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@ -6,7 +6,7 @@
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%Control: page (0) single
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%Control: year (1) truncated
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%Control: production of eprint (0) enabled
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\begin{thebibliography}{179}%
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\begin{thebibliography}{180}%
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\makeatletter
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\providecommand \@ifxundefined [1]{%
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\@ifx{#1\undefined}
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@ -857,6 +857,15 @@
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}\href {\doibase 10.1103/PhysRevA.69.052510} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {69}},\ \bibinfo
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{pages} {052510} (\bibinfo {year} {2004})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Burton}\ and\ \citenamefont
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{Thom}(2019)}]{Burton_2019b}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~G.~A.}\
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\bibnamefont {Burton}}\ and\ \bibinfo {author} {\bibfnamefont {A.~J.~W.}\
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\bibnamefont {Thom}},\ }\href {\doibase 10.1021/acs.jctc.9b00441} {\bibfield
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{journal} {\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo
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{volume} {15}},\ \bibinfo {pages} {4851} (\bibinfo {year}
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{2019})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {L\"owdin}(1955{\natexlab{a}})}]{Lowdin_1955a}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-O.}\ \bibnamefont
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@ -2509,6 +2509,16 @@
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year = {2019},
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Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00289}}
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@article{Burton_2019b,
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author = {Burton, Hugh G. A. and Thom, Alex J. W.},
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doi = {10.1021/acs.jctc.9b00441},
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journal = {J. Chem. Theory Comput.},
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volume = {15},
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pages = {4851},
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title = {General Approach for Multireference Ground and Excited States Using Nonorthogonal Configuration Interaction},
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year = {2019},
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}
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@article{Hiscock_2014,
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author = {Hiscock, Hamish G. and Thom, Alex J. W.},
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doi = {10.1021/ct5007696},
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@ -12,7 +12,7 @@
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\newcommand{\hugh}[1]{\textcolor{hughgreen}{#1}}
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\newcommand{\hughDraft}[1]{\textcolor{orange}{#1}}
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\newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}}
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\newcommand{\trashHB}[1]{\textcolor{orange}{\sout{#1}}}
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\newcommand{\trashHB}[1]{\textcolor{hughgreen}{\sout{#1}}}
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\newcommand{\antoine}[1]{\textcolor{cyan}{#1}}
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\newcommand{\trashantoine}[1]{\textcolor{cyan}{\sout{#1}}}
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@ -81,8 +81,9 @@
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\newcolumntype{Y}{>{\centering\arraybackslash}X}
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% HF rotation angles
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\newcommand{\ta}{\theta_{\alpha}}
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\newcommand{\tb}{\theta_{\beta}}
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\newcommand{\ta}{\theta^{\,\alpha}}
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\newcommand{\tb}{\theta^{\,\beta}}
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\newcommand{\ts}{\theta^{\sigma}}
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% Some constants
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\renewcommand{\i}{\mathrm{i}} % Imaginary unit
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@ -106,7 +107,8 @@
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\newcommand{\Rup}{\mathcal{R}^{\uparrow}}
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\newcommand{\Rdown}{\mathcal{R}^{\downarrow}}
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\newcommand{\Rsi}{\mathcal{R}^{\sigma}}
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\newcommand{\vhf}{v_{\text{HF}}}
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\newcommand{\vhf}{\Hat{v}_{\text{HF}}}
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\newcommand{\whf}{\Psi_{\text{HF}}}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France.}
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@ -127,11 +129,12 @@
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\begin{abstract}
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We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory.
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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.
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We observe that the physics of a quantum system is intimately connected to the position of \hugh{complex-valued} energy singularities
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\trashHB{in the complex plane}, known as exceptional points.
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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.
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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.
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In particular, we highlight the seminal work \trashHB{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.
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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.
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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.
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Each of these points is \trashHB{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.
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\end{abstract}
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\keywords{perturbation theory, complex plane, exceptional point, divergent series, resummation}
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@ -148,7 +151,7 @@ Each of these points is pedagogically illustrated using the Hubbard dimer at hal
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% SPIKE THE READER
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Perturbation theory isn't usually considered in the complex plane.
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Normally it is applied using real numbers as one of very few available tools for
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describing realistic quantum systems where exact solutions of the Schr\"odinger equation are impossible \titou{to find?}.\cite{Dirac_1929}
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describing realistic quantum systems \trashHB{where exact solutions of the Schr\"odinger equation are impossible \titou{to find?}}.\cite{Dirac_1929}
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In particular, time-independent Rayleigh--Schr\"odinger perturbation theory\cite{RayleighBook,Schrodinger_1926}
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has emerged as an instrument of choice among the vast array of methods developed for this purpose.%
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\cite{SzaboBook,JensenBook,CramerBook,HelgakerBook,ParrBook,FetterBook,ReiningBook}
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@ -174,7 +177,7 @@ systematically improvable series largely remains an open challenge.
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% COMPLEX PLANE
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Our conventional view of electronic structure theory is centred around the Hermitian notion of quantised energy levels,
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where the different electronic states of a molecular \antoine{molecule or molecular system?} are discrete and energetically ordered.
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where the different electronic states of a \hugh{molecule} are discrete and energetically ordered.
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The lowest energy state defines the ground electronic state, while higher energy states
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represent electronic excited states.
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However, an entirely different perspective on quantisation can be found by analytically continuing
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@ -217,7 +220,8 @@ microwaves, mechanics, acoustics, atomic systems and optics.\cite{Bittner_2012,C
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% MP THEORY IN THE COMPLEX PLANE
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The MP energy correction can be considered as a function of the perturbation parameter $\lambda$.
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When the domain of $\lambda$ is extended to the complex plane, EPs can also occur in the MP energy.
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Although these EPs generally exist in the complex plane, their positions are intimately related to the
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Although these EPs \hugh{are generally complex-valued} \trashHB{exist in the complex plane},
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their positions are intimately related to the
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convergence of the perturbation expansion on the real axis.%
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\cite{BenderBook,Olsen_1996,Olsen_2000,Olsen_2019,Mihalka_2017a,Mihalka_2017b,Mihalka_2019}
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Furthermore, the existence of an avoided crossing on the real axis is indicative of a nearby EP
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@ -330,7 +334,7 @@ and the electrons localise on opposite sites to minimise their Coulomb repulsion
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This phenomenon is often referred to as Wigner crystallisation. \cite{Wigner_1934}
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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 $\hH(\lambda)$.
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When $\lambda$ is real, the Hamiltonian~\eqref{eq:H_FCI} is Hermitian with the distinct (real-valued) (eigen)energies
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When $\lambda$ is real, the Hamiltonian \trashHB{\eqref{eq:H_FCI}} is Hermitian with the distinct (real-valued) (eigen)energies
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\begin{subequations}
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\begin{align}
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E_{\mp} &= \frac{1}{2} \qty(U \mp \sqrt{ (4 \lambda t)^2 + U^2 } ),
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@ -430,7 +434,7 @@ of the reference and perturbation Hamiltonians in Eq.~\eqref{eq:SchrEq-PT}.\cite
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%
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% LAMBDA IN THE COMPLEX PLANE
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From complex analysis, \cite{BenderBook} the radius of convergence for the energy can be obtained by looking for the
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singularities of $E(\lambda)$ in the complex $\lambda$ plane.
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\hugh{non-analytic} singularities of $E(\lambda)$ in the complex $\lambda$ plane.
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This property arises from the following theorem: \cite{Goodson_2011}
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\begin{quote}
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\it
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@ -464,7 +468,7 @@ Like the exact system in Sec.~\ref{sec:example}, the perturbation energy $E(\lam
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a ``one-to-many'' function with the output elements representing an approximation to both the ground and excited states.
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The most common singularities on $E(\lambda)$ therefore correspond to non-analytic EPs in the complex
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$\lambda$ plane where two states become degenerate.
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Later we will demonstrate how the choice of reference Hamiltonian controls the position of these EPs, and
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Later we will demonstrate how the choice of reference \hugh{wave function} \trashHB{Hamiltonian} controls the position of these EPs, and
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ultimately determines the convergence properties of the perturbation series.
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%===========================================%
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@ -473,10 +477,10 @@ ultimately determines the convergence properties of the perturbation series.
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%===========================================%
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% SUMMARY OF HF
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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.
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In the \trash{Hartree--Fock (HF)} \titou{HF} approximation, the many-electron wave function is approximated as a single Slater determinant $\whf(\vb{x}_1,\ldots,\vb{x}_\Ne)$, where $\vb{x} = (\sigma,\vb{r})$ is a composite vector gathering spin and spatial coordinates.
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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
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\begin{equation}\label{eq:FockOp}
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\Hat{f}(\vb{x}) \phi_p(\vb{x}) = \qty[ \Hat{h}(\vb{x}) + \Hat{v}^{\antoine{\text{HF}}}(\vb{x}) ] \phi_p(\vb{x}) = \epsilon_p \phi_p(\vb{x}).
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\Hat{f}(\vb{x}) \phi_p(\vb{x}) = \qty[ \Hat{h}(\vb{x}) + \vhf(\vb{x}) ] \phi_p(\vb{x}) = \epsilon_p \phi_p(\vb{x}).
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\end{equation}
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Here the (one-electron) core Hamiltonian is
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\begin{equation}
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@ -485,7 +489,7 @@ Here the (one-electron) core Hamiltonian is
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\end{equation}
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and
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\begin{equation}
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\Hat{v}_\text{HF}(\vb{x}) = \sum_i^{\Ne} \qty[ \Hat{J}_i(\vb{x}) - \Hat{K}_i(\vb{x}) ]
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\vhf(\vb{x}) = \sum_i^{\Ne} \qty[ \Hat{J}_i(\vb{x}) - \Hat{K}_i(\vb{x}) ]
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\end{equation}
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is the HF mean-field electron-electron potential with
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\begin{subequations}
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@ -526,8 +530,8 @@ Forcing the spatial part of the orbitals to be the same for spin-up and spin-dow
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while allowing different orbitals for different spins leads to the so-called unrestricted HF (UHF) approach.\cite{StuberPaldus}
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The advantage of the UHF approximation is its ability to correctly describe strongly correlated systems,
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such as antiferromagnetic phases\cite{Slater_1951} or the dissociation of the hydrogen dimer.\cite{Coulson_1949}
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However, by allowing different orbitals for different spins, the UHF is no longer required to be an eigenfunction of
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the total spin operator $\hat{\mathcal{S}}^2$, leading to ``spin-contamination'' in the wave function.
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However, by allowing different orbitals for different spins, the UHF \hugh{wave function} is no longer required to be an eigenfunction of
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the total spin operator $\hat{\mathcal{S}}^2$, leading to ``spin-contamination''.
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%================================================================%
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\subsection{Hartree--Fock in the Hubbard Dimer}
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@ -553,16 +557,16 @@ where we have introduced bonding $\mathcal{B}^{\sigma}$ and anti-bonding $\mathc
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the spin-$\sigma$ electrons as
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\begin{subequations}
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\begin{align}
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\mathcal{B}^{\sigma} & = \hphantom{-} \cos(\frac{\theta_\sigma}{2}) \Lsi + \sin(\frac{\theta_\sigma}{2}) \Rsi,
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\mathcal{B}^{\sigma} & = \hphantom{-} \cos(\frac{\ts}{2}) \Lsi + \sin(\frac{\ts}{2}) \Rsi,
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\\
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\mathcal{A}^{\sigma} & = - \sin(\frac{\theta_\sigma}{2}) \Lsi + \cos(\frac{\theta_\sigma}{2}) \Rsi
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\mathcal{A}^{\sigma} & = - \sin(\frac{\ts}{2}) \Lsi + \cos(\frac{\ts}{2}) \Rsi
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\end{align}
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\label{eq:RHF_orbs}
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\end{subequations}
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In the weak correlation regime $0 \le U \le 2t$, the angles which minimise the HF energy,
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\ie, $\pdv*{E_\text{HF}}{\theta_\sigma} = 0$, are
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\ie, $\pdv*{E_\text{HF}}{\ts} = 0$, are
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\begin{equation}
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\ta^\text{RHF} = \tb^\text{RHF} = \pi/2,
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\ta_\text{RHF} = \tb_\text{RHF} = \pi/2,
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\end{equation}
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giving the symmetry-pure molecular orbitals
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\begin{align}
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@ -572,7 +576,7 @@ giving the symmetry-pure molecular orbitals
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\end{align}
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and the ground-state RHF energy (Fig.~\ref{fig:HF_real})
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\begin{equation}
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E_\text{RHF} \equiv E_\text{HF}(\ta^\text{RHF}, \tb^\text{RHF}) = -2t + \frac{U}{2}.
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E_\text{RHF} \equiv E_\text{HF}(\ta_\text{RHF}, \tb_\text{RHF}) = -2t + \frac{U}{2}.
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\end{equation}
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However, in the strongly correlated regime $U>2t$, the closed-shell orbital restriction prevents RHF from
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modelling the correct physics with the two electrons on opposite sites.
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@ -590,7 +594,7 @@ modelling the correct physics with the two electrons on opposite sites.
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\subcaption{\label{subfig:UHF_cplx_energy}}
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\end{subfigure}
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\caption{%
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(\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$ in the Hubbard dimer for $U/t = 2$.
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(\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta_{\text{UHF}}$ for $\lambda \in \bbC$ in the Hubbard dimer for $U/t = 2$.
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Symmetry-broken solutions correspond to individual sheets and become equivalent at
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the \textit{quasi}-EP $\lambda_{\text{c}}$ (black dot).
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The RHF solution is independent of $\lambda$, giving the constant plane at $\pi/2$.
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@ -608,19 +612,19 @@ This critical point is analogous to the infamous Coulson--Fischer point identifi
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For $U \ge 2t$, the optimal orbital rotation angles for the UHF orbitals become
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\begin{subequations}
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\begin{align}
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\ta^\text{UHF} & = \arctan (-\frac{2t}{\sqrt{U^2 - 4t^2}}),
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\ta_\text{UHF} & = \arctan (-\frac{2t}{\sqrt{U^2 - 4t^2}}),
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\label{eq:ta_uhf}
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\\
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\tb^\text{UHF} & = \arctan (+\frac{2t}{\sqrt{U^2 - 4t^2}}),
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\tb_\text{UHF} & = \arctan (+\frac{2t}{\sqrt{U^2 - 4t^2}}),
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\label{eq:tb_uhf}
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\end{align}
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\end{subequations}
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with the corresponding UHF ground-state energy (Fig.~\ref{fig:HF_real})
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\begin{equation}
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E_\text{UHF} \equiv E_\text{HF}(\ta^\text{UHF}, \tb^\text{UHF}) = - \frac{2t^2}{U}.
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E_\text{UHF} \equiv E_\text{HF}(\ta_\text{UHF}, \tb_\text{UHF}) = - \frac{2t^2}{U}.
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\end{equation}
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Time-reversal symmetry dictates that this UHF wave function must be degenerate with its spin-flipped \titou{pair?}, obtained
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by swapping $\ta^{\text{UHF}}$ and $\tb^{\text{UHF}}$ in Eqs.~\eqref{eq:ta_uhf} and \eqref{eq:tb_uhf}.
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by swapping $\ta_{\text{UHF}}$ and $\tb_{\text{UHF}}$ in Eqs.~\eqref{eq:ta_uhf} and \eqref{eq:tb_uhf}.
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This type of symmetry breaking is also called a spin-density wave in the physics community as the system
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``oscillates'' between the two symmetry-broken configurations. \cite{GiulianiBook}
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Symmetry breaking can also occur in RHF theory when a charge-density wave is formed from an oscillation
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@ -637,7 +641,7 @@ Alternatively, the non-linear terms arising from the Coulomb and exchange operat
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be considered as a perturbation from the core Hamiltonian \eqref{eq:Hcore} by introducing the
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transformation $U \to \lambda\, U$, giving the parametrised Fock operator
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\begin{equation}
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\Hat{f}(\vb{x} ; \lambda) = \Hat{h}(\vb{x}) + \lambda\, \Hat{v}^{\antoine{\text{HF}}}(\vb{x}).
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\Hat{f}(\vb{x} ; \lambda) = \Hat{h}(\vb{x}) + \lambda\, \vhf(\vb{x}).
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\end{equation}
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The orbitals in the reference problem $\lambda=0$ correspond to the symmetry-pure eigenfunctions of the one-electron core
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Hamiltonian, while self-consistent solutions at $\lambda = 1$ represent the orbitals of the true HF solution.
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@ -651,7 +655,7 @@ with coalesence points at
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\label{eq:scaled_fock}
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\end{equation}
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In contrast, when $\lambda$ becomes complex, the HF equations become non-Hermitian and
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each HF solutions can be analytically continued for all $\lambda$ values using
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each HF solution can be analytically continued for all $\lambda$ values using
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the holomorphic HF approach.\cite{Hiscock_2014,Burton_2016,Burton_2018}
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Remarkably, the coalescence point in this analytic continuation emerges as a
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\textit{quasi}-EP on the real $\lambda$ axis (Fig.~\ref{fig:HF_cplx}), where
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@ -675,6 +679,9 @@ a ground-state wave function can be ``morphed'' into an excited-state wave funct
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via a stationary path of HF solutions.
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This novel approach to identifying excited-state wave functions demonstrates the fundamental
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role of \textit{quasi}-EPs in determining the behaviour of the HF approximation.
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\hugh{Furthermore, the complex-scaled Fock operator can be used routinely construct analytic
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continuations of HF solutions beyond the points where real HF solutions
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coalesce and vanish.\cite{Burton_2019b}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{M{\o}ller--Plesset Perturbation Theory in the Complex Plane}
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@ -694,7 +701,7 @@ With the MP partitioning, the parametrised perturbation Hamiltonian becomes
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\hH(\lambda) =
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\sum_{i}^{N} \qty[ - \frac{\grad_i^2}{2} - \sum_{A}^{M} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}} ]
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\\
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+ (1-\lambda) \sum_{i}^{N} v^{\text{HF}}(\vb{x}_i)
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+ (1-\lambda) \sum_{i}^{N} \vhf(\vb{x}_i)
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+ \lambda\sum_{i<j}^{N}\frac{1}{\abs{\vb{r}_i-\vb{r}_j}}.
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\end{multline}
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Any set of orbitals can be used to define the HF Hamiltonian, although either the RHF or UHF orbitals are usually chosen to
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@ -707,9 +714,9 @@ where $E_{\text{MP}}^{(k)}$ is the $k$th-order MP correction and
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\begin{equation}
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E_{\text{MP1}} = E_{\text{MP}}^{(0)} + E_{\text{MP}}^{(1)} = E_\text{HF}.
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\end{equation}
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The second-order MP2 energy is given by
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The second-order MP2 energy correction is given by
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\begin{equation}\label{eq:EMP2}
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E_{\text{MP2}} = \frac{1}{4} \sum_{ij} \sum_{ab} \frac{\abs{\mel{ij}{}{ab}}^2}{\epsilon_i + \epsilon_j - \epsilon_a - \epsilon_b},
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\hugh{E_{\text{MP}}^{(2)}} = \frac{1}{4} \sum_{ij} \sum_{ab} \frac{\abs{\mel{ij}{}{ab}}^2}{\epsilon_i + \epsilon_j - \epsilon_a - \epsilon_b},
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\end{equation}
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where $\mel{pq}{}{rs} = \braket{pq}{rs} - \braket{pq}{sr}$ are the anti-symmetrised two-electron integrals
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in the molecular spin-orbital basis\cite{Gill_1994}
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@ -840,7 +847,7 @@ gradient discontinuities or spurious minima.
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\end{subfigure}
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\caption{
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||||
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 $\lambda$.
|
||||
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 complex $\lambda$.
|
||||
\label{fig:RMP}}
|
||||
\end{figure*}
|
||||
|
||||
@ -879,8 +886,8 @@ The Taylor expansion of the RMP energy can then be evaluated to obtain the $k$th
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% 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}.
|
||||
The RMP series is convergent for $U = 3.5\,t$ with $\rc > 1$, as illustrated for the individual terms at each
|
||||
\hugh{perturbation} order 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
|
||||
@ -961,8 +968,8 @@ for larger $U/t$ as the radius of convergence becomes increasingly close to one
|
||||
% 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 (\antoine{pairs of}) EPs: one connecting the ground state with the
|
||||
For example, while the RMP energy shows only one EP between the ground \trashHB{state} and
|
||||
\trashHB{the} doubly-excited states (Fig.~\ref{fig:RMP}), the UMP energy has two (\antoine{pairs of}) 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.
|
||||
@ -990,7 +997,7 @@ very slowly as the perturbation order is increased.
|
||||
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}
|
||||
29 atomic and molecular systems \trashHB{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
|
||||
@ -1121,11 +1128,11 @@ To understand Stillinger's argument, consider the parametrised MP Hamiltonian in
|
||||
\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{(1-\lambda)\vhf(\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
|
||||
The mean-field potential $\vhf$ 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.
|
||||
@ -1147,7 +1154,7 @@ 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
|
||||
HF potential $\vhf$ 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}
|
||||
@ -1222,7 +1229,7 @@ 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 $\theta_{\alpha}^{\text{RHF}} = \theta_{\beta}^{\text{RHF}} = 0$,
|
||||
orbitals [see Eq.~\eqref{eq:RHF_orbs}] with $\ta_{\text{RHF}} = \tb_{\text{RHF}} = 0$,
|
||||
which corresponds to strictly localising the two electrons on the left site.
|
||||
%and energy
|
||||
%\begin{equation}
|
||||
@ -1378,7 +1385,7 @@ radius of convergence (see Fig.~\ref{fig:RadConv}).
|
||||
|
||||
%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.''}
|
||||
\textit{``the most stupid thing \hugh{to} \trashHB{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.
|
||||
@ -1396,7 +1403,7 @@ We refer the interested reader to more specialised reviews for additional inform
|
||||
%==========================================%
|
||||
|
||||
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
|
||||
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.
|
||||
@ -1416,7 +1423,7 @@ 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)
|
||||
(which are ubiquitous in the singularity structure of \trashHB{a typical} perturbative \hugh{methods} \trashHB{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 $)
|
||||
@ -1603,7 +1610,7 @@ 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
|
||||
For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximants
|
||||
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
|
||||
@ -1634,7 +1641,7 @@ 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
|
||||
or pole-free quadratic approximants, we collect 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
|
||||
@ -1736,7 +1743,7 @@ terms of a perturbation series, even if it diverges.
|
||||
\end{table}
|
||||
|
||||
%==========================================%
|
||||
\subsection{Analytic continuation}
|
||||
\subsection{Analytic Continuation}
|
||||
%==========================================%
|
||||
|
||||
Recently, Mih\'alka \etal\ have studied the effect of different partitionings, such as MP or EN theory, on the position of
|
||||
@ -1782,7 +1789,7 @@ It was then further improved by introducing Cauchy's integral formula\cite{Mihal
|
||||
\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
|
||||
which states that the value of the energy can be computed at $\lambda$ 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
|
||||
@ -1814,7 +1821,7 @@ We began by presenting the fundamental concepts behind non-Hermitian extensions
|
||||
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,
|
||||
Seminal contributions from various research groups \trashHB{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
|
||||
|
||||
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