:FCI_cplx}}
|
|
\end{subfigure}
|
|
\caption{%
|
|
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).
|
|
The contour followed around the EP in order to interchange states is also represented.
|
|
\label{fig:FCI}}
|
|
\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.
|
|
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.
|
|
|
|
Using the (localised) site basis, the Hilbert space of the Hubbard dimer comprises the four configurations
|
|
\begin{align*}
|
|
& \ket{\Lup \Ldown}, & & \ket{\Lup\Rdown}, & & \ket{\Rup\Ldown}, & & \ket{\Rup\Rdown},
|
|
\end{align*}
|
|
where $\Lsi$ ($\Rsi$) denotes an electron with spin $\sigma$ on the left (right) site.
|
|
The exact, or full configuration interaction (FCI), Hamiltonian is then
|
|
\begin{equation}
|
|
\label{eq:H_FCI}
|
|
\bH =
|
|
\begin{pmatrix}
|
|
U & - t & - t & 0 \\
|
|
- t & 0 & 0 & - t \\
|
|
- t & 0 & 0 & - t \\
|
|
0 & - t & - t & U \\
|
|
\end{pmatrix},
|
|
\end{equation}
|
|
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.
|
|
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 large-$U$ (or strong correlation) regime, the electron repulsion term becomes dominant
|
|
and the electrons localise on opposite sites to minimise their Coulomb repulsion.
|
|
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 $t \to \lambda t$ 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
|
|
\begin{subequations}
|
|
\begin{align}
|
|
E_{\mp} &= \frac{1}{2} \qty(U \mp \sqrt{ (4 \lambda t)^2 + U^2 } ),
|
|
\label{eq:singletE}
|
|
\\
|
|
E_{\text{T}} &= 0,
|
|
\\
|
|
E_{\text{S}} &= U.
|
|
\end{align}
|
|
\end{subequations}
|
|
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
|
|
\begin{equation}
|
|
\lambda_{\text{EP}} = \pm \i \frac{U}{4t},
|
|
\end{equation}
|
|
with energy
|
|
\begin{equation}
|
|
\label{eq:E_EP}
|
|
E_\text{EP} = \frac{U}{2}.
|
|
\end{equation}
|
|
These $\lambda$ values correspond to so-called EPs and connect the ground and excited states in the complex 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}).
|
|
This behaviour can be seen by expanding the radicand in Eq.~\eqref{eq:singletE} as a Taylor series around $\lambda_{\text{EP}}$ to give
|
|
\begin{equation}
|
|
E_{\pm} \approx E_{\text{EP}} \pm \sqrt{32t^2 \lambda_{\text{EP}}} \sqrt{\lambda - \lambda_{\text{EP}}}.
|
|
\end{equation}
|
|
Parametrising the complex contour as $\lambda(\theta) = \lambda_{\text{EP}} + R \exp(\i \theta)$ gives the continuous energy pathways
|
|
\begin{equation}
|
|
E_{\pm} \qty(\theta) \approx E_{\text{EP}} \pm \sqrt{32t^2 \lambda_{\text{EP}} R}\, \exp(\i \theta/2)
|
|
\end{equation}
|
|
such that $E_{\pm}(2\pi) = E_{\mp}(0)$ and $E_{\pm}(4\pi) = E_{\pm}(0)$.
|
|
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 can pick up a geometric phase in the process, and four complete loops are required to recover their starting forms.\cite{MoiseyevBook}
|
|
|
|
% LOCATING EPS
|
|
To locate EPs in practice, one must simultaneously solve
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\label{eq:PolChar}
|
|
\det[\hH(\lambda) - E \hI] & = 0,
|
|
\\
|
|
\label{eq:DPolChar}
|
|
\pdv{E}\det[\hH(\lambda) - E \hI] & = 0,
|
|
\end{align}
|
|
\end{subequations}
|
|
where $\hI$ is the identity operator.\cite{Cejnar_2007}
|
|
Equation \eqref{eq:PolChar} is the well-known secular equation providing the (eigen)energies of the system.
|
|
If the energy is also a 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
|
|
for real values of $\lambda$,\cite{Yarkony_1996} or EPs between two states with the
|
|
same symmetry for complex values of $\lambda$.
|
|
|
|
|
|
%============================================================%
|
|
\subsection{Rayleigh--Schr\"odinger Perturbation Theory}
|
|
%============================================================%
|
|
|
|
One of the most common routes to approximately solving the Schr\"odinger equation
|
|
is to introduce a perturbative expansion of the exact energy.
|
|
% SUMMARY OF RS-PT
|
|
Within Rayleigh--Schr\"odinger perturbation theory, the time-independent Schr\"odinger equation
|
|
is recast as
|
|
\begin{equation}
|
|
\hH(\lambda) \Psi(\lambda)
|
|
= \qty(\hH^{(0)} + \lambda \hV ) \Psi(\lambda)
|
|
= E(\lambda) \Psi(\lambda),
|
|
\label{eq:SchrEq-PT}
|
|
\end{equation}
|
|
where $\hH^{(0)}$ is a zeroth-order Hamiltonian and $\hV = \hH - \hH^{(0)}$ represents the perturbation operator.
|
|
Expanding the wave function and energy as power series in $\lambda$ as
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\Psi(\lambda) &= \sum_{k=0}^{\infty} \lambda^{k}\,\Psi^{(k)},
|
|
\label{eq:psi_expansion}
|
|
\\
|
|
E(\lambda) &= \sum_{k=0}^{\infty} \lambda^{k}\,E^{(k)},
|
|
\label{eq:E_expansion}
|
|
\end{align}
|
|
\end{subequations}
|
|
solving the corresponding perturbation equations up to a given order \titou{$n$}, and
|
|
setting $\lambda = 1$ then yields approximate solutions to Eq.~\eqref{eq:SchrEq}.
|
|
|
|
% MATHEMATICAL REPRESENTATION
|
|
Mathematically, Eq.~\eqref{eq:E_expansion} corresponds to a Taylor series expansion of the exact energy
|
|
around the reference system $\lambda = 0$.
|
|
The energy of the target ``physical'' system is recovered at the point $\lambda = 1$.
|
|
However, like all series expansions, Eq.~\eqref{eq:E_expansion} has a radius of convergence $\rc$.
|
|
When $\rc < 1$, the Rayleigh--Schr\"{o}dinger expansion will diverge
|
|
for the physical system.
|
|
The value of $\rc$ can vary significantly between different systems and strongly depends on the particular decomposition
|
|
of the reference and perturbation Hamiltonians in Eq.~\eqref{eq:SchrEq-PT}.\cite{Mihalka_2017b}
|
|
%
|
|
% LAMBDA IN THE COMPLEX PLANE
|
|
From complex analysis, \cite{BenderBook} the radius of convergence for the energy can be obtained by looking for the
|
|
singularities of $E(\lambda)$ in the complex $\lambda$ plane.
|
|
This property arises from the following theorem: \cite{Goodson_2011}
|
|
\begin{quote}
|
|
\it
|
|
``The Taylor series about a point $z_0$ of a function over the complex $z$ plane will converge at a value $z_1$
|
|
if the function is non-singular at all values of $z$ in the circular region centred at $z_0$ with radius $\abs{z_1-z_0}$.
|
|
If the function has a singular point $z_s$ such that $\abs{z_s-z_0} < \abs{z_1-z_0}$,
|
|
then the series will diverge when evaluated at $z_1$.''
|
|
\end{quote}
|
|
As a result, the radius of convergence for a function is equal to the distance from the origin of the closest singularity
|
|
in the complex plane, referred to as the ``dominant'' singularity.
|
|
This singularity may represent a pole of the function, or a branch point (\eg, square-root or logarithmic)
|
|
in a multi-valued function.
|
|
|
|
For example, the simple function
|
|
\begin{equation} \label{eq:DivExample}
|
|
f(x)=\frac{1}{1+x^4}.
|
|
\end{equation}
|
|
is smooth and infinitely differentiable for $x \in \mathbb{R}$, and one might expect that its Taylor series expansion would
|
|
converge in this domain.
|
|
However, this series diverges for $x \ge 1$.
|
|
This divergence occurs because $f(x)$ has four poles 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
|
|
that complex singularities are essential to fully understand the series convergence on the real axis.\cite{BenderBook}
|
|
|
|
The radius of convergence for the perturbation series Eq.~\eqref{eq:E_expansion} is therefore dictated by the magnitude $r_c = \abs{\lambda_c}$ of the
|
|
singularity in $E(\lambda)$ that is closest to the origin.
|
|
Note that when $\abs{\lambda} = r_c$, 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
|
|
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
|
|
$\lambda$ plane where two states become degenerate.
|
|
Later we will demonstrate how the choice of reference Hamiltonian controls the position of these EPs, and
|
|
ultimately determines the convergence properties of the perturbation series.
|
|
|
|
%===========================================%
|
|
\subsection{Hartree--Fock Theory}
|
|
\label{sec:HF}
|
|
%===========================================%
|
|
|
|
% SUMMARY OF HF
|
|
In the \trash{Hartree--Fock (HF)} \titou{HF} approximation, the many-electron wave function is approximated as a single Slater determinant $\Psi^{\text{HF}}(\vb{x}_1,\ldots,\vb{x}_\Ne)$, 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
|
|
\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}).
|
|
\end{equation}
|
|
Here the (one-electron) core Hamiltonian is
|
|
\begin{equation}
|
|
\label{eq:Hcore}
|
|
\Hat{h}(\vb{x}) = -\frac{\grad^2}{2} + \sum_{A}^{\Nn} \frac{Z_A}{\abs{\vb{r}-\vb{R}_A}}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\Hat{v}_\text{HF}(\vb{x}) = \sum_i^{\Ne} \qty[ \Hat{J}_i(\vb{x}) - \Hat{K}_i(\vb{x}) ]
|
|
\end{equation}
|
|
is the HF mean-field electron-electron potential with
|
|
\begin{subequations}
|
|
\begin{gather}
|
|
\label{eq:CoulOp}
|
|
\Hat{J}_i(\vb{x})\phi_j(\vb{x})=\qty(\int \phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_i(\vb{x}') \dd\vb{x}' ) \phi_j(\vb{x}),
|
|
\\
|
|
\label{eq:ExcOp}
|
|
\Hat{K}_i(\vb{x})\phi_j(\vb{x})=\qty(\int \phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_j(\vb{x}') \dd\vb{x}')\phi_i(\vb{x}),
|
|
\end{gather}
|
|
\end{subequations}
|
|
defining the Coulomb and exchange operators (respectively) in the spin-orbital basis.\cite{SzaboBook}
|
|
The HF energy is then defined as
|
|
\begin{equation}
|
|
\label{eq:E_HF}
|
|
E_\text{HF} = \frac{1}{2} \sum_i^{\Ne} \qty( h_i + f_i ),
|
|
\end{equation}
|
|
with the corresponding matrix elements
|
|
\begin{align}
|
|
h_i & = \mel{\phi_i}{\Hat{h}}{\phi_i},
|
|
&
|
|
f_i & = \mel{\phi_i}{\Hat{f}}{\phi_i}.
|
|
\end{align}
|
|
The optimal HF wave function is identified by using the variational principle to minimise the HF energy.
|
|
For any system with more than one electron, the resulting Slater determinant is not an eigenfunction of the exact Hamiltonian $\hH$.
|
|
However, it is by definition an eigenfunction of the approximate many-electron HF Hamiltonian constructed
|
|
from the one-electron Fock operators as
|
|
\begin{equation}\label{eq:HFHamiltonian}
|
|
\hH_{\text{HF}} = \sum_{i}^{\Ne} f(\vb{x}_i).
|
|
\end{equation}
|
|
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.
|
|
|
|
% BRIEF FLAVOURS OF HF
|
|
In the most flexible variant of real HF theory (generalised HF) the one-electron orbitals can be complex-valued
|
|
and contain a mixture of spin-up and spin-down components.\cite{Mayer_1993,Jimenez-Hoyos_2011}
|
|
However, the application of HF \titou{theory} with some level of constraint on the orbital structure is far more common.
|
|
Forcing the spatial part of the orbitals to be the same for spin-up and spin-down electrons leads to restricted HF (RHF) method,
|
|
while allowing different orbitals for different spins leads to the so-called unrestricted HF (UHF) approach.\cite{StuberPaldus}
|
|
The advantage of the UHF approximation is its ability to correctly describe strongly correlated systems,
|
|
such as antiferromagnetic phases\cite{Slater_1951} or the dissociation of the hydrogen dimer.\cite{Coulson_1949}
|
|
However, by allowing different orbitals for different spins, the UHF is no longer required to be an eigenfunction of
|
|
the total spin operator $\hat{\mathcal{S}}^2$, leading to ``spin-contamination'' in the wave function.
|
|
|
|
%================================================================%
|
|
\subsection{Hartree--Fock in the Hubbard Dimer}
|
|
\label{sec:HF_hubbard}
|
|
%================================================================%
|
|
|
|
%%% FIG 2 (?) %%%
|
|
% HF energies as a function of U/t
|
|
%%%%%%%%%%%%%%%%%
|
|
\begin{figure}
|
|
\includegraphics[width=\linewidth]{fig2}
|
|
\caption{\label{fig:HF_real}
|
|
RHF and UHF energies in the Hubbard dimer as a function of the correlation strength $U/t$.
|
|
The symmetry-broken UHF solution emerges at the coalescence point $U=2t$ (black dot), often known as the Coulson--Fischer point.}
|
|
\end{figure}
|
|
%%%%%%%%%%%%%%%%%
|
|
|
|
In the Hubbard dimer, the HF energy can be parametrised using two rotation angles $\ta$ and $\tb$ as
|
|
\begin{equation}
|
|
E_\text{HF}(\ta, \tb) = -t\, \qty( \sin \ta + \sin \tb ) + \frac{U}{2} \qty( 1 + \cos \ta \cos \tb ),
|
|
\end{equation}
|
|
where we have introduced bonding $\mathcal{B}^{\sigma}$ and anti-bonding $\mathcal{A}^{\sigma}$ molecular orbitals for
|
|
the spin-$\sigma$ electrons as
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\mathcal{B}^{\sigma} & = \hphantom{-} \cos(\frac{\theta_\sigma}{2}) \Lsi + \sin(\frac{\theta_\sigma}{2}) \Rsi,
|
|
\\
|
|
\mathcal{A}^{\sigma} & = - \sin(\frac{\theta_\sigma}{2}) \Lsi + \cos(\frac{\theta_\sigma}{2}) \Rsi
|
|
\end{align}
|
|
\label{eq:RHF_orbs}
|
|
\end{subequations}
|
|
In the weak correlation regime $0 \le U \le 2t$, the angles which minimise the HF energy,
|
|
\ie, $\pdv*{E_\text{HF}}{\theta_\sigma} = 0$, are
|
|
\begin{equation}
|
|
\ta^\text{RHF} = \tb^\text{RHF} = \pi/2,
|
|
\end{equation}
|
|
giving the symmetry-pure molecular orbitals
|
|
\begin{align}
|
|
\mathcal{B}_\text{RHF}^{\sigma} & = \frac{\Lsi + \Rsi}{\sqrt{2}},
|
|
&
|
|
\mathcal{A}_\text{RHF}^{\sigma} & = \frac{\Lsi - \Rsi}{\sqrt{2}},
|
|
\end{align}
|
|
and the ground-state RHF energy (Fig.~\ref{fig:HF_real})
|
|
\begin{equation}
|
|
E_\text{RHF} \equiv E_\text{HF}(\ta^\text{RHF}, \tb^\text{RHF}) = -2t + \frac{U}{2}.
|
|
\end{equation}
|
|
However, in the strongly correlated regime $U>2t$, the closed-shell orbital restriction prevents RHF from
|
|
modelling the correct physics with the two electrons on opposite sites.
|
|
|
|
%%% FIG 3 (?) %%%
|
|
% Analytic Continuation of HF
|
|
%%%%%%%%%%%%%%%%%
|
|
\begin{figure*}[t]
|
|
\begin{subfigure}{0.49\textwidth}
|
|
\includegraphics[height=0.65\textwidth,trim={0pt 0pt 0pt -35pt},clip]{fig3a}
|
|
\subcaption{\label{subfig:UHF_cplx_angle}}
|
|
\end{subfigure}
|
|
\begin{subfigure}{0.49\textwidth}
|
|
\includegraphics[height=0.65\textwidth]{fig3b}
|
|
\subcaption{\label{subfig:UHF_cplx_energy}}
|
|
\end{subfigure}
|
|
\caption{%
|
|
(\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$ in the Hubbard dimer for $U/t = 2$.
|
|
Symmetry-broken solutions correspond to individual sheets and become equivalent at
|
|
the \textit{quasi}-EP $\lambda_{\text{c}}$ (black dot).
|
|
The RHF solution is independent of $\lambda$, giving the constant plane at $\pi/2$.
|
|
(\subref{subfig:UHF_cplx_energy}) The corresponding HF energy surfaces show a non-analytic
|
|
point at the \textit{quasi}-EP.
|
|
\label{fig:HF_cplx}}
|
|
\end{figure*}
|
|
%%%%%%%%%%%%%%%%%
|
|
|
|
As the on-site repulsion is increased from 0, the HF approximation reaches a critical value at $U=2t$ where a symmetry-broken
|
|
UHF solution appears with a lower energy than the RHF one.
|
|
Note that the RHF wave function remains a genuine solution of the HF equations for $U \ge 2t$, but corresponds to a saddle point
|
|
of the HF energy rather than a minimum.
|
|
This critical point is analogous to the infamous Coulson--Fischer point identified in the hydrogen dimer.\cite{Coulson_1949}
|
|
For $U \ge 2t$, the optimal orbital rotation angles for the UHF orbitals become
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\ta^\text{UHF} & = \arctan (-\frac{2t}{\sqrt{U^2 - 4t^2}}),
|
|
\label{eq:ta_uhf}
|
|
\\
|
|
\tb^\text{UHF} & = \arctan (+\frac{2t}{\sqrt{U^2 - 4t^2}}),
|
|
\label{eq:tb_uhf}
|
|
\end{align}
|
|
\end{subequations}
|
|
with the corresponding UHF ground-state energy (Fig.~\ref{fig:HF_real})
|
|
\begin{equation}
|
|
E_\text{UHF} \equiv E_\text{HF}(\ta^\text{UHF}, \tb^\text{UHF}) = - \frac{2t^2}{U}.
|
|
\end{equation}
|
|
Time-reversal symmetry dictates that this UHF wave function must be degenerate with its spin-flipped \titou{pair?}, obtained
|
|
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
|
|
``oscillates'' between the two symmetry-broken configurations. \cite{GiulianiBook}
|
|
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}
|
|
|
|
%============================================================%
|
|
\subsection{Self-Consistency as a Perturbation} %OR {Complex adiabatic connection}
|
|
%============================================================%
|
|
|
|
% INTRODUCE PARAMETRISED FOCK HAMILTONIAN
|
|
The inherent non-linearity in the Fock eigenvalue problem arises from self-consistency
|
|
in the HF approximation, and is usually solved through an iterative approach.\cite{Roothaan_1951,Hall_1951}
|
|
Alternatively, the non-linear terms arising from the Coulomb and exchange operators can
|
|
be considered as a perturbation from the core Hamiltonian \eqref{eq:Hcore} by introducing the
|
|
transformation $U \to \lambda\, U$, giving the parametrised Fock operator
|
|
\begin{equation}
|
|
\Hat{f}(\vb{x} ; \lambda) = \Hat{h}(\vb{x}) + \lambda\, \Hat{v}_\text{HF}(\vb{x}).
|
|
\end{equation}
|
|
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.
|
|
|
|
% INTRODUCE COMPLEX ANALYTIC-CONTINUATION
|
|
For real $\lambda$, the self-consistent HF energies at given (real) $U$ and $t$ values
|
|
in the Hubbard dimer directly mirror the energies shown in Fig.~\ref{fig:HF_real},
|
|
with coalesence points at
|
|
\begin{equation}
|
|
\lambda_{\text{c}} = \pm \frac{2t}{U}.
|
|
\label{eq:scaled_fock}
|
|
\end{equation}
|
|
In contrast, when $\lambda$ becomes complex, the HF equations become non-Hermitian and
|
|
each HF solutions can be analytically continued for all $\lambda$ values using
|
|
the holomorphic HF approach.\cite{Hiscock_2014,Burton_2016,Burton_2018}
|
|
Remarkably, the coalescence point in this analytic continuation emerges as a
|
|
\textit{quasi}-EP on the real $\lambda$ axis (Fig.~\ref{fig:HF_cplx}), where
|
|
the different HF solutions become equivalent but not self-orthogonal.\cite{Burton_2019}
|
|
By analogy with perturbation theory, the regime where this \textit{quasi}-EP occurs
|
|
within $\lambda_{\text{c}} \le 1$ can be interpreted as an indication that
|
|
the symmetry-pure reference orbitals no longer provide a qualitatively
|
|
accurate representation for the true HF ground state at $\lambda = 1$.
|
|
For example, in the Hubbard dimer with $U > 2t$, one finds $\lambda_{\text{c}} < 1$ and the symmetry-pure orbitals
|
|
do not provide a good representation of the HF ground state.
|
|
In contrast, $U < 2t$ yields $\lambda_{\text{c}} > 1$ and corresponds to
|
|
the regime where the HF ground state is correctly represented by symmetry-pure orbitals.
|
|
|
|
% COMPLEX ADIABATIC CONNECTION
|
|
We have recently shown that the complex scaled Fock operator \eqref{eq:scaled_fock}
|
|
also allows states of different symmetries to be interconverted by following a well-defined
|
|
contour in the complex $\lambda$-plane.\cite{Burton_2019}
|
|
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}
|
|
a ground-state wave function can be ``morphed'' into an excited-state wave function
|
|
via a stationary path of HF solutions.
|
|
This novel approach to identifying excited-state wave functions demonstrates the fundamental
|
|
role of \textit{quasi}-EPs in determining the behaviour of the HF approximation.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{M{\o}ller--Plesset Perturbation Theory in the Complex Plane}
|
|
\label{sec:MP}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%=====================================================%
|
|
\subsection{Background Theory}
|
|
%=====================================================%
|
|
|
|
In electronic structure, the HF Hamiltonian \eqref{eq:HFHamiltonian} is often used as the zeroth-order Hamiltonian
|
|
to define M\o{}ller--Plesset (MP) perturbation theory.\cite{Moller_1934}
|
|
This approach can recover a large proportion of the electron correlation energy,\cite{Lowdin_1955a,Lowdin_1955b,Lowdin_1955c}
|
|
and provides the foundation for numerous post-HF approximations.
|
|
With the MP partitioning, the parametrised perturbation Hamiltonian becomes
|
|
\begin{multline}\label{eq:MPHamiltonian}
|
|
\hH(\lambda) =
|
|
\sum_{i}^{N} \qty[ - \frac{\grad_i^2}{2} - \sum_{A}^{M} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}} ]
|
|
\\
|
|
+ (1-\lambda) \sum_{i}^{N} v^{\text{HF}}(\vb{x}_i)
|
|
+ \lambda\sum_{i<j}^{N}\frac{1}{\abs{\vb{r}_i-\vb{r}_j}}.
|
|
\end{multline}
|
|
Any set of orbitals can be used to define the HF Hamiltonian, although either the RHF or UHF orbitals are usually chosen to
|
|
define the RMP or UMP series respectively.
|
|
The MP energy at a given order $n$ (\ie, MP$n$) is then defined as
|
|
\begin{equation}
|
|
E_{\text{MP}n}= \sum_{k=0}^n E_{\text{MP}}^{(k)},
|
|
\end{equation}
|
|
where $E_{\text{MP}}^{(k)}$ is the $k$th-order MP correction and
|
|
\begin{equation}
|
|
E_{\text{MP1}} = E_{\text{MP}}^{(0)} + E_{\text{MP}}^{(1)} = E_\text{HF}.
|
|
\end{equation}
|
|
The second-order MP2 energy is given by
|
|
\begin{equation}\label{eq:EMP2}
|
|
E_{\text{MP2}} = \frac{1}{4} \sum_{ij} \sum_{ab} \frac{\abs{\mel{ij}{}{ab}}^2}{\epsilon_i + \epsilon_j - \epsilon_a - \epsilon_b},
|
|
\end{equation}
|
|
where $\mel{pq}{}{rs} = \braket{pq}{rs} - \braket{pq}{sr}$ are the anti-symmetrised two-electron integrals
|
|
in the molecular spin-orbital basis\cite{Gill_1994}
|
|
\begin{equation}
|
|
\braket{pq}{rs}
|
|
= \iint \dd\vb{x}_1\dd\vb{x}_2
|
|
\frac{\phi^{*}_p(\vb{x}_1)\phi^{*}_q(\vb{x}_2)\phi^{\vphantom{*}}_r(\vb{x}_1)\phi^{\vphantom{*}}_s(\vb{x}_2)}%
|
|
{\abs{\vb{r}_1 - \vb{r}_2}}.
|
|
\end{equation}
|
|
|
|
While most practical calculations generally consider only the MP2 or MP3 approximations, higher order terms can
|
|
be computed to understand the convergence of the MP$n$ series.\cite{Handy_1985}
|
|
\textit{A priori}, there is no guarantee that this series will provide the smooth convergence that is desirable for a
|
|
systematically improvable theory.
|
|
In fact, when the reference HF wave function is a poor approximation to the exact wave function,
|
|
for example in multi-configurational systems, MP theory can yield highly oscillatory,
|
|
slowly convergent, or catastrophically divergent results.\cite{Gill_1986,Gill_1988,Handy_1985,Lepetit_1988,Leininger_2000}
|
|
Furthermore, the convergence properties of the MP series can depend strongly on the choice of restricted or
|
|
unrestricted reference orbitals.
|
|
|
|
Although practically convenient for electronic structure calculations, the MP partitioning is not
|
|
the only possibility and alternative partitionings have been considered including:
|
|
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}
|
|
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
|
|
to the convergence properties of the MP expansion.
|
|
|
|
%=====================================================%
|
|
\subsection{Early Studies of M{\o}ller--Plesset Convergence} % in Molecular Systems}
|
|
%=====================================================%
|
|
|
|
% GENERAL DESIRE FOR WELL-BEHAVED CONVERGENCE AND LOW-ORDER TERMS
|
|
Among the most desirable properties of any electronic structure technique is the existence of
|
|
a systematic route to increasingly accurate energies.
|
|
In the context of MP theory, one would like a monotonic convergence of the perturbation
|
|
series towards the exact energy such that the accuracy increases as each term in the series is added.
|
|
If such well-behaved convergence can be established, then our ability to compute individual
|
|
terms in the series becomes the only barrier to computing the exact correlation in a finite basis set.
|
|
Unfortunately, the computational scaling of each term in the MP series increases with the perturbation
|
|
order, and practical calculations must rely on fast convergence
|
|
to obtain high-accuracy results using only the lowest order terms.
|
|
|
|
% INITIAL POSITIVITY AROUND THE CONVERGENCE PROPERTIES AND EARLY WORK SCOPE
|
|
MP theory was first introduced to quantum chemistry through the pioneering
|
|
works of Bartlett \etal\ in the context of many-body perturbation theory,\cite{Bartlett_1975}
|
|
and Pople and co-workers in the context of determinantal expansions.\cite{Pople_1976,Pople_1978}
|
|
Early implementations were restricted to the fourth-order MP4 approach that was considered
|
|
to offer state-of-the-art quantitative accuracy.\cite{Pople_1978,Krishnan_1980}
|
|
However, it was quickly realised that the MP series often demonstrated very slow, oscillatory,
|
|
or erratic convergence, with the UMP series showing particularly slow convergence.\cite{Laidig_1985,Knowles_1985,Handy_1985}
|
|
For example, RMP5 is worse than RMP4 for predicting the homolytic barrier fission of \ce{He2^2+} using a minimal basis set,
|
|
while the UMP series monotonically converges but becomes increasingly slow beyond UMP5.\cite{Gill_1986}
|
|
The first examples of divergent MP series were observed in the \ce{N2} and \ce{F2}
|
|
diatomics, where low-order RMP and UMP expansions give qualitatively wrong binding curves.\cite{Laidig_1987}
|
|
|
|
% SLOW UMP CONVERGENCE AND SPIN CONTAMINATION
|
|
The divergence of RMP expansions for stretched bonds can be easily understood from two perspectives.\cite{Gill_1988a}
|
|
Firstly, the exact wave function becomes increasingly multi-configurational as the bond is stretched, and the
|
|
RHF wave function no longer provides a qualitatively correct reference for the perturbation expansion.
|
|
Secondly, the energy gap between the bonding and antibonding orbitals associated with the stretch becomes
|
|
increasingly small at larger bond lengths, leading to a divergence, for example, in the second-order MP correction \eqref{eq:EMP2}.
|
|
In contrast, the origin of slow UMP convergence is less obvious as the reference UHF energy remains
|
|
qualitatively correct at large bond lengths and the orbital degeneracy is avoided.
|
|
Furthermore, this slow convergence can also be observed in molecules with a UHF ground state at the equilibrium
|
|
geometry (\eg, \ce{CN-}), suggesting a more fundamental link with spin-contamination
|
|
in the reference wave function.\cite{Nobes_1987}
|
|
|
|
Using the UHF framework allows the singlet ground state wave function to mix with triplet wave functions,
|
|
leading to spin contamination where the wave function is no longer an eigenfunction of the $\Hat{\cS}^2$ operator.
|
|
The link between slow UMP convergence and this spin-contamination was first systematically investigated
|
|
by Gill \etal\ using the minimal basis \ce{H2} model.\cite{Gill_1988}
|
|
In this work, the authors %compared titou{the UMP series with the exact RHF- and UHF-based FCI expansions (T2: I don't understand this)} and
|
|
identified that the slow UMP convergence arises from its failure to correctly predict the amplitude of the
|
|
low-lying double excitation.
|
|
This erroneous description of the double excitation amplitude has the same origin as the spin-contamination in the reference
|
|
UHF wave function, creating the first direct link between spin-contamination and slow UMP convergence.\cite{Gill_1988}
|
|
%
|
|
% LEPETIT CHAT
|
|
Lepetit \etal\ later analysed the difference between perturbation convergence using the UMP
|
|
and EN partitionings. \cite{Lepetit_1988}
|
|
They argued that the slow UMP convergence for stretched molecules arises from
|
|
(i) the fact that the MP denominator (see Eq.~\ref{eq:EMP2})
|
|
tends to a constant value instead of vanishing, and (ii) the slow convergence of contributions from the
|
|
singly-excited configurations that strongly couple to the doubly-excited configurations and first
|
|
appear at fourth-order.\cite{Lepetit_1988}
|
|
Drawing these ideas together, we believe that slow UMP convergence occurs because the single excitations must focus on removing
|
|
spin-contamination from the reference wave function, limiting their ability to fine-tune the amplitudes of the higher
|
|
excitations that capture the correlation energy.
|
|
|
|
% SPIN-PROJECTION SCHEMES
|
|
A number of spin-projected extensions have been derived to reduce spin-contamination in the wave function
|
|
and overcome the slow UMP convergence.
|
|
Early versions of these theories, introduced by Schlegel \cite{Schlegel_1986, Schlegel_1988} or
|
|
Knowles and Handy,\cite{Knowles_1988a,Knowles_1988b} exploited the ``projection-after-variation'' philosophy,
|
|
where the spin-projection is applied directly to the UMP expansion.
|
|
These methods succeeded in accelerating the convergence of the projected MP series and were
|
|
considered as highly effective methods for capturing the electron correlation at low computational cost.\cite{Knowles_1988b}
|
|
However, the use of projection-after-variation leads to gradient discontinuities in the vicinity of the UHF symmetry-breaking point,
|
|
and can result in spurious minima along a molecular binding curve.\cite{Schlegel_1986,Knowles_1988a}
|
|
More recent formulations of spin-projected perturbations theories have considered the
|
|
``variation-after-projection'' framework using alternative definitions of the reference
|
|
Hamiltonian.\cite{Tsuchimochi_2014,Tsuchimochi_2019}
|
|
These methods yield more accurate spin-pure energies without
|
|
gradient discontinuities or spurious minima.
|
|
|
|
%==========================================%
|
|
\subsection{Spin-Contamination in the Hubbard Dimer}
|
|
\label{sec:spin_cont}
|
|
%==========================================%
|
|
|
|
%%% FIG 2 %%%
|
|
\begin{figure*}
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.75\textwidth]{fig4a}
|
|
\subcaption{\label{subfig:RMP_3.5} $U/t = 3.5$}
|
|
\end{subfigure}
|
|
%
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.75\textwidth]{fig4b}
|
|
\subcaption{\label{subfig:RMP_cvg}}
|
|
\end{subfigure}
|
|
%
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.75\textwidth]{fig4c}
|
|
\subcaption{\label{subfig:RMP_4.5} $U/t = 4.5$}
|
|
\end{subfigure}
|
|
\caption{
|
|
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 $\lambda$.
|
|
\label{fig:RMP}}
|
|
\end{figure*}
|
|
|
|
The behaviour of the RMP and UMP series observed in \ce{H2} can also be illustrated by considering
|
|
the analytic Hubbard dimer with a complex-valued perturbation strength.
|
|
In this system, the stretching of the \ce{H\bond{-}H} bond is directly mirrored by an increase in the ratio $U/t$.
|
|
Using the ground-state RHF reference orbitals leads to the parametrised RMP Hamiltonian
|
|
\begin{widetext}
|
|
\begin{equation}
|
|
\label{eq:H_RMP}
|
|
\bH_\text{RMP}\qty(\lambda) =
|
|
\begin{pmatrix}
|
|
-2t + U - \lambda U/2 & 0 & 0 & \lambda U/2 \\
|
|
0 & U - \lambda U/2 & \lambda U/2 & 0 \\
|
|
0 & \lambda U/2 & U - \lambda U/2 & 0 \\
|
|
\lambda U/2 & 0 & 0 & 2t + U - \lambda U/2 \\
|
|
\end{pmatrix},
|
|
\end{equation}
|
|
\end{widetext}
|
|
which yields the ground-state energy
|
|
\begin{equation}
|
|
\label{eq:E0MP}
|
|
E_{-}(\lambda) = U - \frac{\lambda U}{2} - \frac{1}{2} \sqrt{(4t)^2 + \lambda ^2 U^2}.
|
|
\end{equation}
|
|
From this expression, the EPs can be identified as $\lep = \pm \i 4t / U$,
|
|
giving the radius of convergence
|
|
\begin{equation}
|
|
\rc = \abs{\frac{4t}{U}}.
|
|
\end{equation}
|
|
Remarkably, these EPs are identical to the exact EPs discussed in Sec.~\ref{sec:example}.
|
|
The Taylor expansion of the RMP energy can then be evaluated to obtain the $k$th-order MP correction
|
|
\begin{equation}
|
|
E_\text{RMP}^{(k)} = U \delta_{0,k} - \frac{1}{2} \frac{U^k}{(4t)^{k-1}} \mqty( 1/2 \\ k/2).
|
|
\end{equation}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% RADIUS OF CONVERGENCE PLOTS
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
The RMP series is convergent for $U = 3.5\,t$ with $\rc > 1$, as illustrated for the individual terms at each order
|
|
of perturbation in Fig.~\ref{subfig:RMP_cvg}.
|
|
In contrast, for $U = 4.5t$ one finds $\rc < 1$, and the RMP series becomes divergent.
|
|
The corresponding Riemann surfaces for $U = 3.5\,t$ and $4.5\,t$ are shown in Figs.~\ref{subfig:RMP_3.5} and
|
|
\ref{subfig:RMP_4.5}, respectively, with the single EP at $\lep$ (black dot) and the radius of convergence indicated
|
|
by the vertical cylinder of unit radius.
|
|
For the divergent case, the $\lep$ lies inside this cylinder of convergence, while in the convergent case $\lep$ lies
|
|
outside this cylinder.
|
|
In both cases, the EP connects the ground state with the doubly-excited state, and thus the convergence behaviour
|
|
for the two states using the ground-state RHF orbitals is identical.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% RADIUS OF CONVERGENCE PLOTS
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{figure}[htb]
|
|
\includegraphics[width=\linewidth]{fig5}
|
|
\caption{
|
|
Radius of convergence $r_c$ for the RMP ground state (red), the UMP ground state (blue), and the UMP excited state (orange)
|
|
series as functions of the ratio $U/t$.
|
|
\label{fig:RadConv}}
|
|
\end{figure}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
|
|
%%% FIG 3 %%%
|
|
\begin{figure*}
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.75\textwidth]{fig6a}
|
|
\subcaption{\label{subfig:UMP_3} $U/t = 3$}
|
|
\end{subfigure}
|
|
%
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.75\textwidth]{fig6b}
|
|
\subcaption{\label{subfig:UMP_cvg}}
|
|
\end{subfigure}
|
|
%
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.75\textwidth]{fig6c}
|
|
\subcaption{\label{subfig:UMP_7} $U/t = 7$}
|
|
\end{subfigure} \caption{
|
|
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 $\lambda$.
|
|
\label{fig:UMP}}
|
|
\end{figure*}
|
|
|
|
The behaviour of the UMP series is more subtle than the RMP series as the spin-contamination in the wave function
|
|
introduces additional coupling between the singly- and doubly-excited configurations.
|
|
Using the ground-state UHF reference orbitals in the Hubbard dimer yields the parametrised UMP Hamiltonian
|
|
\begin{widetext}
|
|
\begin{equation}
|
|
\label{eq:H_UMP}
|
|
\bH_\text{UMP}\qty(\lambda) =
|
|
\begin{pmatrix}
|
|
-2t^2 \lambda/U & 0 & 0 & 2t^2 \lambda/U \\
|
|
0 & U - 2t^2 \lambda/U & 2t^2\lambda/U & 2t \sqrt{U^2 - (2t)^2} \lambda/U \\
|
|
0 & 2t^2\lambda/U & U - 2t^2 \lambda/U & -2t \sqrt{U^2 - (2t)^2} \lambda/U \\
|
|
2t^2 \lambda/U & 2t \sqrt{U^2 - (2t)^2} \lambda/U & -2t \sqrt{U^2 - (2t)^2} \lambda/U & 2U(1-\lambda) + 6t^2\lambda/U \\
|
|
\end{pmatrix}.
|
|
\end{equation}
|
|
\end{widetext}
|
|
While a closed-form expression for the ground-state energy exists, it is cumbersome and we eschew reporting it.
|
|
Instead, the radius of convergence of the UMP series can be obtained numerically as a function of $U/t$, as shown
|
|
in Fig.~\ref{fig:RadConv}.
|
|
These numerical values reveal that the UMP ground-state series has $\rc > 1$ for all $U/t$ and always converges.
|
|
However, in the strong correlation limit (large $U/t$), this radius of convergence tends to unity, indicating that
|
|
the convergence of the corresponding UMP series becomes increasingly slow.
|
|
Furthermore, the doubly-excited state using the ground-state UHF orbitals has $\rc < 1$ for almost any value
|
|
of $U/t$, reaching the limiting value of $1/2$ for $U/t \to \infty$. Hence, the
|
|
excited-state UMP series will always diverge.
|
|
|
|
% DISCUSSION OF UMP RIEMANN SURFACES
|
|
The convergence behaviour can be further elucidated by considering the full structure of the UMP energies
|
|
in the complex $\lambda$-plane (see Figs.~\ref{subfig:UMP_3} and \ref{subfig:UMP_7}).
|
|
These Riemann surfaces are illustrated for $U = 3t$ and $7t$ alongside the perturbation terms at each order
|
|
in Fig.~\ref{subfig:UMP_cvg}.
|
|
At $U = 3t$, the RMP series is convergent, while RMP becomes divergent for $U=7t$.
|
|
The ground-state UMP expansion is convergent in both cases, although the rate of convergence is significantly slower
|
|
for larger $U/t$ as the radius of convergence becomes increasingly close to one (Fig.~\ref{fig:RadConv}).
|
|
|
|
% EFFECT OF SYMMETRY BREAKING
|
|
As the UHF orbitals break the spin symmetry, new coupling terms emerge between the electronic states that
|
|
cause fundamental changes to the structure of EPs in the complex $\lambda$-plane.
|
|
For example, while the RMP energy shows only one EP between the ground state and
|
|
the doubly-excited state (Fig.~\ref{fig:RMP}), the UMP energy has two EPs: one connecting the ground state with the
|
|
singly-excited open-shell singlet, and the other connecting this single excitation to the
|
|
doubly-excited second excitation (Fig.~\ref{fig:UMP}).
|
|
This new ground-state EP always appears outside the unit cylinder and guarantees convergence of the ground-state energy.
|
|
However, the excited-state EP is moved within the unit cylinder and causes the
|
|
convergence of the excited-state UMP series to deteriorate.
|
|
Our interpretation of this effect is that the symmetry-broken orbital optimisation has redistributed the strong
|
|
coupling between the ground- and doubly-excited states into weaker couplings between all states, and has thus
|
|
sacrificed convergence of the excited-state series so that the ground-state convergence can be maximised.
|
|
|
|
Since the UHF ground state already provides a good approximation to the exact energy, the ground-state sheet of
|
|
the UMP energy is relatively flat and the corresponding EP in the Hubbard dimer always lies outside the unit cylinder.
|
|
The slow convergence observed in stretched \ce{H2}\cite{Gill_1988} can then be seen as this EP
|
|
moves increasingly close to the unit cylinder at large $U/t$ and $\rc$ approaches one (from above).
|
|
Furthermore, the majority of the UMP expansion in this regime is concerned with removing spin-contamination from the wave
|
|
function rather than improving the energy.
|
|
It is well-known that the spin-projection needed to remove spin-contamination can require non-linear combinations
|
|
of highly-excited determinants,\cite{Lowdin_1955c} and thus it is not surprising that this process proceeds
|
|
very slowly as the perturbation order is increased.
|
|
|
|
%==========================================%
|
|
\subsection{Classifying Types of Convergence} % Behaviour} % Further insights from a two-state model}
|
|
%==========================================%
|
|
|
|
% CREMER AND HE
|
|
As computational implementations of higher-order MP terms improved, the systematic investigation
|
|
of convergence behaviour in a broader class of molecules became possible.
|
|
Cremer and He introduced an efficient MP6 approach and used it to analyse the RMP convergence of
|
|
29 atomic and molecular systems with respect to the FCI energy.\cite{Cremer_1996}
|
|
They established two general classes: ``class A'' systems that exhibit monotonic convergence;
|
|
and ``class B'' systems for which convergence is erratic after initial oscillations.
|
|
By analysing the different cluster contributions to the MP energy terms, they proposed that
|
|
class A systems generally include well-separated and weakly correlated electron pairs, while class B systems
|
|
are characterised by dense electron clustering in one or more spatial regions.\cite{Cremer_1996}
|
|
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.
|
|
|
|
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}
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\Delta E_{\text{A}}
|
|
&= \Emp^{(2)} + \Emp^{(3)} + \Emp^{(4)}
|
|
+ \frac{\Emp^{(5)}}{1 - (\Emp^{(6)} / \Emp^{(5)})},
|
|
\\[5pt]
|
|
\Delta E_{\text{B}}
|
|
&= \Emp^{(2)} + \Emp^{(3)} + \qty(\Emp^{(4)} + \Emp^{(5)}) \exp(\Emp^{(6)} / \Emp^{(5)}).
|
|
\end{align}
|
|
\end{subequations}
|
|
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 Sec.~\ref{sec:Resummation}, we consider more advanced extrapolation routines that take account of EPs in the complex $\lambda$-plane.
|
|
|
|
In the late 90's, Olsen \etal\ discovered an even more concerning behaviour of the MP series. \cite{Olsen_1996}
|
|
They showed that the series could be divergent even in systems that were considered to be well understood,
|
|
such as \ce{Ne} or the \ce{HF} molecule. \cite{Olsen_1996, Christiansen_1996}
|
|
Cremer and He had already studied these two systems and classified them as \textit{class B} systems.\cite{Cremer_1996}
|
|
However, Olsen and co-workers performed their analysis in larger basis sets containing diffuse functions,
|
|
finding that the corresponding MP series becomes divergent at (very) high order.
|
|
The discovery of this divergent behaviour is particularly worrying as large basis sets
|
|
are required to get meaningful and accurate energies.\cite{Loos_2019d,Giner_2019}
|
|
Furthermore, diffuse functions are particularly important for anions and/or Rydberg excited states, where the wave function
|
|
is inherently more diffuse than the ground state.\cite{Loos_2018a,Loos_2020a}
|
|
|
|
Olsen \etal\ investigated the causes of these divergences and the different types of convergence by
|
|
analysing the relation between the dominant singularity (\ie, the closest singularity to the origin)
|
|
and the convergence behaviour of the series.\cite{Olsen_2000}
|
|
Their analysis is based on Darboux's theorem: \cite{Goodson_2011}
|
|
\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. ''}
|
|
\end{quote}
|
|
Following this theory, a singularity in the unit circle is designated as an intruder state,
|
|
with a front-door (or back-door) intruder state if the real part of the singularity is positive (or negative).
|
|
|
|
Using their observations in Ref.~\onlinecite{Olsen_1996}, Olsen and collaborators proposed
|
|
a simple method that performs a scan of the real axis to detect the avoided crossing responsible
|
|
for the dominant singularities in the complex plane. \cite{Olsen_2000}
|
|
By modelling this avoided crossing using a two-state Hamiltonian, one can obtain an approximation for
|
|
the dominant singularities as the EPs of the two-state matrix
|
|
\begin{equation}
|
|
\label{eq:Olsen_2x2}
|
|
\underbrace{\mqty(\alpha & \delta \\ \delta & \beta )}_{\bH}
|
|
= \underbrace{\mqty(\alpha + \alpha_{\text{s}} & 0 \\ 0 & \beta + \beta_{\text{s}} )}_{\bH^{(0)}}
|
|
+ \underbrace{\mqty( -\alpha_{\text{s}} & \delta \\ \delta & - \beta_{\text{s}})}_{\bV},
|
|
\end{equation}
|
|
where the diagonal matrix is the unperturbed Hamiltonian matrix $\bH^{(0)}$ with level shifts
|
|
$\alpha_{\text{s}}$ and $\beta_{\text{s}}$, and $\bV$ represents the perturbation.
|
|
|
|
The authors first considered molecules with low-lying doubly-excited states with the same spatial
|
|
and spin symmetry as the ground state. \cite{Olsen_2000}
|
|
In these systems, the exact wave function has a non-negligible contribution from the doubly-excited states,
|
|
and thus the low-lying excited states are likely to become intruder states.
|
|
For \ce{CH_2} in a diffuse, yet rather small basis set, the series is convergent at least up to the 50th order, and
|
|
the dominant singularity lies close (but outside) the unit circle, causing slow convergence of the series.
|
|
These intruder-state effects are analogous to the EP that dictates the convergence behaviour of
|
|
the RMP series for the Hubbard dimer (Fig.~\ref{fig:RMP}).
|
|
Furthermore, the authors demonstrated that the divergence for \ce{Ne} is due to a back-door intruder state
|
|
that arise when the ground state undergoes sharp avoided crossings with highly diffuse excited states.
|
|
This divergence is related to a more fundamental critical point in the MP energy surface that we will
|
|
discuss in Sec.~\ref{sec:MP_critical_point}.
|
|
|
|
Finally, Ref.~\onlinecite{Olsen_1996} proved that the extrapolation formulas of Cremer and He \cite{Cremer_1996}
|
|
are not mathematically motivated when considering the complex singularities causing the divergence, and therefore
|
|
cannot be applied for all systems.
|
|
For example, the \ce{HF} molecule contains both back-door intruder states and low-lying doubly-excited states that
|
|
result in alternating terms up to 10th order.
|
|
The series becomes monotonically convergent at higher orders since
|
|
the two pairs of singularities are approximately the same distance from the origin.
|
|
|
|
More recently, this two-state model has been extended to non-symmetric Hamiltonians as\cite{Olsen_2019}
|
|
\begin{equation}
|
|
\underbrace{\mqty(\alpha & \delta_1 \\ \delta_2 & \beta)}_{\bH} = \underbrace{\mqty(\alpha & 0 \\ 0 & \beta + \gamma )}_{\bH^{(0)}} + \underbrace{\mqty( 0 & \delta_2 \\ \delta_1 & - \gamma)}_{\bV}.
|
|
\end{equation}
|
|
This extension allows various choices of perturbation to be analysed, including coupled cluster
|
|
perturbation expansions \cite{Pawlowski_2019a,Pawlowski_2019b,Pawlowski_2019c,Pawlowski_2019d,Pawlowski_2019e}
|
|
and other non-Hermitian perturbation methods.
|
|
Note that new forms of perturbation expansions only occur when the sign of $\delta_1$ and $\delta_2$ differ.
|
|
Using this 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.
|
|
%
|
|
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.
|
|
This analysis highlights the importance of the primary critical point in controlling the high-order convergence,
|
|
regardless of whether this point is inside or outside the complex unit circle. \cite{Handy_1985,Olsen_2000}
|
|
|
|
%=======================================
|
|
\subsection{M{\o}ller--Plesset Critical Point}
|
|
\label{sec:MP_critical_point}
|
|
%=======================================
|
|
|
|
% STILLINGER INTRODUCES THE CRITICAL POINT
|
|
In the early 2000's, Stillinger reconsidered the mathematical origin behind the divergent series with odd-even
|
|
sign alternation.\cite{Stillinger_2000}
|
|
This type of convergence behaviour corresponds to Cremer and He's class B systems with closely spaced
|
|
electron pairs and includes \ce{Ne}, \ce{HF}, \ce{F-}, and \ce{H2O}.\cite{Cremer_1996}
|
|
Stillinger proposed that these series diverge due to a dominant singularity
|
|
on the negative real $\lambda$ axis, corresponding to a multielectron autoionisation threshold.\cite{Stillinger_2000}
|
|
To understand Stillinger's argument, consider the parametrised MP Hamiltonian in the form
|
|
\begin{multline}
|
|
\label{eq:HamiltonianStillinger}
|
|
\hH(\lambda) =
|
|
\sum_{i}^{\Ne} \Bigg[
|
|
\overbrace{-\frac{1}{2}\grad_i^2
|
|
- \sum_{A}^{\Nn} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}}}^{\text{independent of $\lambda$}}
|
|
\\
|
|
+ \underbrace{(1-\lambda)v^{\text{HF}}(\vb{x}_i)}_{\text{repulsive for $\lambda < 1$}}
|
|
+ \underbrace{\lambda\sum_{i<j}^{\Ne}\frac{1}{|\vb{r}_i-\vb{r}_j|}}_{\text{attractive for $\lambda < 0$}}
|
|
\Bigg].
|
|
\end{multline}
|
|
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 negative critical point $\lc$ 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}
|
|
While these critical points are singularities on the real axis, their exact mathematical form is difficult
|
|
to identify and remains an open question.
|
|
|
|
% CLASSIFICATIONS BY GOODSOON AND SERGEEV
|
|
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 systems corresponds to a EP with a positive real component,
|
|
where the magnitude of the imaginary component controls the oscillations in the signs of successive MP
|
|
terms.\cite{Goodson_2000a,Goodson_2000b}
|
|
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
|
|
The existence of the MP critical point can also explain why the divergence observed by Olsen \etal\ in the \ce{Ne} atom
|
|
and the \ce{HF} molecule occurred 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 occupying the ghost atom orbitals.\cite{Sergeev_2005}
|
|
This effect explains the origin of the divergence in the \ce{HF} molecule as the fluorine valence electrons jump to hydrogen at
|
|
a sufficiently negative $\lambda$ value.\cite{Sergeev_2005}
|
|
Furthermore, the two-state model of Olsen and collaborators \cite{Olsen_2000} was simply too minimal to understand the complexity of
|
|
divergences caused by the MP critical point.
|
|
|
|
% RELATIONSHIP TO QUANTUM PHASE TRANSITION
|
|
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).%
|
|
\cite{Heiss_1988,Heiss_2002,Borisov_2015,Sindelka_2017,CarrBook,Vojta_2003,SachdevBook,GilmoreBook}
|
|
Meanwhile, as an avoided crossing becomes increasingly sharp, the corresponding EPs move increasingly close to the real axis, eventually forming a critical point.
|
|
The existence of an EP \emph{on} the real axis is therefore diagnostic of a QPT.\cite{Cejnar_2005, Cejnar_2007a}
|
|
Since the MP critical point corresponds to a singularity on the real $\lambda$ axis, it can immediately be
|
|
recognised as a QPT with respect to varying the perturbation parameter $\lambda$.
|
|
However, a conventional QPT can only occur in the thermodynamic limit, which here is analogous to the complete
|
|
basis set limit.\cite{Kais_2006}
|
|
The MP critical point and corresponding $\beta$ singularities in a finite basis must therefore be modelled by pairs of EPs
|
|
that tend towards the real axis, exactly as described by Sergeev \etal\cite{Sergeev_2005}
|
|
In contrast, $\alpha$ singularities correspond to large avoided crossings that are indicative of low-lying excited
|
|
states which share the symmetry of the ground state,\cite{Goodson_2004} and are thus not manifestations of a QPT.
|
|
|
|
%=======================================
|
|
\subsection{Critical Points in the Hubbard Dimer}
|
|
\label{sec:critical_point_hubbard}
|
|
%=======================================
|
|
|
|
%------------------------------------------------------------------%
|
|
% Figure on the RMP critical point
|
|
%------------------------------------------------------------------%
|
|
\begin{figure*}[t]
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.75\textwidth]{fig7a}
|
|
\subcaption{\label{subfig:rmp_cp}}
|
|
\end{subfigure}
|
|
%
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.75\textwidth]{fig7b}
|
|
\subcaption{\label{subfig:rmp_cp_surf}}
|
|
\end{subfigure}
|
|
%
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.75\textwidth]{fig7c}
|
|
\subcaption{\label{subfig:rmp_ep_to_cp}}
|
|
\end{subfigure}
|
|
\caption{%
|
|
RMP critical point using the asymmetric Hubbard dimer with $\epsilon = 2.5 U$.
|
|
(\subref{subfig:rmp_cp}) Exact critical points with $t=0$ occur on the negative real $\lambda$ axis (dashed).
|
|
(\subref{subfig:rmp_cp_surf}) Modelling a finite basis using $t=0.1$ yields complex-conjugate EPs close to the
|
|
real axis, giving a sharp avoided crossing on the real axis (solid).
|
|
(\subref{subfig:rmp_ep_to_cp}) Convergence of the ground-state EP onto the real axis in the limit $t \to 0$.
|
|
\label{fig:RMP_cp}}
|
|
\end{figure*}
|
|
%------------------------------------------------------------------%
|
|
|
|
% INTRODUCING THE MODEL
|
|
The simplified site basis of the Hubbard dimer makes explicilty modelling the ionisation continuum impossible.
|
|
Instead, we can use an asymmetric version of the Hubbard dimer \cite{Carrascal_2015,Carrascal_2018}
|
|
where we consider one of the sites as a ``ghost atom'' that acts as a
|
|
destination for ionised electrons being originally localised on the other site.
|
|
To mathematically model this scenario in this asymmetric Hubbard dimer, we introduce a one-electron potential $-\epsilon$ on the left site to
|
|
represent the attraction between the electrons and the model ``atomic'' nucleus, where we define $\epsilon > 0$.
|
|
The reference Slater determinant for a doubly-occupied atom can be represented using RHF
|
|
orbitals [see Eq.~\eqref{eq:RHF_orbs}] with $\theta_{\alpha}^{\text{RHF}} = \theta_{\beta}^{\text{RHF}} = 0$,
|
|
which corresponds to strictly localising the two electrons on the left site.
|
|
%and energy
|
|
%\begin{equation}
|
|
% E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon).
|
|
%\end{equation}
|
|
With this representation, the parametrised asymmetric RMP Hamiltonian becomes
|
|
\begin{widetext}
|
|
\begin{equation}
|
|
\label{eq:H_asym}
|
|
\bH_\text{asym}\qty(\lambda) =
|
|
\begin{pmatrix}
|
|
2(U-\epsilon) - \lambda U & -\lambda t & -\lambda t & 0 \\
|
|
-\lambda t & (U-\epsilon) - \lambda U & 0 & -\lambda t \\
|
|
-\lambda t & 0 & (U-\epsilon) -\lambda U & -\lambda t \\
|
|
0 & -\lambda t & -\lambda t & \lambda U \\
|
|
\end{pmatrix}.
|
|
\end{equation}
|
|
\end{widetext}
|
|
|
|
% DERIVING BEHAVIOUR OF THE CRITICAL SITE
|
|
For the ghost site to perfectly represent ionised electrons, the hopping term between the two sites must vanish (\ie, $t=0$).
|
|
This limit corresponds to the dissociative regime in the asymmetric Hubbard dimer as discussed in Ref.~\onlinecite{Carrascal_2018},
|
|
and the RMP energies become
|
|
\begin{subequations}
|
|
\begin{align}
|
|
E_{-} &= 2(U - \epsilon) - \lambda U,
|
|
\\
|
|
E_{\text{S}} &= (U - \epsilon) - \lambda U,
|
|
\\
|
|
E_{+} &= U \lambda,
|
|
\end{align}
|
|
\end{subequations}
|
|
as shown in Fig.~\ref{subfig:rmp_cp} (dashed lines).
|
|
The RMP critical point then corresponds to the intersection $E_{-} = E_{+}$, giving the critical $\lambda$ value
|
|
\begin{equation}
|
|
\lc = 1 - \frac{\epsilon}{U}.
|
|
\end{equation}
|
|
Clearly the radius of convergence $\rc = \abs{\lc}$ is controlled directly by the ratio $\epsilon / U$,
|
|
with a convergent RMP series occurring for $\epsilon > 2 U$.
|
|
The on-site repulsion $U$ controls the strength of the HF potential localised around the ``atomic site'', with a
|
|
stronger repulsion encouraging the electrons to be ionised at a less negative value of $\lambda$.
|
|
Large $U$ can be physically interpreted as strong electron repulsion effects in electron dense molecules.
|
|
In contrast, smaller $\epsilon$ gives a weaker attraction to the atomic site,
|
|
representing strong screening of the nuclear attraction by core and valence electrons,
|
|
and again a less negative $\lambda$ is required for ionisation to occur.
|
|
Both of these factors are common in atoms on the right-hand side of the periodic table, \eg\ \ce{F},
|
|
\ce{O}, \ce{Ne}.
|
|
Molecules containing these atoms are therefore often class $\beta$ systems with
|
|
a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_2006}
|
|
|
|
% EXACT VERSUS APPROXIMATE
|
|
The critical point in the exact case $t=0$ lies on the negative real $\lambda$ axis (Fig.~\ref{subfig:rmp_cp}: dashed lines),
|
|
mirroring the behaviour of a quantum phase transition.\cite{Kais_2006}
|
|
However, in practical calculations performed with a finite basis set, the critical point is modelled as a cluster
|
|
of branch points close to the real axis.
|
|
The use of a finite basis can be modelled in the asymmetric dimer by making the second site a less
|
|
idealised destination for the ionised electrons with a non-zero (yet small) hopping term $t$.
|
|
Taking the value $t=0.1$ (Fig.~\ref{subfig:rmp_cp}: solid lines), the critical point becomes a
|
|
sharp avoided crossing with a complex-conjugate pair of EPs close to the real axis (Fig.~\ref{subfig:rmp_cp_surf}).
|
|
In the limit $t \to 0$, these EPs approach the real axis (Fig.~\ref{subfig:rmp_ep_to_cp}),
|
|
mirroring Sergeev's discussion on finite basis
|
|
set representations of the MP critical point.\cite{Sergeev_2006}
|
|
|
|
%------------------------------------------------------------------%
|
|
% Figure on the UMP critical point
|
|
%------------------------------------------------------------------%
|
|
\begin{figure*}[t]
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.75\textwidth,trim={0pt 5pt -10pt 15pt},clip]{fig8a}
|
|
\subcaption{\label{subfig:ump_cp}}
|
|
\end{subfigure}
|
|
%
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.75\textwidth]{fig8b}
|
|
\subcaption{\label{subfig:ump_cp_surf}}
|
|
\end{subfigure}
|
|
%
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.75\textwidth]{fig8c}
|
|
\subcaption{\label{subfig:ump_ep_to_cp}}
|
|
\end{subfigure}
|
|
% \includegraphics[height=0.65\textwidth,trim={0pt 5pt 0pt 15pt}, clip]{ump_critical_point}
|
|
\caption{%
|
|
The UMP ground-state EP in the symmetric Hubbard dimer becomes a critical point in the strong correlation limit (\ie, large $U/t$).
|
|
(\subref{subfig:ump_cp}) As $U/t$ increases, the avoided crossing on the real $\lambda$ axis
|
|
becomes increasingly sharp.
|
|
(\subref{subfig:ump_cp_surf}) Complex energy surfaces for $U = 5t$.
|
|
(\subref{subfig:ump_ep_to_cp}) Convergence of the EPs at $\lep$ onto the real axis for $U/t \to \infty$.
|
|
%mirrors the formation of the RMP critical point and other QPTs in the complete basis set limit.
|
|
\label{fig:UMP_cp}}
|
|
|
|
\end{figure*}
|
|
%------------------------------------------------------------------%
|
|
|
|
% RELATIONSHIP BETWEEN QPT AND UMP
|
|
Returning to the symmetric Hubbard dimer, we showed in Sec.~\ref{sec:spin_cont} that the slow
|
|
convergence of the strongly correlated UMP series
|
|
was due to a complex-conjugate pair of EPs just outside the radius of convergence.
|
|
These EPs have positive real components and small imaginary components (see Fig.~\ref{fig:UMP}), suggesting a potential
|
|
connection to MP critical points and QPTs (see Sec.~\ref{sec:MP_critical_point}).
|
|
For $\lambda>1$, the HF potential becomes an attractive component in Stillinger's
|
|
Hamiltonian displayed in Eq.~\eqref{eq:HamiltonianStillinger}, while the explicit electron-electron interaction
|
|
becomes increasingly repulsive.
|
|
Closed--shell critical points along the positive real $\lambda$ axis then represent
|
|
points where the two-electron repulsion overcomes the attractive HF potential
|
|
and a single electron dissociates from the molecule (see Ref.~\onlinecite{Sergeev_2006})
|
|
|
|
In contrast, symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons.
|
|
Consider one of the two reference UHF solutions where the spin-up and spin-down electrons are localised on the left and right sites respectively.
|
|
The spin-up HF potential will then be a repulsive interaction from the spin-down electron
|
|
density that is centred around the right site (and vice-versa).
|
|
As $\lambda$ becomes greater than 1 and the HF potentials become attractive, there will be a sudden
|
|
driving force for the electrons to swap sites.
|
|
This swapping process can also be represented as a double excitation, and thus an avoided crossing will occur
|
|
for $\lambda \geq 1$ (Fig.~\ref{subfig:ump_cp}).
|
|
While this appears to be an avoided crossing between the ground and first-excited state,
|
|
the presence of an earlier excited-state avoided crossing means that the first-excited state qualitatively
|
|
represents the reference double excitation for $\lambda > 1/2$.
|
|
|
|
% SHARPNESS AND QPT
|
|
The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$.
|
|
For small $U/t$, the HF potentials will be weak and the electrons will delocalise over the two sites,
|
|
both in the UHF reference and the exact wave function.
|
|
This delocalisation dampens the electron swapping process and leads to a ``shallow'' avoided crossing
|
|
that corresponds to EPs with non-zero imaginary components (solid lines in Fig.~\ref{subfig:ump_cp}).
|
|
As $U/t$ becomes larger, the HF potentials become stronger and the on-site repulsion dominates the hopping
|
|
term to make electron delocalisation less favourable.
|
|
In other words, the electrons localise on individual sites to form a Wigner crystal.
|
|
These effects create a stronger driving force for the electrons to swap sites until eventually this swapping
|
|
occurs exactly at $\lambda = 1$.
|
|
In this limit, the ground-state EPs approach the real axis (Fig.~\ref{subfig:ump_ep_to_cp}) and the avoided
|
|
crossing creates a gradient discontinuity in the ground-state energy (dashed lines in Fig.~\ref{subfig:ump_cp}).
|
|
We therefore find that, in the strong correlation limit, the symmetry-broken ground-state EP becomes
|
|
a new type of MP critical point and represents a QPT as the perturbation parameter $\lambda$ is varied.
|
|
Furthermore, this argument explains why the dominant UMP singularity lies so close, but always outside, the
|
|
radius of convergence (see Fig.~\ref{fig:RadConv}).
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Resummation Methods}
|
|
\label{sec:Resummation}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%
|
|
\begin{figure*}
|
|
\includegraphics[height=0.23\textheight]{fig9a}
|
|
\includegraphics[height=0.23\textheight]{fig9b}
|
|
\caption{\label{fig:PadeRMP}
|
|
RMP ground-state energy as a function of $\lambda$ obtained using various resummation
|
|
techniques at $U/t = 3.5$ (left) and $U/t = 4.5$ (right).}
|
|
\end{figure*}
|
|
%%%%%%%%%%%%%%%%%
|
|
|
|
|
|
%As frequently claimed by Carl Bender,
|
|
It is frequently stated that
|
|
\textit{``the most stupid thing that one can do with a series is to sum it.''}
|
|
Nonetheless, quantum chemists are basically doing this on a daily basis.
|
|
As we have seen throughout this review, the MP series can often show erratic,
|
|
slow, or divergent behaviour.
|
|
In these cases, estimating the correlation energy by simply summing successive
|
|
low-order terms is almost guaranteed to fail.
|
|
Here, we discuss alternative tools that can be used to sum slowly convergent or divergent series.
|
|
These so-called ``resummation'' techniques form a vast field of research and thus we will
|
|
provide details for only the most relevant methods.
|
|
We refer the interested reader to more specialised reviews for additional information.%
|
|
\cite{Goodson_2011,Goodson_2019}
|
|
|
|
|
|
%==========================================%
|
|
\subsection{Pad\'e Approximant}
|
|
%==========================================%
|
|
|
|
The failure of a Taylor series for correctly modelling the MP energy function $E(\lambda)$
|
|
arises because one is trying to model a complicated function containing multiple branches, branch points and
|
|
singularities using a simple polynomial of finite order.
|
|
A truncated Taylor series can only predict a single sheet and does not have enough
|
|
flexibility to adequately describe functions such as the MP energy.
|
|
Alternatively, the description of complex energy functions can be significantly improved
|
|
by introducing Pad\'e approximants, \cite{Pade_1892} and related techniques. \cite{BakerBook,BenderBook}
|
|
|
|
A Pad\'e approximant can be considered as the best approximation of a function by a
|
|
rational function of given order.
|
|
More specifically, a $[d_A/d_B]$ Pad\'e approximant is defined as
|
|
\begin{equation}
|
|
\label{eq:PadeApp}
|
|
E_{[d_A/d_B]}(\lambda) = \frac{A(\lambda)}{B(\lambda)}
|
|
= \frac{\sum_{k=0}^{d_A} a_k\, \lambda^k}{1 + \sum_{k=1}^{d_B} b_k\, \lambda^k},
|
|
\end{equation}
|
|
where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting terms for each 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 appear at 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).
|
|
Despite this limitation, the successive diagonal Pad\'e approximants (\ie, $d_A = d_B $)
|
|
often define a convergent perturbation series in cases where the Taylor series expansion diverges.
|
|
|
|
\begin{table}[b]
|
|
\caption{RMP ground-state energy estimate at $\lambda = 1$ provided by various truncated Taylor
|
|
series and Pad\'e approximants at $U/t = 3.5$ and $4.5$.
|
|
We also report the distance of the closest pole to the origin $\abs{\lc}$ provided by the diagonal Pad\'e approximants.
|
|
\label{tab:PadeRMP}}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{lccccc}
|
|
& & \mc{2}{c}{$\abs{\lc}$} & \mc{2}{c}{$E_{-}(\lambda = 1)$} \\
|
|
\cline{3-4} \cline{5-6}
|
|
Method & Degree & $U/t = 3.5$ & $U/t = 4.5$ & $U/t = 3.5$ & $U/t = 4.5$ \\
|
|
\hline
|
|
Taylor & 2 & & & $-1.01563$ & $-1.01563$ \\
|
|
& 3 & & & $-1.01563$ & $-1.01563$ \\
|
|
& 4 & & & $-0.86908$ & $-0.61517$ \\
|
|
& 5 & & & $-0.86908$ & $-0.61517$ \\
|
|
& 6 & & & $-0.92518$ & $-0.86858$ \\
|
|
\hline
|
|
Pad\'e & [1/1] & $2.29$ & $1.78$ & $-1.61111$ & $-2.64286$ \\
|
|
& [2/2] & $2.29$ & $1.78$ & $-0.82124$ & $-0.48446$ \\
|
|
& [3/3] & $1.73$ & $1.34$ & $-0.91995$ & $-0.81929$ \\
|
|
& [4/4] & $1.47$ & $1.14$ & $-0.90579$ & $-0.74866$ \\
|
|
& [5/5] & $1.35$ & $1.05$ & $-0.90778$ & $-0.76277$ \\
|
|
\hline
|
|
Exact & & $1.14$ & $0.89$ & $-0.90754$ & $-0.76040$ \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table}
|
|
|
|
Figure~\ref{fig:PadeRMP} illustrates the improvement provided by diagonal Pad\'e
|
|
approximants 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 as the truncation
|
|
degree increases at $U/t = 3.5$, the Pad\'e approximants yield much more accurate results.
|
|
Furthermore, the distance of the closest pole to the origin $\abs{\lc}$ in the Pad\'e approximants
|
|
indicate that they provide a relatively good approximation to the position of the
|
|
true branch point singularity in the RMP energy.
|
|
For $U/t = 4.5$, the Taylor series expansion performs worse and eventually diverges,
|
|
while the Pad\'e approximants still offer relatively accurate energies and recovers
|
|
a convergent series.
|
|
|
|
%%%%%%%%%%%%%%%%%
|
|
\begin{figure}[t]
|
|
\includegraphics[width=\linewidth]{fig10}
|
|
\caption{\label{fig:QuadUMP}
|
|
UMP energies as a function of $\lambda$ obtained using various resummation techniques at $U/t = 3$.}
|
|
\end{figure}
|
|
%%%%%%%%%%%%%%%%%
|
|
|
|
We can expect the UMP energy function to be much more challenging
|
|
to model properly as it contains three connected branches
|
|
(see Figs.~\ref{subfig:UMP_3} and \ref{subfig:UMP_7}).
|
|
Figure~\ref{fig:QuadUMP} and Table~\ref{tab:QuadUMP} indicate that this is indeed the case.
|
|
In particular, Fig.~\ref{fig:QuadUMP} illustrates that the Pad\'e approximants are trying to model
|
|
the square root branch point that lies close to $\lambda = 1$ by placing a pole on the real axis
|
|
(\eg, [3/3]) or with a very small imaginary component (\eg, [4/4]).
|
|
The proximity of these poles to the physical point $\lambda = 1$ means that any error in the Pad\'e
|
|
functional form becomes magnified in the estimate of the exact energy, as seen for the low-order
|
|
approximants in Table~\ref{tab:QuadUMP}.
|
|
However, with sufficiently high degree polynomials, one obtains
|
|
accurate estimates for the position of the closest singularity and the ground-state energy at $\lambda = 1$,
|
|
even in cases where the convergence of the UMP series is incredibly slow
|
|
(see Fig.~\ref{subfig:UMP_cvg}).
|
|
|
|
%==========================================%
|
|
\subsection{Quadratic Approximant}
|
|
%==========================================%
|
|
Quadratic approximants are designed 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}
|
|
\begin{equation}
|
|
\label{eq:QuadApp}
|
|
E(\lambda) = \frac{1}{2 Q(\lambda)} \qty[ P(\lambda) \pm \sqrt{P^2(\lambda) - 4 Q(\lambda) R(\lambda)} ],
|
|
\end{equation}
|
|
with the polynomials
|
|
\begin{align}
|
|
\label{eq:PQR}
|
|
P(\lambda) & = \sum_{k=0}^{d_P} p_k \lambda^k,
|
|
&
|
|
Q(\lambda) & = \sum_{k=0}^{d_Q} q_k \lambda^k,
|
|
&
|
|
R(\lambda) & = \sum_{k=0}^{d_R} r_k \lambda^k,
|
|
\end{align}
|
|
defined such that $d_P + d_Q + d_R = n - 1$, and $n$ is the truncation order of the Taylor series of $E(\lambda)$.
|
|
Recasting Eq.~\eqref{eq:QuadApp} as a second-order expression in $E(\lambda)$, \ie,
|
|
\begin{equation}
|
|
Q(\lambda) E^2(\lambda) - P(\lambda) E(\lambda) + R(\lambda) \sim \order*{\lambda^{n+1}},
|
|
\end{equation}
|
|
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
|
|
$P^2(\lambda) - 4 Q(\lambda) R(\lambda)$ and $d_q$ poles at the roots of $Q(\lambda)$.
|
|
|
|
Generally, the diagonal sequence of quadratic approximant,
|
|
\ie, $[0/0,0]$, $[1/0,0]$, $[1/0,1]$, $[1/1,1]$, $[2/1,1]$,
|
|
is of particular interest as the order of the corresponding Taylor series increases on each step.
|
|
However, while a quadratic approximant can reproduce multiple branch points, it can only describe
|
|
a total of two branches.
|
|
%\titou{Since every branch points must therefore correspond to a degeneracy of the same two branches,}
|
|
This constraint
|
|
can hamper the faithful description of more complicated singularity structures such as the MP energy surface.
|
|
Despite this limitation, Ref.~\onlinecite{Goodson_2000a} demonstrates that 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}
|
|
|
|
As a note of caution, Ref.~\onlinecite{Goodson_2019} suggests that low-order
|
|
quadratic approximants can struggle to correctly model the singularity structure when
|
|
the energy function has poles in both the positive and negative half-planes.
|
|
In such a scenario, the quadratic approximant will tend to place its branch points in-between, potentially introducing singularities quite close to the origin.
|
|
The remedy for this problem involves applying a suitable transformation of the complex plane (such as a bilinear conformal mapping) which leaves the points at $\lambda = 0$ and $\lambda = 1$ unchanged. \cite{Feenberg_1956}
|
|
|
|
\begin{table}[b]
|
|
\caption{Estimate for the distance of the closest singularity (pole or branch point) to the origin $\abs{\lc}$
|
|
in 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.
|
|
\label{tab:QuadUMP}}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{lccccccc}
|
|
& & & & \mc{2}{c}{$\abs{\lc}$} & \mc{2}{c}{$E_{-}(\lambda = 1)$} \\
|
|
\cline{5-6}\cline{7-8}
|
|
\mc{2}{c}{Method} & $n$ & $n_\text{bp}$ & $U/t = 3$ & $U/t = 7$ & $U/t = 3$ & $U/t = 7$ \\
|
|
\hline
|
|
Taylor & & 2 & & & & $-0.74074$ & $-0.29155$ \\
|
|
& & 3 & & & & $-0.78189$ & $-0.29690$ \\
|
|
& & 4 & & & & $-0.82213$ & $-0.30225$ \\
|
|
& & 5 & & & & $-0.85769$ & $-0.30758$ \\
|
|
& & 6 & & & & $-0.88882$ & $-0.31289$ \\
|
|
\hline
|
|
Pad\'e & [1/1] & 2 & & $9.000$ & $49.00$ & $-0.75000$ & $-0.29167$ \\
|
|
& [2/2] & 4 & & $0.974$ & $1.003$ & $\hphantom{-}0.75000$ & $-17.9375$ \\
|
|
& [3/3] & 6 & & $1.141$ & $1.004$ & $-1.10896$ & $-1.49856$ \\
|
|
& [4/4] & 8 & & $1.068$ & $1.003$ & $-0.85396$ & $-0.33596$ \\
|
|
& [5/5] & 10 & & $1.122$ & $1.004$ & $-0.97254$ & $-0.35513$ \\
|
|
\hline
|
|
Quadratic & [2/1,2] & 6 & 4 & $1.086$ & $1.003$ & $-1.01009$ & $-0.53472$ \\
|
|
& [2/2,2] & 7 & 4 & $1.082$ & $1.003$ & $-1.00553$ & $-0.53463$ \\
|
|
& [3/2,2] & 8 & 6 & $1.082$ & $1.001$ & $-1.00568$ & $-0.52473$ \\
|
|
& [3/2,3] & 9 & 6 & $1.071$ & $1.002$ & $-0.99973$ & $-0.53102$ \\
|
|
& [3/3,3] & 10 & 6 & $1.071$ & $1.002$ & $-0.99966$ & $-0.53103$ \\[0.5ex]
|
|
(pole-free) & [3/0,2] & 6 & 6 & $1.059$ & $1.003$ & $-1.13712$ & $-0.57199$ \\
|
|
& [3/0,3] & 7 & 6 & $1.073$ & $1.002$ & $-1.00335$ & $-0.53113$ \\
|
|
& [3/0,4] & 8 & 6 & $1.071$ & $1.002$ & $-1.00074$ & $-0.53116$ \\
|
|
& [3/0,5] & 9 & 6 & $1.070$ & $1.002$ & $-1.00042$ & $-0.53114$ \\
|
|
& [3/0,6] & 10 & 6 & $1.070$ & $1.002$ & $-1.00039$ & $-0.53113$ \\
|
|
\hline
|
|
Exact & & & & $1.069$ & $1.002$ & $-1.00000$ & $-0.53113$ \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table}
|
|
|
|
\begin{figure*}
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.85\textwidth]{fig11a}
|
|
\subcaption{\label{subfig:322quad} [3/2,2] Quadratic}
|
|
\end{subfigure}
|
|
%
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.85\textwidth]{fig11b}
|
|
\subcaption{\label{subfig:exact} Exact}
|
|
\end{subfigure}
|
|
%
|
|
\begin{subfigure}{0.32\textwidth}
|
|
\includegraphics[height=0.85\textwidth]{fig11c}
|
|
\subcaption{\label{subfig:304quad} [3/0,4] Quadratic}
|
|
\end{subfigure}
|
|
\caption{%
|
|
Comparison of the [3/2,2] and [3/0,4] quadratic approximants with the exact UMP energy surface in the complex $\lambda$
|
|
plane with $U/t = 3$.
|
|
Both quadratic approximants correspond to the same truncation degree of the Taylor series and model the branch points
|
|
using a radicand polynomial of the same order.
|
|
However, the [3/2,2] approximant introduces poles into the surface that limits it accuracy, while the [3/0,4] approximant
|
|
is free of poles.}
|
|
\label{fig:nopole_quad}
|
|
\end{figure*}
|
|
|
|
For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximant
|
|
are quite poor approximations, but the $[1/0,1]$ version perfectly models the RMP energy
|
|
function by predicting a single pair of EPs at $\lambda_\text{EP} = \pm \i 4t/U$.
|
|
This is expected from the form of the RMP energy [see Eq.~\eqref{eq:E0MP}], which matches
|
|
the ideal target for quadratic approximants.
|
|
Furthermore, the greater flexibility of the diagonal quadratic approximants provides a significantly
|
|
improved model of the UMP energy in comparison to the Pad\'e approximants or Taylor series.
|
|
In particular, these quadratic approximants provide an effective model for the avoided crossings
|
|
(Fig.~\ref{fig:QuadUMP}) and an improved estimate for the distance of the
|
|
closest branch point to the origin.
|
|
Table~\ref{tab:QuadUMP} shows that they provide remarkably accurate
|
|
estimates of the ground-state energy at $\lambda = 1$.
|
|
|
|
While the diagonal quadratic approximants provide significanty improved estimates of the
|
|
ground-state energy, we can use our knowledge of the UMP singularity structure to develop
|
|
even more accurate results.
|
|
We have seen in previous sections that the UMP energy surface
|
|
contains only square-root branch cuts that approach the real axis in the limit $U/t \to \infty$.
|
|
Since there are no true poles on this surface, we can obtain more accurate quadratic approximants by
|
|
taking $d_q = 0$ and increasing $d_r$ to retain equivalent accuracy in the square-root term [see Eq.\eqref{eq:QuadApp}].
|
|
Figure~\ref{fig:nopole_quad} illustrates this improvement for the pole-free [3/0,4] quadratic
|
|
approximant compared to the [3/2,2] approximant with the same truncation degree in the Taylor
|
|
expansion.
|
|
Clearly, modelling the square-root branch point using $d_q = 2$ has the negative effect of
|
|
introducing spurious poles in the energy, while focussing purely on the branch point with $d_q = 0$
|
|
leads to a significantly improved model.
|
|
Table~\ref{tab:QuadUMP} shows that these pole-free quadratic approximants
|
|
provide a rapidly convergent series with essentially exact energies at low order.
|
|
|
|
|
|
Finally, to emphasise the improvement that can be gained by using either Pad\'e, diagonal quadratic,
|
|
or pole-free quadratic approximants, we consider the energy and error obtained using only the first 10 terms of the UMP
|
|
Taylor series in Table~\ref{tab:UMP_order10}.
|
|
The accuracy of these approximants reinforces how our understanding of the MP
|
|
energy surface in the complex plane can be leveraged to significantly improve estimates of the exact
|
|
energy using low-order perturbation expansions.
|
|
|
|
\begin{table}[h]
|
|
\caption{
|
|
Estimate and associated error of the exact UMP energy at $U/t = 7$ for
|
|
various approximants using up to ten terms in the Taylor expansion.
|
|
\label{tab:UMP_order10}}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{lccc}
|
|
\mc{2}{c}{Method} & $E_{-}(\lambda = 1)$ & \% Abs.\ Error \\
|
|
\hline
|
|
Taylor & 10 & $-0.33338$ & $37.150$ \\
|
|
Pad\'e & [5/5] & $-0.35513$ & $33.140$ \\
|
|
Quadratic (diagonal) & [3/3,3] & $-0.53103$ & $\hphantom{0}0.019$ \\
|
|
Quadratic (pole-free)& [3/0,6] & $-0.53113$ & $\hphantom{0}0.005$ \\
|
|
\hline
|
|
Exact & & $-0.53113$ & \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table}
|
|
|
|
%==========================================%
|
|
\subsection{Shanks Transformation}
|
|
\label{sec:Shanks}
|
|
%==========================================%
|
|
|
|
While the Pad\'e and quadratic approximants can yield a convergent series representation
|
|
in cases where the standard MP series diverges, there is no guarantee that the rate of convergence
|
|
will be fast enough for low-order approximations to be useful.
|
|
However, these low-order partial sums or approximants often contain a remarkable amount of information
|
|
that can be used to extract further information about the exact result.
|
|
The Shanks transformation presents one approach for extracting this information
|
|
and accelerating the rate of convergence of a sequence.\cite{Shanks_1955,BenderBook}
|
|
|
|
Consider the partial sums
|
|
$S_n = \sum_{k=0}^{n} s_k$
|
|
defined from the truncated summation of an infinite series
|
|
$S = \sum_{k=0}^{\infty} s_k$.
|
|
If the series converges, then the partial sums will tend to the exact result
|
|
\begin{equation}
|
|
\lim_{n \to \infty} S_n = S.
|
|
\end{equation}
|
|
The Shanks transformation attempts to generate increasingly accurate estimates of this
|
|
limit by defining a new series as
|
|
\begin{equation}
|
|
T(S_n) = \frac{S_{n+1} S_{n-1} - S_{n}^2}{S_{n+1} + S_{n-1} - 2 S_{n}}.
|
|
\end{equation}
|
|
This series can converge faster than the original partial sums and can thus provide greater
|
|
accuracy using only the first few terms in the series.
|
|
However, it is only designed to accelerate converging partial sums with
|
|
the approximate form
|
|
\begin{equation}
|
|
S_n \approx S + \alpha\,\beta^n.
|
|
\end{equation}
|
|
Furthermore, while this transformation can accelerate the convergence of a series,
|
|
there is no guarantee that this acceleration will be fast enough to significantly
|
|
improve the accuracy of low-order approximations.
|
|
|
|
To the best of our knowledge, the Shanks transformation has never previously been applied
|
|
to accelerate the convergence of the MP series.
|
|
We have therefore applied it to the convergent Taylor series, Pad\'e approximants, and quadratic
|
|
approximants for RMP and UMP in the symmetric Hubbard dimer.
|
|
The UMP approximants converge too slowly for the Shanks transformation
|
|
to provide any improvement, even in the case where the quadratic approximants are already
|
|
very accurate.
|
|
In contrast, acceleration of the diagonal Pad\'e approximants for the RMP cases
|
|
can significantly improve the estimate of the energy using low-order perturbation terms,
|
|
as shown in Table~\ref{tab:RMP_shank}.
|
|
Even though the RMP series diverges at $U/t = 4.5$, the combination
|
|
of diagonal Pad\'e approximants with the Shanks transformation reduces the absolute error in
|
|
the best energy estimate to 0.002\,\% using only the first 10 terms in the Taylor series.
|
|
This remarkable result indicates just how much information is contained in the first few
|
|
terms of a perturbation series, even if it diverges.
|
|
|
|
\begin{table}[th]
|
|
\caption{
|
|
Acceleration of the diagonal Pad\'e approximant sequence for the RMP energy
|
|
using the Shanks transformation.
|
|
\label{tab:RMP_shank}}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{lcccc}
|
|
& & & \mc{2}{c}{$E_{-}(\lambda = 1)$} \\
|
|
\cline{4-5}
|
|
Method & Degree & Series Term & $U/t = 3.5$ & $U/t = 4.5$ \\
|
|
\hline
|
|
Pad\'e & [1/1] & $S_1$ & $-1.61111$ & $-2.64286$ \\
|
|
& [2/2] & $S_2$ & $-0.82124$ & $-0.48446$ \\
|
|
& [3/3] & $S_3$ & $-0.91995$ & $-0.81929$ \\
|
|
& [4/4] & $S_4$ & $-0.90579$ & $-0.74866$ \\
|
|
& [5/5] & $S_5$ & $-0.90778$ & $-0.76277$ \\
|
|
\hline
|
|
Shanks & & $T(S_2)$ & $-0.90898$ & $-0.77432$ \\
|
|
& & $T(S_3)$ & $-0.90757$ & $-0.76096$ \\
|
|
& & $T(S_4)$ & $-0.90753$ & $-0.76042$ \\
|
|
\hline
|
|
Exact & & & $-0.90754$ & $-0.76040$ \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table}
|
|
|
|
%==========================================%
|
|
\subsection{Analytic continuation}
|
|
%==========================================%
|
|
|
|
Recently, Mih\'alka \etal\ have studied the effect of different partitionings, such as MP or EN theory, on the position of
|
|
branch points and the convergence properties of Rayleigh--Schr\"odinger perturbation theory\cite{Mihalka_2017b} (see also
|
|
Ref.~\onlinecite{Surjan_2000}).
|
|
Taking the equilibrium and stretched water structures as an example, they estimated the radius of convergence using quadratic
|
|
Pad\'e approximants.
|
|
The EN partitioning provided worse convergence properties than the MP partitioning, which is believed to be
|
|
because the EN denominators are generally smaller than the MP denominators.
|
|
To remedy the situation, they showed that introducing a suitably chosen level shift parameter can turn a
|
|
divergent series into a convergent one by increasing the magnitude of these denominators.\cite{Mihalka_2017b}
|
|
However, like the UMP series in stretched \ce{H2},\cite{Lepetit_1988}
|
|
the cost of larger denominators is an overall slower rate of convergence.
|
|
|
|
\begin{figure}
|
|
\includegraphics[width=\linewidth]{fig12}
|
|
\caption{%
|
|
Comparison of the scaled RMP10 Taylor expansion with the exact RMP energy as a function
|
|
of $\lambda$ for the symmetric Hubbard dimer at $U/t = 4.5$.
|
|
The two functions correspond closely within the radius of convergence.
|
|
}
|
|
\label{fig:rmp_anal_cont}
|
|
\end{figure}
|
|
|
|
In a later study by the same group, they used analytic continuation techniques
|
|
to resum a divergent MP series such as a stretched water molecule.\cite{Mihalka_2017a}
|
|
Any MP series truncated at a given order $n$ can be used to define the scaled function
|
|
\begin{equation}
|
|
E_{\text{MP}n}(\lambda) = \sum_{k=0}^{n} \lambda^{k} E_\text{MP}^{(k)}.
|
|
\end{equation}
|
|
Reliable estimates of the energy can be obtained for values of $\lambda$ where the MP series is rapidly
|
|
convergent (\ie, for $\abs{\lambda} < \rc$), as shown in Fig.~\ref{fig:rmp_anal_cont} for the RMP10 series
|
|
of the symmetric Hubbard dimer with $U/t = 4.5$.
|
|
These values can then be analytically continued using a polynomial- or Pad\'e-based fit to obtain an
|
|
estimate of the exact energy at $\lambda = 1$.
|
|
However, choosing the functional form for the best fit remains a difficult and subtle challenge.
|
|
|
|
This technique was first generalised using complex scaling parameters to construct an analytic
|
|
continuation by solving the Laplace equations.\cite{Surjan_2018}
|
|
It was then further improved by introducing Cauchy's integral formula\cite{Mihalka_2019}
|
|
\begin{equation}
|
|
\label{eq:Cauchy}
|
|
E(\lambda) = \frac{1}{2\pi \i} \oint_{\mathcal{C}} \frac{E(\lambda')}{\lambda' - \lambda},
|
|
\end{equation}
|
|
which states that the value of the energy can be computed at $\lambda_1$ inside the complex
|
|
contour $\mathcal{C}$ using only the values along the same contour.
|
|
Starting from a set of points in a ``trusted'' region where the MP series is convergent, their approach
|
|
self-consistently refines estimates of the $E(\lambda')$ values on a contour that includes the physical point
|
|
$\lambda = 1$.
|
|
The shape of this contour is arbitrary, but there must be no branch points or other singularities inside
|
|
the contour.
|
|
Once the contour values of $E(\lambda')$ are converged, Cauchy's integral formula Eq.~\eqref{eq:Cauchy} can
|
|
be invoked to compute the value at $E(\lambda=1)$ and obtain a final estimate of the exact energy.
|
|
The authors illustrate this protocol for the dissociation curve of \ce{LiH} and the stretched water
|
|
molecule to obtain encouragingly accurate results.\cite{Mihalka_2019}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%
|
|
\section{Concluding Remarks}
|
|
\label{sec:ccl}
|
|
%%%%%%%%%%%%%%%%%%%%
|
|
|
|
% INTRO TO CONC.
|
|
To accurately model chemical systems, one must choose a computational protocol from an ever growing
|
|
collection of theoretical methods.
|
|
Until the Sch\"odinger equation is solved exactly, this choice must make a compromise on the accuracy
|
|
of certain properties depending on the system that is being studied.
|
|
It is therefore essential that we understand the strengths and weaknesses of different methods,
|
|
and why one might fail in cases where others work beautifully.
|
|
In this review, we have seen that the success and failure of perturbation-based methods are
|
|
directly connected to the position of exceptional point singularities in the complex plane.
|
|
|
|
% HISTORICAL OVERVIEW
|
|
We began by presenting the fundamental concepts behind non-Hermitian extensions of quantum chemistry into the complex plane,
|
|
including the Hartree--Fock approximation and Rayleigh--Schr\"odinger perturbation theory.
|
|
We then provided a comprehensive review of the various research that has been performed
|
|
around the physics of complex singularities in perturbation theory, with a particular focus on M{\o}ller--Plesset theory.
|
|
Seminal contributions from various research groups around the world have revealed highly oscillatory,
|
|
slowly convergent, or catastrophically divergent behaviour of the restricted and/or unrestricted MP perturbation series.%
|
|
\cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988}
|
|
In particular, the spin-symmetry-broken unrestricted MP series is notorious
|
|
for giving incredibly slow convergence.\cite{Gill_1986,Nobes_1987,Gill_1988a,Gill_1988}
|
|
All these behaviours can be rationalised and explained by the position of exceptional points
|
|
and other singularities that arise when perturbation theory is extended across the complex plane.
|
|
|
|
% CLASSIFICATIONS
|
|
The classifications of different convergence types developed by Cremer and He,\cite{Cremer_1996}
|
|
Olsen \etal,\cite{Christiansen_1996,Olsen_1996,Olsen_2000,Olsen_2019}
|
|
or Sergeev and Goodson\cite{Goodson_2000a,Goodson_2000b,Goodson_2004,Sergeev_2005,Sergeev_2006} are particularly
|
|
worth highlighting.
|
|
In Cremer and He's original classification, ``class A'' systems exhibit monotonic convergence and generally
|
|
correspond to weakly correlated electron pairs, while ``class B'' systems show erratic convergence after initial
|
|
oscillations and generally contain spatially dense electron clusters.\cite{Cremer_1996}
|
|
Further insights were provided by Olsen and coworkers
|
|
who employed a two-state model to understand the various convergence behaviours of Hermitian and non-Hermitian
|
|
perturbation series.\cite{Christiansen_1996,Olsen_1996,Olsen_2000,Olsen_2019}
|
|
The careful analysis from Sergeev and Goodson later refined these classes depending on the position of the
|
|
singularity closest to the origin, giving $\alpha$ singularities which have large imaginary component,
|
|
and $\beta$ singularities which have a very small imaginary component.%
|
|
\cite{Goodson_2000a,Goodson_2000b,Goodson_2004,Sergeev_2005,Sergeev_2006}
|
|
Remarkably, the position of $\beta$ singularities close to the real axis can be justified as a critical
|
|
point where one (or more) electron is ionised from the molecule, creating a quantum phase transition.\cite{Stillinger_2000}
|
|
We have shown that the slow convergence of symmetry-broken MP approximations can also be driven by a $\beta$
|
|
singularity and is closely related to these quantum phase transitions.
|
|
|
|
% RESUMMATION
|
|
We have also discussed several resummation techniques that can be used to improve energy estimates
|
|
for both convergent and divergent series, including Pad\'e and quadratic approximants.
|
|
Furthermore, we have provided the first illustration of how the Shanks transformation can accelerate
|
|
convergence of MP approximants to improve the accuracy of low-order approximations.
|
|
Using these resummation and acceleration methods to turn low-order truncated MP series into convergent and
|
|
systematically improvable series can dramatically improve the accuracy and applicability of these perturbative methods.
|
|
However, the application of these approaches requires the evaluation of higher-order MP coefficients
|
|
(\eg, MP3, MP4, MP5, etc) that are generally expensive to compute in practice.
|
|
There is therefore a strong demand for computationally efficient approaches to evaluate general terms in the MP
|
|
series, and the development of stochastic,\cite{Thom_2007,Neuhauser_2012,Willow_2012,Takeshita_2017,Li_2019}
|
|
or linear-scaling approximations\cite{Rauhut_1998,Schutz_1999}
|
|
may prove fruitful avenues in this direction.
|
|
|
|
% ORBITAL OPTIMISATION EXCITED STATES
|
|
The present review has only considered the convergence of the MP series using the RHF or UHF
|
|
reference orbitals.
|
|
However, numerous recent studies have shown that the use of orbitals optimised in the presence of the MP2
|
|
correction\cite{Bozkaya_2011,Neese_2009,Lee_2018} or Kohn--Sham density-functional theory (DFT) orbitals
|
|
can significantly improve the accuracy of the MP3 correction,\cite{Bertels_2019,Rettig_2020}
|
|
particularly in the presence of symmetry-breaking.
|
|
Beyond intuitive heuristics, it is not clear why these alternative orbitals provide such accurate results,
|
|
and a detailed investigation of their MP energy function in the complex plane is therefore bound to provide
|
|
fascinating insights.
|
|
Furthermore, the convergence properties of the excited-state MP series using orbital-optimised higher energy
|
|
HF solutions\cite{Gilbert_2008} remains entirely unexplored.\cite{Lee_2019,CarterFenk_2020}
|
|
|
|
% HUBBARD
|
|
Finally, the physical concepts and mathematical tools presented in this manuscript have been illustrated
|
|
on the symmetric (or asymmetric in one occasion) Hubbard dimer at half-filling.
|
|
Although extremely simple, these illustrations highlight the incredible versatility of the Hubbard model
|
|
for understanding the subtle features of perturbation theory in the complex plane, alongisde other examples
|
|
such as Kohn-Sham DFT, \cite{Carrascal_2015,Cohen_2016} linear-response theory,\cite{Carrascal_2018}
|
|
many-body perturbation theory,\cite{Romaniello_2009,Romaniello_2012,DiSabatino_2015,Hirata_2015,Tarantino_2017,Olevano_2019}
|
|
ensemble DFT, \cite{Deur_2017,Deur_2018,Senjean_2018,Sagredo_2018,Fromager_2020} thermal DFT,\cite{Smith_2016,Smith_2018}
|
|
coupled cluster theory,\cite{Stein_2014,Henderson_2015,Shepherd_2016} and many more.
|
|
In particular, we have shown that the Hubbard dimer contains sufficient flexibility to describe
|
|
the effects of symmetry breaking, the MP critical point, and resummation techniques, in contrast to the more
|
|
minimalistic models considered previously.
|
|
We therefore propose that the Hubbard dimer provides the ideal arena for further developing our fundamental understanding
|
|
and applications of perturbation theory.
|
|
|
|
% DIRECTIONS
|
|
Perturbation theory isn't usually considered in the complex plane.
|
|
But when it is, a lot can be learnt about the performance of perturbation theory on the real axis.
|
|
These insights can allow incredibly accurate results to be obtained using only the lowest-order terms in a perturbation series.
|
|
Yet perturbation theory represents only one method for approximating the exact energy, and few other methods
|
|
have been considered through similar complex non-Hermitian extensions.
|
|
There is therefore much still to be discovered about the existence and consequences of exceptional points
|
|
throughout electronic structure theory.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{acknowledgements}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
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).
|
|
HGAB gratefully acknowledges New College, Oxford for funding through the Astor Junior Research Fellowship.
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\end{acknowledgements}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\bibliography{EPAWTFT}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\end{document}
|