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125
Manuscript/EPAWTFT.bbl
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%Control: page (0) single
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\begin{thebibliography}{84}%
\begin{thebibliography}{93}%
\makeatletter
\providecommand \@ifxundefined [1]{%
\@ifx{#1\undefined}
@ -487,6 +487,15 @@
{Goodson}},\ }\href {\doibase 10.1002/wcms.92} {\bibfield {journal}
{\bibinfo {journal} {{WIREs} Comput. Mol. Sci.}\ }\textbf {\bibinfo {volume}
{2}},\ \bibinfo {pages} {743} (\bibinfo {year} {2012})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Cejnar}\ \emph {et~al.}(2007)\citenamefont {Cejnar},
\citenamefont {Heinze},\ and\ \citenamefont {Macek}}]{Cejnar_2007}%
\BibitemOpen
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{Cejnar}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Heinze}}, \
and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Macek}},\ }\href
{\doibase 10.1103/PhysRevLett.99.100601} {\bibfield {journal} {\bibinfo
{journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {99}},\ \bibinfo
{pages} {100601} (\bibinfo {year} {2007})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Szabo}\ and\ \citenamefont
{Ostlund}(1989)}]{SzaboBook}%
\BibitemOpen
@ -503,6 +512,99 @@
{Fischer}},\ }\href {\doibase 10.1080/14786444908521726} {\bibfield
{journal} {\bibinfo {journal} {1949}\ }\textbf {\bibinfo {volume} {40}},\
\bibinfo {pages} {386}}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Hiscock}\ and\ \citenamefont
{Thom}(2014)}]{Hiscock_2014}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~G.}\ \bibnamefont
{Hiscock}}\ and\ \bibinfo {author} {\bibfnamefont {A.~J.~W.}\ \bibnamefont
{Thom}},\ }\href {\doibase 10.1021/ct5007696} {\bibfield {journal} {\bibinfo
{journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {10}},\
\bibinfo {pages} {4795} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Burton}\ \emph {et~al.}(2018)\citenamefont {Burton},
\citenamefont {Gross},\ and\ \citenamefont {Thom}}]{Burton_2018}%
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{\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume}
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{NoStop}%
\bibitem [{\citenamefont {Langreth}\ and\ \citenamefont
{Perdew}(1979)}]{Langreth_1975}%
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{Perdew}},\ }\href {\doibase https://doi.org/10.1016/0038-1098(79)90254-0}
{\bibfield {journal} {\bibinfo {journal} {Solid State Commun.}\ }\textbf
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{journal} {\bibinfo {journal} {J. Phys. Chem. A}\ }\textbf {\bibinfo
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\bibitem [{\citenamefont {Thom}\ and\ \citenamefont
{{Head-Gordon}}(2008)}]{Thom_2008}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~J.~W.}\
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{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf
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\bibitem [{\citenamefont {Shea}\ and\ \citenamefont
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\bibnamefont {Shea}}\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Neuscamman}},\ }\href {\doibase 10.1063/1.5045056} {\bibfield {journal}
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {149}},\
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\bibitem [{\citenamefont {M{\o}ller}\ and\ \citenamefont
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\BibitemOpen
@ -591,18 +693,6 @@
{Epstein}},\ }\href {\doibase 10.1103/PhysRev.28.695} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {28}},\
\bibinfo {pages} {695} (\bibinfo {year} {1926})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Seidl}\ \emph {et~al.}(2018)\citenamefont {Seidl},
\citenamefont {Giarrusso}, \citenamefont {Vuckovic}, \citenamefont
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{author} {\bibfnamefont {P.}~\bibnamefont {Gori-Giorgi}},\ }\href {\doibase
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Phys.}\ }\textbf {\bibinfo {volume} {149}},\ \bibinfo {pages} {241101}
(\bibinfo {year} {2018})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Cremer}\ and\ \citenamefont
{He}(1996)}]{Cremer_1996}%
\BibitemOpen
@ -752,15 +842,6 @@
}\href {\doibase 10.1103/PhysRevC.71.011304} {\bibfield {journal} {\bibinfo
{journal} {Phys. Rev. C}\ }\textbf {\bibinfo {volume} {71}},\ \bibinfo
{pages} {011304} (\bibinfo {year} {2005})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Cejnar}\ \emph {et~al.}(2007)\citenamefont {Cejnar},
\citenamefont {Heinze},\ and\ \citenamefont {Macek}}]{Cejnar_2007}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
{Cejnar}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Heinze}}, \
and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Macek}},\ }\href
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{pages} {100601} (\bibinfo {year} {2007})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Cejnar}\ and\ \citenamefont
{Jolie}(2009)}]{Cejnar_2009}%
\BibitemOpen

190
Manuscript/EPAWTFT.bib
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@ -1,13 +1,146 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-11-14 21:44:16 +0100
%% Created for Pierre-Francois Loos at 2020-11-18 21:23:03 +0100
%% Saved with string encoding Unicode (UTF-8)
@article{Zhang_2004,
author = {Zhang, Fan and Burke, Kieron},
date-added = {2020-11-18 21:23:02 +0100},
date-modified = {2020-11-18 21:23:02 +0100},
doi = {10.1103/PhysRevA.69.052510},
journal = {Phys. Rev. A},
pages = {052510},
title = {Adiabatic connection for near degenerate excited states},
volume = {69},
year = {2004},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.69.052510},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.69.052510}}
@article{Gunnarsson_1976,
author = {Gunnarsson, O. and Lundqvist, B. I.},
date-added = {2020-11-18 21:22:53 +0100},
date-modified = {2020-11-18 21:22:53 +0100},
doi = {10.1103/PhysRevB.13.4274},
issue = {10},
journal = {Phys. Rev. B},
month = {May},
numpages = {0},
pages = {4274--4298},
publisher = {American Physical Society},
title = {Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism},
url = {https://link.aps.org/doi/10.1103/PhysRevB.13.4274},
volume = {13},
year = {1976},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.13.4274},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.13.4274}}
@article{Langreth_1975,
author = {D.C. Langreth and J.P. Perdew},
date-added = {2020-11-18 21:22:40 +0100},
date-modified = {2020-11-18 21:22:40 +0100},
doi = {https://doi.org/10.1016/0038-1098(79)90254-0},
issn = {0038-1098},
journal = {Solid State Commun.},
number = {8},
pages = {567 - 571},
title = {The gradient approximation to the exchange-correlation energy functional: A generalization that works},
url = {http://www.sciencedirect.com/science/article/pii/0038109879902540},
volume = {31},
year = {1979},
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/0038109879902540},
Bdsk-Url-2 = {https://doi.org/10.1016/0038-1098(79)90254-0}}
@article{Shea_2018,
author = {J. A. R. Shea and E. Neuscamman},
date-added = {2020-11-18 21:17:15 +0100},
date-modified = {2020-11-18 21:17:15 +0100},
doi = {10.1063/1.5045056},
journal = {J. Chem. Phys.},
pages = {081101},
title = {A mean field platform for excited state quantum chemistry},
volume = {149},
year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5045056}}
@article{Shea_2017,
author = {Shea, Jacqueline A. R. and Neuscamman, Eric},
date-added = {2020-11-18 21:17:15 +0100},
date-modified = {2020-11-18 21:17:15 +0100},
doi = {10.1021/acs.jctc.7b00923},
issn = {1549-9618, 1549-9626},
journal = {J. Chem. Theory Comput.},
month = dec,
number = {12},
pages = {6078-6088},
title = {Size {{Consistent Excited States}} via {{Algorithmic Transformations}} between {{Variational Principles}}},
volume = {13},
year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b00923}}
@article{Thom_2008,
author = {Thom, Alex J. W. and {Head-Gordon}, Martin},
date-added = {2020-11-18 21:17:05 +0100},
date-modified = {2020-11-18 21:17:05 +0100},
doi = {10.1103/PhysRevLett.101.193001},
file = {/Users/loos/Zotero/storage/HVKYKGQU/Thom and Head-Gordon - 2008 - Locating Multiple Self-Consistent Field Solutions.pdf},
issn = {0031-9007, 1079-7114},
journal = {Phys. Rev. Lett.},
month = nov,
number = {19},
pages = {193001},
shorttitle = {Locating {{Multiple Self}}-{{Consistent Field Solutions}}},
title = {Locating {{Multiple Self}}-{{Consistent Field Solutions}}: {{An Approach Inspired}} by {{Metadynamics}}},
volume = {101},
year = {2008},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.101.193001}}
@article{Gilbert_2008,
author = {A. T. B. Gilbert and N. A. Besley and P. M. W. Gill},
date-added = {2020-11-18 21:16:52 +0100},
date-modified = {2020-11-18 21:16:52 +0100},
doi = {10.1021/jp801738f},
journal = {J. Phys. Chem. A},
pages = {13164},
title = {Self-Consistent Field Calculations of Excited States Using the Maximum Overlap Method {(MOM)}},
volume = {112},
year = {2008},
Bdsk-Url-1 = {https://doi.org/10.1021/jp801738f}}
@article{Burton_2018,
abstract = {We explore the existence and behavior of holomorphic restricted Hartree-Fock (h-RHF) solutions for two-electron problems. Through algebraic geometry, the exact number of solutions with n basis functions is rigorously identified as 1/2(3n - 1), proving that states must exist for all molecular geometries. A detailed study on the h-RHF states of HZ (STO3G) then demonstrates both the conservation of holomorphic solutions as geometry or atomic charges are varied and the emergence of complex h-RHF solutions at coalescence points. Using catastrophe theory, the nature of these coalescence points is described, highlighting the influence of molecular symmetry. The h-RHF states of HHeH2+ and HHeH (STO-3G) are then compared, illustrating the isomorphism between systems with two electrons and two electron holes. Finally, we explore the h-RHF states of ethene (STO-3G) by considering the $\pi$ electrons as a two-electron problem and employ NOCI to identify a crossing of the lowest energy singlet and triplet states at the perpendicular geometry.},
author = {Burton, Hugh G. A. and Gross, Mark and Thom, Alex J. W.},
date-added = {2020-11-18 21:16:36 +0100},
date-modified = {2020-11-18 21:16:36 +0100},
doi = {10.1021/acs.jctc.7b00980},
file = {/Users/loos/Zotero/storage/E9FNMAU8/Burton et al. - 2018 - Holomorphic Hartree--Fock Theory The Nature of Two.pdf},
issn = {1549-9618, 1549-9626},
journal = {J. Chem. Theory Comput.},
month = feb,
number = {2},
pages = {607-618},
shorttitle = {Holomorphic {{Hartree}}\textendash{{Fock Theory}}},
title = {Holomorphic {{Hartree}}\textendash{{Fock Theory}}: {{The Nature}} of {{Two}}-{{Electron Problems}}},
volume = {14},
year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b00980}}
@article{Burton_2016,
author = {H. G. A. Burton and A. J. W. Thom},
date-added = {2020-11-18 21:16:36 +0100},
date-modified = {2020-11-18 21:16:36 +0100},
doi = {10.1021/acs.jctc.5b01005},
journal = {J. Chem. Theory Comput.},
pages = {167},
title = {Holomorphic {Hartree--Fock} Theory: An Inherently Multireference Approach},
volume = {12},
year = {2016},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.5b01005}}
@article{Carrascal_2015,
abstract = {This review explains the relationship between density functional theory and strongly correlated models using the simplest possible example, the two-site Hubbard model. The relationship to traditional quantum chemistry is included. Even in this elementary example, where the exact ground-state energy and site occupations can be found analytically, there is much to be explained in terms of the underlying logic and aims of density functional theory. Although the usual solution is analytic, the density functional is given only implicitly. We overcome this difficulty using the Levy\textendash{}Lieb construction to create a parametrization of the exact function with negligible errors. The symmetric case is most commonly studied, but we find a rich variation in behavior by including asymmetry, as strong correlation physics vies with charge-transfer effects. We explore the behavior of the gap and the many-body Green's function, demonstrating the `failure' of the Kohn\textendash{}Sham (KS) method to reproduce the fundamental gap. We perform benchmark calculations of the occupation and components of the KS potentials, the correlation kinetic energies, and the adiabatic connection. We test several approximate functionals (restricted and unrestricted Hartree\textendash{}Fock and Bethe ansatz local density approximation) to show their successes and limitations. We also discuss and illustrate the concept of the derivative discontinuity. Useful appendices include analytic expressions for density functional energy components, several limits of the exact functional (weak- and strong-coupling, symmetric and asymmetric), various adiabatic connection results, proofs of exact conditions for this model, and the origin of the Hubbard model from a minimal basis model for stretched H2.},
author = {Carrascal, D J and Ferrer, J and Smith, J C and Burke, K},
@ -1170,35 +1303,34 @@
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.103.123008}}
@article{Cejnar_2003,
title = {Ground-State Shape Phase Transitions in Nuclei: {{Thermodynamic}} Analogy and Finite-\${{N}}\$ Effects},
author = {Cejnar, Pavel and Heinze, Stefan and Jolie, Jan},
year = {2003},
volume = {68},
pages = {034326},
publisher = {{American Physical Society}},
doi = {10.1103/PhysRevC.68.034326},
journal = {Phys. Rev. C}}
author = {Cejnar, Pavel and Heinze, Stefan and Jolie, Jan},
doi = {10.1103/PhysRevC.68.034326},
journal = {Phys. Rev. C},
pages = {034326},
publisher = {{American Physical Society}},
title = {Ground-State Shape Phase Transitions in Nuclei: {{Thermodynamic}} Analogy and Finite-\${{N}}\$ Effects},
volume = {68},
year = {2003},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevC.68.034326}}
@article{Cejnar_2000,
title = {Quantum Phase Transitions Studied within the Interacting Boson Model},
author = {Cejnar, Pavel and Jolie, Jan},
year = {2000},
volume = {61},
pages = {6237--6247},
publisher = {{American Physical Society}},
doi = {10.1103/PhysRevE.61.6237},
journal = {Phys. Rev. E}}
author = {Cejnar, Pavel and Jolie, Jan},
doi = {10.1103/PhysRevE.61.6237},
journal = {Phys. Rev. E},
pages = {6237--6247},
publisher = {{American Physical Society}},
title = {Quantum Phase Transitions Studied within the Interacting Boson Model},
volume = {61},
year = {2000},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevE.61.6237}}
@article{Cejnar_2007a,
title = {Phase Structure of Interacting Boson Models in Arbitrary Dimension},
author = {Cejnar, Pavel and Iachello, Francesco},
year = {2007},
volume = {40},
pages = {581--595},
publisher = {{IOP Publishing}},
doi = {10.1088/1751-8113/40/4/001},
journal = {J. Phys. A: Math. Theor.}}
author = {Cejnar, Pavel and Iachello, Francesco},
doi = {10.1088/1751-8113/40/4/001},
journal = {J. Phys. A: Math. Theor.},
pages = {581--595},
publisher = {{IOP Publishing}},
title = {Phase Structure of Interacting Boson Models in Arbitrary Dimension},
volume = {40},
year = {2007},
Bdsk-Url-1 = {https://doi.org/10.1088/1751-8113/40/4/001}}

18
Manuscript/EPAWTFT.out
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@ -6,12 +6,12 @@
\BOOKMARK [1][-]{section*.7}{Perturbation theory}{section*.2}% 6
\BOOKMARK [2][-]{section*.8}{Rayleigh-Schr\366dinger perturbation theory}{section*.7}% 7
\BOOKMARK [2][-]{section*.9}{The Hartree-Fock Hamiltonian}{section*.7}% 8
\BOOKMARK [2][-]{section*.10}{M\370ller-Plesset perturbation theory}{section*.7}% 9
\BOOKMARK [1][-]{section*.11}{Historical overview}{section*.2}% 10
\BOOKMARK [2][-]{section*.12}{Behavior of the M\370ller-Plesset series}{section*.11}% 11
\BOOKMARK [2][-]{section*.15}{Insights from a two-state model}{section*.11}% 12
\BOOKMARK [2][-]{section*.16}{The singularity structure}{section*.11}% 13
\BOOKMARK [2][-]{section*.17}{The physics of quantum phase transitions}{section*.11}% 14
\BOOKMARK [1][-]{section*.18}{Conclusion}{section*.2}% 15
\BOOKMARK [1][-]{section*.19}{Acknowledgments}{section*.2}% 16
\BOOKMARK [1][-]{section*.20}{References}{section*.2}% 17
\BOOKMARK [2][-]{section*.11}{M\370ller-Plesset perturbation theory}{section*.7}% 9
\BOOKMARK [1][-]{section*.12}{Historical overview}{section*.2}% 10
\BOOKMARK [2][-]{section*.13}{Behavior of the M\370ller-Plesset series}{section*.12}% 11
\BOOKMARK [2][-]{section*.16}{Insights from a two-state model}{section*.12}% 12
\BOOKMARK [2][-]{section*.17}{The singularity structure}{section*.12}% 13
\BOOKMARK [2][-]{section*.18}{The physics of quantum phase transitions}{section*.12}% 14
\BOOKMARK [1][-]{section*.19}{Conclusion}{section*.2}% 15
\BOOKMARK [1][-]{section*.20}{Acknowledgments}{section*.2}% 16
\BOOKMARK [1][-]{section*.21}{References}{section*.2}% 17

52
Manuscript/EPAWTFT.tex
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@ -243,6 +243,7 @@ E_{\text{S}} &= U.
\end{align}
\end{subequations}
While the open-shell triplet ($E_{\text{T}}$) and singlet ($E_{\text{S}}$) are independent of $\lambda$, the closed-shell singlet ground state ($E_{-}$) and doubly-excited state ($E_{+}$) couple strongly to form an avoided crossing at $\lambda=0$ (see Fig.~\ref{subfig:FCI_real}).
At non-zero values of $U$ and $t$, these closed-shell singlets can only become degenerate at a pair of complex conjugate points in the complex $\lambda$ plane
\begin{equation}
\lambda_{\text{EP}} = \pm \i \frac{U}{4t},
@ -329,15 +330,15 @@ Expanding the wave function and energy as power series in $\lambda$ as
\label{eq:E_expansion}
\end{align}
\end{subequations}
and solving the corresponding perturbation equations up to a given order $k$ then
yields approximate solutions to Eq.~\eqref{eq:SchrEq}.
and solving the corresponding perturbation equations up to a given order $k$, then
yields approximate solutions to Eq.~\eqref{eq:SchrEq} \titou{by setting $\lambda = 1$}.
% MATHEMATICAL REPRESENTATION
Mathematically, Eq.~\eqref{eq:E_expansion} corresponds to a Taylor series expansion of the exact energy
around the reference system $\lambda = 0$.
The energy of the target ``physical'' system is then recovered at the point $\lambda = 1$.
However, like all series expansions, the Eq.~\eqref{eq:E_expansion} has a radius of convergence $\rc$.
When $\rc < 1$, the Rayleigh--Sch\"{r}odinger expansion will diverge
When $\rc \titou{\le} 1$, the Rayleigh--Sch\"{r}odinger expansion will diverge
for the physical system.
The value of $\rc$ can vary significantly between different systems and strongly depends on the particular decomposition
of the reference and perturbation Hamiltonians in Eq.~\eqref{eq:SchrEq-PT}.\cite{Mihalka_2017b}
@ -366,15 +367,31 @@ This divergence occurs because $f(x)$ has four singularities in the complex
($\e^{\i\pi/4}$, $\e^{-\i\pi/4}$, $\e^{\i3\pi/4}$, and $\e^{-\i3\pi/4}$) with a modulus equal to $1$, demonstrating
that complex singularities are essential to fully understand the series convergence on the real axis.
The radius of convergence of the perturbation series is therefore dictated by the magnitude $|\lambda_0|$ of the
The radius of convergence of the perturbation series is therefore dictated by the magnitude $\abs{\lambda_0}$ of the
singularity in $E(\lambda)$ that is closest to the origin.
\hugh{Like the exact system in Section~\ref{sec:example}, the perturbation energy $E(\lambda)$ represents
Like the exact system in Section~\ref{sec:example}, the perturbation energy $E(\lambda)$ represents
a ``one-to-many'' function with the output elements representing an approximation to both the ground and excited states.
The most common singularities on $E(\lambda)$ therefore correspond to non-analytic EPs in the complex
$\lambda$ plane where two states become degenerate.
We will demonstrate how the choice of reference Hamiltonian controls the position of these EPs, and
ultimately determines the convergence properties of the perturbation series.
}
\titou{Practically, to locate EPs in a more complicated systems, one must solve simultaneously the following equations: \cite{Cejnar_2007}
\begin{subequations}
\begin{align}
\label{eq:PolChar}
\det[E\hI-\hH(\lambda)] & = 0,
\\
\label{eq:DPolChar}
\pdv{E}\det[E\hI-\hH(\lambda)] & = 0,
\end{align}
\end{subequations}
where $\hI$ is the identity operator.
Equation \eqref{eq:PolChar} is the well-known secular equation providing us with the (eigen)energies of the system.
If an energy is also solution of Eq.~\eqref{eq:DPolChar}, then this energy value is, at least, two-fold degenerate.
These degeneracies can be conical intersections between two states with different symmetries for real values of $\lambda$ \cite{Yarkony_1996} or EPs between two states with the same symmetry for complex values of $\lambda$.}
%============================================================%
\subsection{The Hartree-Fock Hamiltonian}
@ -428,14 +445,14 @@ from the one-electron Fock operators as
From hereon, $i$ and $j$ denote occupied orbitals, $a$ and $b$ denote unoccupied (or virtual) orbitals, while $p$, $q$, $r$, and $s$ denote arbitrary orbitals.
% BRIEF FLAVOURS OF HF
\hugh{In the most flexible variant of real HF theory (generalised HF) the one-electron orbitals can be complex-valued
In the most flexible variant of real HF theory (generalised HF) the one-electron orbitals can be complex-valued
and contain a mixture of spin-up and spin-down components.
However, the application of HF with some level of constraint on the orbital structure is far more common.
Forcing the spatial part of the orbitals to be the same for spin-up and spin-down electrons leads to restricted HF (RHF) theory, while allowing different for different spins leads to the so-called unrestricted HF (UHF) approach.
The advantage of the UHF approximation is its ability to correctly describe strongly correlated systems,
such as the dissociation of the hydrogen dimer.\cite{Coulson_1949}
However, by allowing different orbitals for different spins, the UHF is no longer required to be an eigenfunction of
the total spin $\hat{\mathcal{S}}^2$ operator, leading to so-called ``spin-contamination'' in the wave function.}
the total spin $\hat{\mathcal{S}}^2$ operator, leading to so-called ``spin-contamination'' in the wave function.
%
%The spatial part of the RHF wave function is then
@ -470,7 +487,7 @@ giving the symmetry-pure molecular orbitals
\end{align}
and the ground-state RHF energy
\begin{equation}
E_\text{RHF} = -2t + \frac{U}{2}
E_\text{RHF} \equiv E_\text{HF}(\ta^\text{RHF}, \tb^\text{RHF}) = -2t + \frac{U}{2}
\end{equation}
However, in the strongly correlated regime (large $U$), the closed-shell restriction on the orbitals prevents RHF from
correctly modelling the physics of the system with the two electrons on opposing sites.
@ -488,13 +505,28 @@ For $U \ge 2t$, the optimal orbital rotation angles for the UHF orbitals become
\end{align}
with the corresponding UHF ground-state energy
\begin{equation}
E_\text{UHF} = - \frac{2t^2}{U}.
E_\text{UHF} \equiv E_\text{HF}(\ta^\text{UHF}, \tb^\text{UHF}) = - \frac{2t^2}{U}.
\end{equation}
Time-reversal symmetry dictates that this UHF wave function must be degenerate with its spin-flipped pair, obtained
by swapping $\ta^{\text{UHF}}$ and $\tb^{\text{UHF}}$ in Eqs.~\eqref{eq:ta_uhf} and \eqref{eq:tb_uhf}.
Note that the RHF wave function remains a genuine solution of the HF equations for $U \ge 2t$, but corresponds to a saddle point
of the HF energy rather than a minimum.
\titou{In a recent paper, \cite{Burton_2019} using holomorphic Hartree-Fock (h-HF) \cite{Hiscock_2014,Burton_2018,Burton_2016} as an analytic continuation of conventional HF theory, we have demonstrated, on a simple model, that one can interconvert states of different symmetries and natures by following well-defined contours in the complex $\lambda$-plane, where $\lambda$ is the strength of the electron-electron interaction (see Fig.~\ref{fig:iAC}).
In particular, by slowly varying $\lambda$ in a similar (yet different) manner to an adiabatic connection in density-functional theory, \cite{Langreth_1975,Gunnarsson_1976,Zhang_2004} one can ``morph'' a ground-state wave function into an excited-state wave function via a stationary path of HF solutions. \cite{Seidl_2018}
In such a way, we could obtain a doubly-excited state wave function starting from the ground state wave function, a process which is not as easy as one might think. \cite{Gilbert_2008,Thom_2008,Shea_2018}
One of the fundamental discovery we made was that Coulson-Fisher points (where multiple symmetry-broken solutions coalesce) play a central role and can be classified as \textit{quasi}-exceptional points, as the wave functions do not become self-orthogonal.
The findings reported in Ref.~\onlinecite{Burton_2019} represent the very first study of non-Hermitian quantum mechanics for the exploration of multiple solutions at the HF level.
It perfectly illustrates the deeper topology of electronic states revealed using a complex-scaled electron-electron interaction.
Through the introduction of non-Hermiticity, we have provided a more general framework in which the complex and diverse characteristics of multiple solutions can be explored and understood.}
\begin{figure}
\includegraphics[width=\linewidth]{iAC}
\caption{
\label{fig:iAC}
An example of complex adiabatic connection. \cite{Burton_2019}}
\end{figure}
%=====================================================%
\subsection{M{\o}ller-Plesset perturbation theory}
%=====================================================%

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