From ed53e8a3be6a0d3516f2e0a9984a36539143fa6e Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 1 Dec 2020 17:09:05 +0100 Subject: [PATCH 1/7] Done with IIID --- Manuscript/EPAWTFT.tex | 13 +++---------- 1 file changed, 3 insertions(+), 10 deletions(-) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 719e479..362f668 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -972,15 +972,11 @@ exact correlation energy $\Delta E$ using terms up to MP6\cite{Cremer_1996} &= \Emp^{(2)} + \Emp^{(3)} + \qty(\Emp^{(4)} + \Emp^{(5)}) \exp(\Emp^{(6)} / \Emp^{(5)}). \end{align} \end{subequations} -%As one can only compute the first terms of the MP series, a smart way of getting more accurate results is to use extrapolation formula, \ie, estimating the limit of the series with only few terms. -%Cremer and He proved that using specific extrapolation formulas of the MP series for class A and class B systems improves the precision of the results compared to the formula used without resorting to classes. \cite{Cremer_1996} 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 Section~\ref{sec:Resummation}, we consider more advanced extrapolation routines that take account of EPs in the complex $\lambda$-plane. -%The mean absolute deviation taking the FCI correlation energies as reference is $0.3$ millihartree with the class-specific formula whereas the deviation increases to 12 millihartree using the general formula. -%Even if there were still shaded areas in their analysis and that their classification was incomplete, the work of Ref.~\onlinecite{Cremer_1996} clearly evidenced that understanding the origin of the different modes of convergence could potentially lead to a more rationalised use of MP perturbation theory and, hence, to more accurate correlation energy estimates. +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, @@ -1007,7 +1003,7 @@ Using their observations in Ref.~\onlinecite{Olsen_1996}, Olsen and collaborator 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 $2\times2$ matrix +the dominant singularities as the EPs of the two-state matrix \begin{equation} \label{eq:Olsen_2x2} \underbrace{\mqty(\alpha & \delta \\ \delta & \beta )}_{\bH} @@ -1027,10 +1023,8 @@ These intruder-state effects are analogous to the EP that dictates the convergen 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. -%They used their two-state model on this avoided crossings and the model was actually predicting the divergence of the series. -%They concluded that the divergence of the series was due to the interaction with a highly diffuse excited state. This divergence is related to a more fundamental critical point in the MP energy surface that we will -discuss in Section~\ref{sec:MP_critical_point}. +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 @@ -1053,7 +1047,6 @@ according to a so-called ``archetype'' that defines the overall ``shape'' of the 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. -%Other features characterising the convergence behaviour of a perturbation method are its rate of convergence, its length of recurring period, and its sign pattern. 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} From d3c0f4f55299e70bb22cd44ce9f3fd624cc7ac42 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 1 Dec 2020 17:16:26 +0100 Subject: [PATCH 2/7] Done with IIIE --- Manuscript/EPAWTFT.tex | 21 ++------------------- 1 file changed, 2 insertions(+), 19 deletions(-) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 362f668..47e7b5a 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -1096,7 +1096,7 @@ processes.\cite{Sergeev_2005} 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 dominant EP with a positive real component, -with the magnitude of the imaginary component controlling the oscillations in the signs of successive MP +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. @@ -1117,26 +1117,9 @@ Alternatively, Sergeev \etal\ demonstrated that the inclusion of a ``ghost'' ato 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 \etal{}\cite{Olsen_2000} was simply too minimal to understand the complexity of +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. -% BASIS SET DEPENDENCE (INCLUDE?) -%Finally, it was shown that $\beta$ singularities are very sensitive to changes in the basis set but not to the bond stretching. -%On the contrary, $\alpha$ singularities are relatively insensitive to the basis sets but very sensitive to bond stretching. -%According to Goodson, \cite{Goodson_2004} the singularity structure of stretched molecules is difficult because there is more than one significant singularity. -%This is consistent with the observation of Olsen and coworkers \cite{Olsen_2000} on the \ce{HF} molecule at equilibrium and stretched geometries. -%To the best of our knowledge, the effect of bond stretching on singularities, its link with spin contamination and symmetry breaking of the wave function has not been as well understood as the ionization phenomenon and its link with diffuse functions. - -%In the 2000's, Sergeev and Goodson \cite{Sergeev_2005, Sergeev_2006} analysed this problem from a more mathematical point of view by looking at the whole singularity structure where Olsen and collaborators were trying to find the dominant singularity causing the divergence. \cite{Olsen_1996,Olsen_2000,Olsen_2019} -%Singularities of $\alpha$-type are related to large avoided crossing between the ground and low-lying excited states, whereas $\beta$ singularities come from a sharp avoided crossing between the ground state and a highly diffuse state. -%They succeeded to explain the divergence of the series caused by $\beta$ singularities following previous work by Stillinger. \cite{Stillinger_2000} - - -%This reasoning is done on the exact Hamiltonian and energy, \ie, the Hamiltonian in the complete Hilbert space, this is the exact energy which exhibits this singularity on the negative real axis. -%However, in a finite basis set which does not span the complete Hilbert space, one can prove that, for a Hermitian Hamiltonian, the singularities of $E(\lambda)$ occurs in complex conjugate pairs with non-zero imaginary parts. -%Sergeev and Goodson proved, \cite{Sergeev_2005} as predicted by Stillinger, \cite{Stillinger_2000} that in a finite basis set the critical point on the real axis is modelled by a cluster of sharp avoided crossings with diffuse functions, equivalently by a cluster of $\beta$ singularities in the negative half plane. -%This explains that Olsen \textit{et al.}, because they used a simple two-state model, only observed the first singularity of this cluster of singularities causing the divergence. \cite{Olsen_2000} - % 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).% From 0fd95f136f234d5f3a70551a2d2b22aa771f1038 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 1 Dec 2020 17:22:20 +0100 Subject: [PATCH 3/7] Done with IIIF --- Manuscript/EPAWTFT.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 47e7b5a..33fa43c 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -1175,7 +1175,7 @@ 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 [see Eq.~\eqref{eq:H_FCI}], where we define $\epsilon > 0$. -The reference Slater determinant for a doubly-occupied atom can be represented using the RHF +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 From fdbd7f2136be680b8eac442a9ebad1885204f86a Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 1 Dec 2020 21:00:09 +0100 Subject: [PATCH 4/7] abstract --- Manuscript/EPAWTFT.tex | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 33fa43c..c5a9433 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -146,9 +146,11 @@ \begin{abstract} In this review, we explore the 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 exceptionnal points. +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 a presentation of the fundamental notions of quantum chemistry in the complex plane, such as the mean-field Hartree--Fock approximation and Rayleigh-Schr\"odinger perturbation theory, and their illustration with the ubiquitous (symmetric) Hubbard dimer at half filling, 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 apparent link with quantum phase transitions. +Each of these points is further illustrated with the Hubbard dimer. +Finally, we discuss several resummation techniques (such as Pad\'e and quadratic approximants) alongside concrete examples. \end{abstract} \maketitle From e0367edf5e53df786684010e4a550456ac5f5458 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 1 Dec 2020 21:24:59 +0100 Subject: [PATCH 5/7] adding references in intro and slight modifications --- Manuscript/EPAWTFT.bbl | 167 ++++++++++++++++++++++++---- Manuscript/EPAWTFT.bib | 239 +++++++++++++++++++++++++++++++++++++++-- Manuscript/EPAWTFT.tex | 9 +- 3 files changed, 386 insertions(+), 29 deletions(-) diff --git a/Manuscript/EPAWTFT.bbl b/Manuscript/EPAWTFT.bbl index 3b0fe9f..84d396f 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}{133}% +\begin{thebibliography}{148}% \makeatletter \providecommand \@ifxundefined [1]{% \@ifx{#1\undefined} @@ -50,6 +50,154 @@ \providecommand \BibitemShut [1]{\csname bibitem#1\endcsname}% \let\auto@bib@innerbib\@empty % +\bibitem [{\citenamefont {Szabo}\ and\ \citenamefont + {Ostlund}(1989)}]{SzaboBook}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont + {Szabo}}\ and\ \bibinfo {author} {\bibfnamefont {N.~S.}\ \bibnamefont + {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 {Jensen}(2017)}]{JensenBook}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont + 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Rev.}, + pages = {1022-1043}, + publisher = {The Royal Society of Chemistry}, + title = {The Bethe--Salpeter equation in chemistry: relations with TD-DFT{,} applications and challenges}, + volume = {47}, + year = {2018}, + Bdsk-Url-1 = {http://dx.doi.org/10.1039/C7CS00049A}} + +@article{Blase_2020, + author = {X. Blase and I. Duchemin and D. Jacquemin and P. F. Loos}, + date-added = {2020-12-01 21:12:31 +0100}, + date-modified = {2020-12-01 21:12:31 +0100}, + doi = {10.1021/acs.jpclett.0c01875}, + journal = {J. Phys. Chem. Lett.}, + pages = {7371}, + title = {The Bethe-Salpeter Formalism: From Physics to Chemistry}, + volume = {11}, + year = {2020}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.0c01875}} + +@article{Ghosh_2018, + author = {Ghosh, Soumen and Verma, Pragya and Cramer, Christopher J. and Gagliardi, Laura and Truhlar, Donald G.}, + date-added = {2020-12-01 21:12:15 +0100}, + date-modified = {2020-12-01 21:12:15 +0100}, + doi = {10.1021/acs.chemrev.8b00193}, + journal = {Chem. Rev.}, + pages = {7249--7292}, + title = {Combining Wave Function Methods with Density Functional Theory for Excited States}, + volume = {118}, + year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.chemrev.8b00193}} + +@article{Adamo_2013, + author = {Adamo, C. and Jacquemin, D.}, + date-added = {2020-12-01 21:11:58 +0100}, + date-modified = {2020-12-01 21:15:50 +0100}, + doi = {10.1039/C2CS35394F}, + journal = {Chem. Soc. Rev.}, + pages = {845--856}, + title = {The Calculations of Excited-State Properties with Time-Dependent Density Functional Theory}, + volume = {42}, + year = {2013}} + +@article{Laurent_2013, + author = {Laurent, Ad{\`e}le D. and Jacquemin, Denis}, + date-added = {2020-12-01 21:11:49 +0100}, + date-modified = {2020-12-01 21:15:13 +0100}, + doi = {10.1002/qua.24438}, + journal = {Int. J. Quantum Chem.}, + pages = {2019--2039}, + title = {TD-DFT Benchmarks: A Review}, + volume = {113}, + year = {2013}} + +@article{Gonzales_2012, + author = {Gonz{\'a}lez, Leticia and Escudero, D. and Serrano-Andr\`es, L.}, + date-added = {2020-12-01 21:11:38 +0100}, + date-modified = {2020-12-01 21:11:38 +0100}, + doi = {10.1002/cphc.201100200}, + journal = {ChemPhysChem}, + pages = {28--51}, + title = {Progress and Challenges in the Calculation of Electronic Excited States}, + volume = {13}, + year = {2012}, + Bdsk-Url-1 = {https://doi.org/10.1002/cphc.201100200}} + +@article{Sneskov_2012, + abstract = {Abstract We review coupled cluster (CC) theory for electronically excited states. We outline the basics of a CC response theory framework that allows the transfer of the attractive accuracy and convergence properties associated with CC methods over to the calculation of electronic excitation energies and properties. Key factors affecting the accuracy of CC excitation energy calculations are discussed as are some of the key CC models in this field. To aid both the practitioner as well as the developer of CC excited state methods, we also briefly discuss the key computational steps in a working CC response implementation. Approaches aimed at extending the application range of CC excited state methods either in terms of molecular size and phenomena or in terms of environment (solution and proteins) are also discussed. {\copyright} 2011 John Wiley \& Sons, Ltd. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods}, + author = {Sneskov, Kristian and Christiansen, Ove}, + date-added = {2020-12-01 21:11:24 +0100}, + date-modified = {2020-12-01 21:14:26 +0100}, + doi = {https://doi.org/10.1002/wcms.99}, + journal = {WIREs Comput. Mol. Sci.}, + pages = {566--584}, + title = {Excited State Coupled Cluster Methods}, + volume = {2}, + year = {2012}, + Bdsk-Url-1 = {https://onlinelibrary.wiley.com/doi/abs/10.1002/wcms.99}, + Bdsk-Url-2 = {https://doi.org/10.1002/wcms.99}} + +@article{Krylov_2006, + author = {Krylov, Anna I.}, + date-added = {2020-12-01 21:10:56 +0100}, + date-modified = {2020-12-01 21:14:02 +0100}, + doi = {10.1021/ar0402006}, + journal = {Acc. Chem. Res.}, + pages = {83-91}, + title = {Spin-Flip Equation-of-Motion Coupled-Cluster Electronic Structure Method for a Description of Excited States, Bond Breaking, Diradicals, and Triradicals}, + volume = {39}, + year = {2006}, + Bdsk-Url-1 = {https://doi.org/10.1021/ar0402006}} + +@article{Dreuw_2005, + author = {Dreuw, Andreas and Head-Gordon, Martin}, + date-added = {2020-12-01 21:10:39 +0100}, + date-modified = {2020-12-01 21:10:39 +0100}, + doi = {10.1021/cr0505627}, + file = {/Users/loos/Zotero/storage/WKGXAHGE/Dreuw_2005.pdf}, + issn = {0009-2665, 1520-6890}, + journal = {Chem. Rev.}, + language = {en}, + pages = {4009--4037}, + title = {Single-{{Reference}} Ab {{Initio Methods}} for the {{Calculation}} of {{Excited States}} of {{Large Molecules}}}, + volume = {105}, + year = {2005}, + Bdsk-Url-1 = {https://dx.doi.org/10.1021/cr0505627}} + +@article{Piecuch_2002, + author = {Piotr Piecuch and Karol Kowalski and Ian S. O. Pimienta and Michael J. Mcguire}, + date-added = {2020-12-01 21:10:26 +0100}, + date-modified = {2020-12-01 21:13:27 +0100}, + doi = {10.1080/0144235021000053811}, + journal = {Int. Rev. Phys. Chem.}, + pages = {527-655}, + publisher = {Taylor & Francis}, + title = {Recent advances in electronic structure theory: Method of moments of coupled-cluster equations and renormalized coupled-cluster approaches}, + volume = {21}, + year = {2002}, + Bdsk-Url-1 = {https://doi.org/10.1080/0144235021000053811}} + +@book{AveryBook, + address = {Dordrecht}, + author = {J. Avery}, + date-added = {2020-12-01 21:06:44 +0100}, + date-modified = {2020-12-01 21:06:44 +0100}, + publisher = {Kluwer Academic}, + title = {Hyperspherical harmonics: applications in quantum theory}, + year = {1989}} + +@book{CramerBook, + author = {C. J. Cramer}, + date-added = {2020-12-01 21:06:44 +0100}, + date-modified = {2020-12-01 21:06:44 +0100}, + keywords = {qmech}, + publisher = {Wiley}, + title = {Essentials of Computational Chemistry: Theories and Models}, + year = {2004}} + +@book{FetterBook, + author = {A. L. Fetter and J. D. Waleck}, + date-added = {2020-12-01 21:06:44 +0100}, + date-modified = {2020-12-01 21:06:44 +0100}, + publisher = {McGraw Hill, San Francisco}, + title = {Quantum Theory of Many Particle Systems}, + year = {1971}} + +@book{HerzbergBook, + author = {K. P. Huber and G. Herzberg}, + date-added = {2020-12-01 21:06:44 +0100}, + date-modified = {2020-12-01 21:06:44 +0100}, + publisher = {van Nostrand Reinhold Company}, + title = {Molecular Spectra and Molecular Structure: IV. Constants of diatomic molecules}, + year = {1979}} + +@book{JensenBook, + address = {New York}, + author = {F. Jensen}, + date-added = {2020-12-01 21:06:44 +0100}, + date-modified = {2020-12-01 21:06:44 +0100}, + edition = {3rd}, + keywords = {qmech}, + publisher = {Wiley}, + title = {Introduction to Computational Chemistry}, + year = {2017}} + +@book{NISTbook, + address = {New York}, + date-added = {2020-12-01 21:06:44 +0100}, + date-modified = {2020-12-01 21:06:44 +0100}, + editor = {F. W. J. Olver and D. W. Lozier and R. F. Boisvert and C. W. Clark}, + keywords = {maths}, + publisher = {Cambridge University Press}, + title = {NIST Handbook of Mathematical Functions}, + year = {2010}} + +@book{ParrBook, + address = {Clarendon Press}, + author = {R. G. Parr and W. Yang}, + date-added = {2020-12-01 21:06:44 +0100}, + date-modified = {2020-12-01 21:06:44 +0100}, + keywords = {dft; qmech}, + publisher = {Oxford}, + title = {Density-Functional Theory of Atoms and Molecules}, + year = {1989}} + +@book{ReiningBook, + author = {Martin, R.M. and Reining, L. and Ceperley, D.M.}, + date-added = {2020-12-01 21:06:44 +0100}, + date-modified = {2020-12-01 21:06:44 +0100}, + isbn = {0521871506}, + publisher = {Cambridge University Press}, + title = {Interacting Electrons: Theory and Computational Approaches}, + year = {2016}} + +@book{Schuck_Book, + author = {P. Ring and P. Schuck}, + date-added = {2020-12-01 21:06:44 +0100}, + date-modified = {2020-12-01 21:06:44 +0100}, + publisher = {Springer}, + title = {The Nuclear Many-Body Problem}, + year = {2004}} + +@book{Stefanucci_2013, + abstract = {"The Green's function method is one of the most powerful and versatile formalisms in physics, and its nonequilibrium version has proved invaluable in many research fields. This book provides a unique, self-contained introduction to nonequilibrium many-body theory. Starting with basic quantum mechanics, the authors introduce the equilibrium and nonequilibrium Green's function formalisms within a unified framework called the contour formalism. The physical content of the contour Green's functions and the diagrammatic expansions are explained with a focus on the time-dependent aspect. Every result is derived step-by-step, critically discussed and then applied to different physical systems, ranging from molecules and nanostructures to metals and insulators. With an abundance of illustrative examples, this accessible book is ideal for graduate students and researchers who are interested in excited state properties of matter and nonequilibrium physics"--}, + address = {Cambridge}, + author = {Stefanucci, Gianluca and van Leeuwen, Robert}, + date-added = {2020-12-01 21:06:44 +0100}, + date-modified = {2020-12-01 21:06:44 +0100}, + isbn = {978-0-521-76617-3}, + keywords = {Many-body problem,Quantum theory,Green's functions,Mathematics,SCIENCE / Physics}, + lccn = {QC174.17.G68 S74 2013}, + publisher = {{Cambridge University Press}}, + shorttitle = {Nonequilibrium Many-Body Theory of Quantum Systems}, + title = {Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction}, + year = {2013}} + +@book{HelgakerBook, + author = {T. Helgaker and P. J{\o}rgensen and J. Olsen}, + date-added = {2020-12-01 21:06:11 +0100}, + date-modified = {2020-12-01 21:06:17 +0100}, + owner = {joshua}, + publisher = {John Wiley \& Sons, Inc.}, + timestamp = {2014.11.24}, + title = {Molecular Electronic-Structure Theory}, + year = {2013}} + @article{Feenberg_1956, author = {Feenberg, Eugene}, date-added = {2020-12-01 13:27:51 +0100}, @@ -1584,12 +1815,6 @@ title = {Modern quantum chemistry: {Introduction} to advanced electronic structure}, year = {1989}} -@book{JensenBook, - author = {F. Jensen}, - publisher = {Wiley}, - title = {Introduction to computational chemistry}, - year = {2017}} - @article{Lepetit_1988, author = {Lepetit, M. B. and P{\'e}lissier, M. and Malrieu, J. P.}, date-modified = {2020-08-22 22:14:08 +0200}, diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index c5a9433..28415d8 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -164,13 +164,14 @@ Finally, we discuss several resummation techniques (such as Pad\'e and quadratic %%%%%%%%%%%%%%%%%%%%%%% Due to the ubiquitous influence of processes involving electronic states in physics, chemistry, and biology, their faithful description from first principles has been one of the grand challenges faced by theoretical chemists since the dawn of computational chemistry. -Accurately predicting ground- and excited-state energies (hence excitation energies) is particularly valuable in this context, and it has concentrated most of the efforts within the community. -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 methodologies, their main goal being to get the most accurate energies (and properties) at the lowest possible computational cost in the most general context. +Accurately predicting ground- and excited-state energies (hence excitation energies) is particularly valuable in this context, and it has concentrated most of the efforts within the community. +An armada of theoretical and computational methods have been developed to this end, each of them being plagued by its own flaws. \cite{SzaboBook,JensenBook,CramerBook,HelgakerBook,ParrBook,FetterBook,ReiningBook} +The fact that none of these methods is successful in every chemical scenario has encouraged chemists and physicists to carry on the development of new methodologies, their main goal being to get the most accurate energies (and properties) at the lowest possible computational cost in the most general context. +In particular, the design of an affordable, black-box method performing well in both the weak and strong correlation regimes is still elusive. 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. +For example, many current methods study excited states using only ground-state information, creating a ground-state bias that leads to incorrect excitation energies.\cite{Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Ghosh_2018,Blase_2020,Loos_2020a} 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 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} From 433cf4e8b36c9e6222587ef69d6c37e6a41d9ad6 Mon Sep 17 00:00:00 2001 From: Hugh Burton Date: Wed, 2 Dec 2020 09:46:16 +0000 Subject: [PATCH 6/7] local changes --- Manuscript/EPAWTFT.tex | 125 +++++++++++++++++++++++++++++++---------- 1 file changed, 94 insertions(+), 31 deletions(-) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 33fa43c..0657a79 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -1364,33 +1364,65 @@ radius of convergence (see Fig.~\ref{fig:RadConv}). \label{sec:Resummation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -As frequently claimed by Carl Bender, \textit{``the most stupid thing that one can do with a series is to sum it.''} +%As frequently claimed by Carl Bender, +\hugh{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 exactly this on a daily basis. -Here, we discuss tools that can be used to sum divergent series. -Resummation techniques is a vast field of research and, below, we provide details for a non-exhaustive list of these techniques. -We refer the interested reader to more specialised reviews for additional information. \cite{Goodson_2011,Goodson_2019} +\hugh{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. +\hugh{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} +\subsection{Pad\'e Approximant} %==========================================% -The inability of Taylor series to model properly the energy function $E(\lambda$) can be simply understood by the fact that one aims at modelling a complicated function with potentially poles and singularities by a simple polynomial of finite order. +\hugh{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 branch points and +singularities} using a simple polynomial of finite order. A truncated Taylor series just does not have enough flexibility to do the job properly. -Nonetheless, the description of complex energy functions can be significantly improved thanks to Pad\'e approximant, \cite{Pade_1892} and related techniques. \cite{BakerBook,BenderBook} +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} -According to Wikipedia, \textit{``a Pad\'e approximant is the best approximation of a function by a rational function of given order''}. +\hugh{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}{\sum_{k=0}^{d_B} b_k \lambda^k} + 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} -(with $b_0 = 1$), where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting terms according to power of $\lambda$. +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 appears at the locations of 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) for example. +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). +\hugh{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.} -Figure \ref{fig:PadeRMP} illustrates the improvement brought by diagonal (\ie, $d_A = d_B$) Pad\'e approximants as compared to the usual Taylor expansion in cases where the RMP series of the Hubbard dimer converges ($U/t = 3.5$) and diverges ($U/t = 4.5$). +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 at $U/t = 3.5$ when one increases the truncation degree, the Pad\'e approximants yield much more accurate results with, additionally, a rather good estimate of the radius of convergence of the RMP series. -For $U/t = 4.5$, the struggles of the truncated Taylor expansions are magnified and the Pad\'e approximants still provide quite accurate energies even outside the radius of convergence of the RMP series. +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. +\hugh{Furthermore, the Pad\'e approximants provide a rather good estimate of the radius of convergence of the RMP series.} +For $U/t = 4.5$, the Taylor series expansion performs worse (and eventually diverges), +while the Pad\'e approximants still offer relaitively accurate energies even outside the radius of convergence of the RMP series. + +\hugh{% +We can expect that the singularity structure of the UMP energy will be much more challenging to model properly as the UMP energy function 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. +However, with sufficiently high degree polynomials, one obtains +accurate estimates of both the radius of convergence 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}). +In Figure \ref{fig:QuadUMP}, it becomes clear 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 +(for [3/3]) or with a very small imaginary component (for [4/4]). +The proximity of these poles to the radius of convergence means that any error in the Pad\'e +functional form becomes magnified in the estimate of energy at $\lambda = 1$. +} \begin{table} \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 $4.5$. @@ -1428,14 +1460,14 @@ For $U/t = 4.5$, the struggles of the truncated Taylor expansions are magnified %%%%%%%%%%%%%%%%% %==========================================% -\subsection{Quadratic approximant} +\subsection{Quadratic Approximant} %==========================================% -In a nutshell, the idea behind quadratic approximant is 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} +Quadratic approximants \hugh{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} -where +with the polynomials \begin{align} \label{eq:PQR} P(\lambda) & = \sum_{k=0}^{d_P} p_k \lambda^k, @@ -1444,7 +1476,7 @@ where & R(\lambda) & = \sum_{k=0}^{d_R} r_k \lambda^k \end{align} -are polynomials, such that $d_P + d_Q + d_R = n - 1$, and $n$ is the truncation order of the Taylor series of $E(\lambda)$. +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}} @@ -1452,17 +1484,11 @@ Recasting Eq.~\eqref{eq:QuadApp} as a second-order expression in $E(\lambda)$, \ 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 $P^2(\lambda) - 4 Q(\lambda) R(\lambda)$. 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. -Note that, by construction, a quadratic approximant has only two branches which hampers the faithful description of more complicated singularity structures. +However, by construction, a quadratic approximant has only two branches, which hampering the faithful description of more complicated singularity structures. As shown in Ref.~\onlinecite{Goodson_2000a}, 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} -For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximant are quite poor approximation, but its $[1/0,1]$ version already model perfectly the RMP energy function by predicting a single pair of EPs at $\lambda_\text{EP} = \pm i 4t/U$. -This is expected knowing the form of the RMP energy [see Eq.~\eqref{eq:E0MP}] which perfectly suits the purpose of quadratic approximants. - -We can anticipate that the singularity structure of the UMP energy function is going to be much more challenging to model properly, and this is indeed the case as the UMP energy function contains three branches (see Figs.~\ref{subfig:UMP_3} and \ref{subfig:UMP_7}). -However, by ramping up high enough the degree of the polynomials, one is able to get both, as shown in Fig.~\ref{fig:QuadUMP} and Table \ref{tab:QuadUMP}, accurate estimates of the radius of convergence of the UMP series and of the ground-state energy at $\lambda = 1$, even in cases where the convergence of the UMP series is painfully slow (see Fig.~\ref{subfig:UMP_cvg}). -Figure \ref{fig:QuadUMP} evidences that the Pad\'e approximants are trying to model the square root singularity by placing a pole on the real axis (for [3/3]) or just off the real axis (for [4/4]). -Thanks to greater flexibility, the quadratic approximants are able to model nicely the avoided crossing and the location of the singularities. -Besides, they provide accurate estimates of the ground-state energy at $\lambda = 1$ (see Table \ref{tab:QuadUMP}). +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. %%%%%%%%%%%%%%%%% \begin{figure} @@ -1482,7 +1508,9 @@ Besides, they provide accurate estimates of the ground-state energy at $\lambda \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 - Pad\'e & [3/3] & 6 & & $1.141$ & $1.004$ & $-1.10896$ & $-1.49856$ \\ + 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 & & $1.122$ & $1.004$ & $-0.97254$ & $-0.35513$ \\ Quadratic & [2/1,2] & 6 & 4 & $1.086$ & $1.003$ & $-1.01009$ & $-0.53472$ \\ @@ -1495,10 +1523,45 @@ Besides, they provide accurate estimates of the ground-state energy at $\lambda \end{tabular} \end{ruledtabular} \end{table} +\hugh{On the other hand, the greater flexibility of the quadratic approximants provides a significantly +improved model of the UMP energy in comparison to the Pad\' approximants or Taylor series. +In particular, the quadratic approximants provide an effect model for the avoided crossings +(Fig.~\ref{fig:QuadUMP}) and a far better estimate for the location of the branch point singularities. +Furthermore, they provide remarkably accurate estimates of the ground-state energy at $\lambda = 1$, +as shown in Table~\ref{tab:QuadUMP}} + +However, 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} + + +%==========================================% +\subsection{Shanks Transformation} +%==========================================% + +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} + +Consider the partial sums $S_N$ defined from the truncated summation of an infinite series +\begin{equation} + S_N = \sum_{k=0}^{N} a_k. +\end{equation} +If the series converges, then the partial sums will tend to the exact result in the limit $N\rightarrow \infty$. +The Shanks transformation attempts to generate increasingly accurate estimates of the +exact result 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. -An interesting point raised in Ref.~\onlinecite{Goodson_2019} suggests that low-order quadratic approximants might struggle to model the correct singularity structure when the energy function has poles in both the positive and negative half-planes. -In such a scenario, the quadratic approximant will have the tendency to place its branch points in-between, potentially introducing singularities quite close to the origin. -A simple potential cure for this consists in applying a judicious transformation (like a bilinear conformal mapping) which does not affect the points at $\lambda = 0$ and $\lambda = 1$. \cite{Feenberg_1956} %==========================================% \subsection{Analytic continuation} From f9239557fb6c6e451a29879a2cf76df798893b91 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Wed, 2 Dec 2020 10:50:11 +0100 Subject: [PATCH 7/7] comment in conclusion --- Manuscript/EPAWTFT.tex | 7 ++++++- 1 file changed, 6 insertions(+), 1 deletion(-) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 28415d8..6ddb0cb 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -1526,7 +1526,7 @@ The authors illustrate this protocol on the dissociation curve of \ce{LiH} and t \section{Conclusion} %%%%%%%%%%%%%%%%%%%% -In order to model accurately chemical systems, one must choose, in a ever growing zoo of methods, which computational protocol is adapted to the system of interest. +In order to model accurately chemical systems, one must choose, in an ever growing zoo of methods, which computational protocol is adapted to the system of interest. This choice can be, moreover, motivated by the type of properties that one is interested in. That means that one must understand the strengths and weaknesses of each method, \ie, why one method might fail in some cases and work beautifully in others. We have seen that for methods relying on perturbation theory, their successes and failures are directly connected to the position of EPs in the complex plane. @@ -1544,6 +1544,11 @@ We have found that the $\beta$ singularities modelling the ionisation phenomenon Indeed, those singularities close to the real axis are connected to quantum phase transitions and symmetry breaking, and theoretical physics have demonstrated that the behaviour of the EPs depends of the type of transitions from which the EPs result (first or higher orders, ground state or excited state transitions). To conclude, this work shows that our understanding of the singularity structure of the energy is still incomplete but we hope that it opens new perspectives for the understanding of the physics of EPs in electronic structure theory. +\titou{The Hubbard dimer is clearly a very versatile model to understand perturbation theory and could be used for further developments around PT. +Comment on approximants are requiring higher order MP which is expensive. +Paragraph on Paola's stuff.} + + %%%%%%%%%%%%%%%%%%%%%%%% \begin{acknowledgements} %%%%%%%%%%%%%%%%%%%%%%%%