From 500325e744f2a1295e56eb2cc6d981094a0a01af Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 17 Nov 2020 16:12:57 +0100 Subject: [PATCH] first draft for HF --- Manuscript/EPAWTFT.bbl | 16 +++- Manuscript/EPAWTFT.blg | 73 ++++++++--------- Manuscript/EPAWTFT.tex | 173 +++++++++++++++++------------------------ 3 files changed, 122 insertions(+), 140 deletions(-) diff --git a/Manuscript/EPAWTFT.bbl b/Manuscript/EPAWTFT.bbl index f663c60..fed22c5 100644 --- a/Manuscript/EPAWTFT.bbl +++ b/Manuscript/EPAWTFT.bbl @@ -6,7 +6,7 @@ %Control: page (0) single %Control: year (1) truncated %Control: production of eprint (0) enabled -\begin{thebibliography}{82}% +\begin{thebibliography}{84}% \makeatletter \providecommand \@ifxundefined [1]{% \@ifx{#1\undefined} @@ -475,6 +475,12 @@ 10.1140/epjb/e2018-90114-9} {\bibfield {journal} {\bibinfo {journal} {Eur. Phys. J. B}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo {pages} {142} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Wigner}(1934)}]{Wigner_1934}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont + {Wigner}},\ }\href {\doibase 10.1103/PhysRev.46.1002} {\bibfield {journal} + {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {46}},\ + \bibinfo {pages} {1002} (\bibinfo {year} {1934})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Goodson}(2012)}]{Goodson_2012}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~Z.}\ \bibnamefont @@ -489,6 +495,14 @@ {Ostlund}},\ }\href@noop {} {\emph {\bibinfo {title} {Modern quantum chemistry: {Introduction} to advanced electronic structure}}}\ (\bibinfo {publisher} {McGraw-Hill},\ \bibinfo {year} {1989})\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Coulson}\ and\ \citenamefont + {Fischer}()}]{Coulson_1949}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~A.}\ \bibnamefont + {Coulson}}\ and\ \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont + {Fischer}},\ }\href {\doibase 10.1080/14786444908521726} {\bibfield + {journal} {\bibinfo {journal} {1949}\ }\textbf {\bibinfo {volume} {40}},\ + \bibinfo {pages} {386}}\BibitemShut {NoStop}% \bibitem [{\citenamefont {M{\o}ller}\ and\ \citenamefont {Plesset}(1934)}]{Moller_1934}% \BibitemOpen diff --git a/Manuscript/EPAWTFT.blg b/Manuscript/EPAWTFT.blg index 9c828fa..3944800 100644 --- a/Manuscript/EPAWTFT.blg +++ b/Manuscript/EPAWTFT.blg @@ -1,5 +1,5 @@ -This is BibTeX, Version 0.99d (TeX Live 2016) -Capacity: max_strings=35307, hash_size=35307, hash_prime=30011 +This is BibTeX, Version 0.99d (TeX Live 2020) +Capacity: max_strings=200000, hash_size=200000, hash_prime=170003 The top-level auxiliary file: EPAWTFT.aux The style file: apsrev4-1.bst Reallocated singl_function (elt_size=4) to 100 items from 50. @@ -23,45 +23,46 @@ Control: production of article title (-1) disabled Control: page (0) single Control: year (1) truncated Control: production of eprint (0) enabled -You've used 84 entries, +Warning--empty year in Coulson_1949 +You've used 86 entries, 5847 wiz_defined-function locations, - 2218 strings with 30556 characters, -and the built_in function-call counts, 82528 in all, are: -= -- 5275 -> -- 2197 -< -- 560 -+ -- 698 -- -- 535 -* -- 12816 -:= -- 8105 -add.period$ -- 82 -call.type$ -- 84 -change.case$ -- 330 -chr.to.int$ -- 87 -cite$ -- 84 -duplicate$ -- 7655 -empty$ -- 6137 -format.name$ -- 1282 -if$ -- 16484 + 2232 strings with 30838 characters, +and the built_in function-call counts, 84131 in all, are: += -- 5391 +> -- 2225 +< -- 571 ++ -- 708 +- -- 541 +* -- 13040 +:= -- 8275 +add.period$ -- 84 +call.type$ -- 86 +change.case$ -- 338 +chr.to.int$ -- 89 +cite$ -- 87 +duplicate$ -- 7807 +empty$ -- 6261 +format.name$ -- 1300 +if$ -- 16801 int.to.chr$ -- 6 -int.to.str$ -- 91 -missing$ -- 1009 -newline$ -- 298 -num.names$ -- 246 -pop$ -- 3145 +int.to.str$ -- 93 +missing$ -- 1030 +newline$ -- 304 +num.names$ -- 252 +pop$ -- 3216 preamble$ -- 1 -purify$ -- 410 +purify$ -- 419 quote$ -- 0 -skip$ -- 3020 +skip$ -- 3083 stack$ -- 0 -substring$ -- 2212 -swap$ -- 7288 -text.length$ -- 260 +substring$ -- 2260 +swap$ -- 7422 +text.length$ -- 263 text.prefix$ -- 0 top$ -- 10 -type$ -- 1166 -warning$ -- 1 -while$ -- 251 +type$ -- 1191 +warning$ -- 2 +while$ -- 257 width$ -- 0 -write$ -- 703 -(There was 1 warning) +write$ -- 718 +(There were 2 warnings) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index f28c7cb..299ce01 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -112,7 +112,6 @@ \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.} -\newcommand{\VU}{Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling, FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands.} \begin{document} \title{Perturbation theory in the complex plane: Exceptional points and where to find them} @@ -120,11 +119,8 @@ \author{Antoine \surname{Marie}} \affiliation{\LCPQ} \author{Hugh G.~A.~\surname{Burton}} -%\affiliation{\UCAM} \email{hugh.burton@chem.ox.ac.uk} \affiliation{\UOX} -%\author{Paola \surname{Gori-Giorgi}} -%\affiliation{\VU} \author{Pierre-Fran\c{c}ois \surname{Loos}} \email{loos@irsamc.ups-tlse.fr} \affiliation{\LCPQ} @@ -152,13 +148,13 @@ Accurately predicting ground- and excited-state energies (hence excitation energ An armada of theoretical and computational methods have been developed to this end, each of them being plagued by its own flaws. The fact that none of these methods is successful in every chemical scenario has encouraged chemists to carry on the development of new excited-state methodologies, their main goal being to get the most accurate excitation energies (and properties) at the lowest possible computational cost in the most general context. -One common feature of all these methods is that they rely on the notion of quantized energy levels of Hermitian quantum mechanics, in which the different electronic states of a molecule or an atom are energetically ordered, the lowest being the ground state while the higher ones are excited states. -Within this quantized paradigm, electronic states look completely disconnected from one another. +One common feature of all these methods is that they rely on the notion of quantised energy levels of Hermitian quantum mechanics, in which the different electronic states of a molecule or an atom are energetically ordered, the lowest being the ground state while the higher ones are excited states. +Within this quantised paradigm, electronic states look completely disconnected from one another. Many current methods study excited states using only ground-state information, creating a ground-state bias that leads to incorrect excitation energies. -However, one can gain a different perspective on quantization extending quantum chemistry into the complex domain. +However, one can gain a different perspective on quantisation extending quantum chemistry into the complex domain. In a non-Hermitian complex picture, the energy levels are \textit{sheets} of a more complicated topological manifold called \textit{Riemann surface}, and they are smooth and continuous \textit{analytic continuation} of one another. -In other words, our view of the quantized nature of conventional Hermitian quantum mechanics arises only from our limited perception of the more complex and profound structure of its non-Hermitian variant. \cite{MoiseyevBook,BenderPTBook} -The realization that ground and excited states both emerge from one single mathematical structure with equal importance suggests that excited-state energies can be computed from first principles in their own right. +In other words, our view of the quantised nature of conventional Hermitian quantum mechanics arises only from our limited perception of the more complex and profound structure of its non-Hermitian variant. \cite{MoiseyevBook,BenderPTBook} +The realisation that ground and excited states both emerge from one single mathematical structure with equal importance suggests that excited-state energies can be computed from first principles in their own right. One could then exploit the structure of these Riemann surfaces to develop methods that directly target excited-state energies without needing ground-state information. \cite{Burton_2019,Burton_2019a} By analytically continuing the electronic energy $E(\lambda)$ in the complex domain (where $\lambda$ is a coupling parameter), the ground and excited states of a molecule can be smoothly connected. @@ -168,8 +164,8 @@ Amazingly, this smooth and continuous transition from one state to another has r Exceptional points (EPs) are 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} They are the non-Hermitian analogs of conical intersections, \cite{Yarkony_1996} which are ubiquitous in non-adiabatic processes and play a key role in photo-chemical mechanisms. -In the case of auto-ionizing resonances, EPs have a role in deactivation processes similar to conical intersections in the decay of bound excited states. \cite{Benda_2018} -Although Hermitian and non-Hermitian Hamiltonians are closely related, the behavior of their eigenvalues near degeneracies is starkly different. +In the case of auto-ionising resonances, EPs have a role in deactivation processes similar to conical intersections in the decay of bound excited states. \cite{Benda_2018} +Although Hermitian and non-Hermitian Hamiltonians are closely related, the behaviour of their eigenvalues near degeneracies is starkly different. For example, encircling non-Hermitian degeneracies at EPs leads to an interconversion of states, and two loops around the EP are necessary to recover the initial energy. \cite{MoiseyevBook,Heiss_2016,Benda_2018} Additionally, the wave function picks up a geometric phase (also known as Berry phase \cite{Berry_1984}) and four loops are required to recover the initial wave function. In contrast, encircling Hermitian degeneracies at conical intersections only introduces a geometric phase while leaving the states unchanged. @@ -197,7 +193,9 @@ More importantly here, although EPs usually lie off the real axis, these singula \end{figure*} To illustrate the concepts discussed throughout this article, we will consider the symmetric Hubbard dimer at half filling, \ie\ with two opposite-spin fermions. -Using the localised site basis, the Hilbert space for this system comprises the four configurations +Simple systems that are analytically solvable are of great importance in theoretical chemistry and physics as they can be employed to illustrate concepts and test new methods as the mathematics are easier than in realistic systems (such as molecules or solids) but they retain much of the key physics. + +Using the (localised) site basis, the (singlet) 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} @@ -215,7 +213,10 @@ The exact Hamiltonian is then \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 $U$ dictates the correlation regime. +In the weak correlation regime (\ie, small $U$), the kinetic energy dominates and the electrons are delocalised over both sites. +For large $U$ (or strong correlation regime), the electron repulsion term drives the physics and the electrons localise on opposite sites to minimise their Coulomb repulsion. +This phenomenon is sometimes referred to as a Wigner crystallisation. \cite{Wigner_1934} To illustrate the formation of an exceptional points, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t\rightarrow \lambda t$. When $\lambda$ is real, the Hamiltonian~\eqref{eq:H_FCI} is Hermitian with the distinct (real-valued) eigenvalues @@ -262,67 +263,6 @@ such that 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 pick up a geometric phase in the process, and four complete loops are required to recover their starting forms.\cite{MoiseyevBook} -%In order to highlight the general properties of EPs mentioned above, we propose to consider the ubiquitous symmetric Hubbard dimer at half filling (\ie, with two opposite-spin fermions) whose Hamiltonian reads in the singlet configuration state function basis -%\begin{align} -% \ket{1\up1\dw} & \ket{1\up2\dw} & \ket{1\dw2\up} & \ket{2\up2\dw} \\ -% \uddot \quad \vac & \updot \quad \dwdot & \dwdot \quad \updot & \vac \quad \uddot \\ -%\end{align} -%\begin{equation} -%\label{eq:H_FCI} -% \bH = -% \begin{pmatrix} -% -2t + U & 0 & U/2 \\ -% 0 & U & 0 \\ -% U/2 & 0 & -2t + 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. -%We will consistently use this system to illustrate the different concepts discussed in the present review article. - -%For real $U$, the Hamiltonian \eqref{eq:H_FCI} is clearly Hermitian, and it becomes non-Hermitian %for any complex $U$ value. -%The eigenvalues associated with its singlet ground state and singlet doubly-excited state are -%\begin{equation} -%\label{eq:E_FCI} -% E_{\pm} = \frac{1}{2} \qty( U \pm \sqrt{(4t^2) + U^2} ). -%\end{equation} -%and they are represented as a function of $U$ in Fig.~\ref{fig:FCI} together with the energy of the %singlet open-shell configuration of energy $U$. - - -%One notices that these two states become degenerate only for a pair of complex conjugate values of $U$ -%\begin{equation} -%\label{eq:lambda_EP} -% U_\text{EP} = \pm 4 i t, -%\end{equation} -%with energy -%\begin{equation} -%\label{eq:E_EP} -% E_\text{EP} = \pm 2 i t, -%\end{equation} -%which correspond to square-root singularities in the complex-$U$ plane [see Fig.~\eqref{fig:FCI}]. -%These two $U$ values, given by Eq.~\eqref{eq:lambda_EP}, are the so-called EPs and one can clearly see that they connect the ground and excited states. -%Starting from $U_\text{EP}$, two square-root branch cuts run on the imaginary axis towards $\pm i \infty$. -%In the real $U$ axis, the point for which the states are the closest ($U = 0$) is called an avoided crossing and this occurs at $U = \Re(U_\text{EP})$. -%The ``shape'' of the avoided crossing is linked to the magnitude of $\Im(U_\text{EP})$: the smaller $\Im(U_\text{EP})$, the sharper the avoided crossing is. - -%Around $U = U_\text{EP}$, Eq.~\eqref{eq:E_FCI} behaves as \cite{MoiseyevBook} -%\begin{equation} %\label{eq:E_EP} -% E_{\pm} = E_\text{EP} \pm \sqrt{2U_\text{EP}} \sqrt{U - U_\text{EP}}, -%\end{equation} -%and following a complex contour around the EP, \ie, $U = U_\text{EP} + R \exp(i\theta)$, yields -%\begin{equation} -% E_{\pm}(\theta) = E_\text{EP} \pm \sqrt{2U_\text{EP} R} \exp(i\theta/2), -%\end{equation} -%and we have -%\begin{align} -% E_{\pm}(2\pi) & = E_{\mp}(0), -% & -% E_{\pm}(4\pi) & = E_{\pm}(0). \notag -%\end{align} -%This evidences that encircling non-Hermitian degeneracies at EPs leads to an interconversion of states, and two loops around the EP are necessary to recover the initial energy. -%Additionally, the wave function picks up a geometric phase and four loops are required to recover the starting wave function. -%\cite{MoiseyevBook} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Perturbation theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -335,7 +275,7 @@ Within the Born-Oppenheimer approximation, \begin{equation}\label{eq:ExactHamiltonian} \hH = - \frac{1}{2} \sum_{i}^{n} \grad_i^2 - \sum_{i}^{n} \sum_{A}^{N} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}} + \sum_{i