degreenified Hugh stuff
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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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\begin{thebibliography}{94}%
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\begin{thebibliography}{91}%
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\makeatletter
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\providecommand \@ifxundefined [1]{%
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\@ifx{#1\undefined}
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@ -529,6 +529,15 @@
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{Thom}},\ }\href {\doibase 10.1021/ct5007696} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {10}},\
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\bibinfo {pages} {4795} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Burton}\ and\ \citenamefont
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{Thom}(2016)}]{Burton_2016}%
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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.5b01005} {\bibfield
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{journal} {\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo
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{volume} {12}},\ \bibinfo {pages} {167} (\bibinfo {year} {2016})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Burton}\ \emph {et~al.}(2018)\citenamefont {Burton},
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\citenamefont {Gross},\ and\ \citenamefont {Thom}}]{Burton_2018}%
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\BibitemOpen
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@ -539,15 +548,6 @@
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{\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume}
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{14}},\ \bibinfo {pages} {607} (\bibinfo {year} {2018})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Burton}\ and\ \citenamefont
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{Thom}(2016)}]{Burton_2016}%
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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.5b01005} {\bibfield
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{journal} {\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo
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{volume} {12}},\ \bibinfo {pages} {167} (\bibinfo {year} {2016})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Langreth}\ and\ \citenamefont
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{Perdew}(1979)}]{Langreth_1975}%
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\BibitemOpen
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@ -574,46 +574,6 @@
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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 {Seidl}\ \emph {et~al.}(2018)\citenamefont {Seidl},
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\citenamefont {Giarrusso}, \citenamefont {Vuckovic}, \citenamefont
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{Fabiano},\ and\ \citenamefont {Gori-Giorgi}}]{Seidl_2018}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{Seidl}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Giarrusso}},
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\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Vuckovic}}, \bibinfo
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{author} {\bibfnamefont {E.}~\bibnamefont {Fabiano}}, \ and\ \bibinfo
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{author} {\bibfnamefont {P.}~\bibnamefont {Gori-Giorgi}},\ }\href {\doibase
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10.1063/1.5078565} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
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Phys.}\ }\textbf {\bibinfo {volume} {149}},\ \bibinfo {pages} {241101}
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(\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Gilbert}\ \emph {et~al.}(2008)\citenamefont
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{Gilbert}, \citenamefont {Besley},\ and\ \citenamefont
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{Gill}}]{Gilbert_2008}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~T.~B.}\
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\bibnamefont {Gilbert}}, \bibinfo {author} {\bibfnamefont {N.~A.}\
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\bibnamefont {Besley}}, \ and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\
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\bibnamefont {Gill}},\ }\href {\doibase 10.1021/jp801738f} {\bibfield
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{journal} {\bibinfo {journal} {J. Phys. Chem. A}\ }\textbf {\bibinfo
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{volume} {112}},\ \bibinfo {pages} {13164} (\bibinfo {year}
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{2008})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Thom}\ and\ \citenamefont
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{{Head-Gordon}}(2008)}]{Thom_2008}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~J.~W.}\
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\bibnamefont {Thom}}\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{{Head-Gordon}}},\ }\href {\doibase 10.1103/PhysRevLett.101.193001}
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{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf
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{\bibinfo {volume} {101}},\ \bibinfo {pages} {193001} (\bibinfo {year}
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{2008})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Shea}\ and\ \citenamefont
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{Neuscamman}(2018)}]{Shea_2018}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.~R.}\
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\bibnamefont {Shea}}\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
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{Neuscamman}},\ }\href {\doibase 10.1063/1.5045056} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {149}},\
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\bibinfo {pages} {081101} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {M{\o}ller}\ and\ \citenamefont
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{Plesset}(1934)}]{Moller_1934}%
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\BibitemOpen
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@ -702,6 +662,18 @@
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{Epstein}},\ }\href {\doibase 10.1103/PhysRev.28.695} {\bibfield {journal}
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{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {28}},\
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\bibinfo {pages} {695} (\bibinfo {year} {1926})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Seidl}\ \emph {et~al.}(2018)\citenamefont {Seidl},
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\citenamefont {Giarrusso}, \citenamefont {Vuckovic}, \citenamefont
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{Fabiano},\ and\ \citenamefont {Gori-Giorgi}}]{Seidl_2018}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{Seidl}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Giarrusso}},
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\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Vuckovic}}, \bibinfo
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{author} {\bibfnamefont {E.}~\bibnamefont {Fabiano}}, \ and\ \bibinfo
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{author} {\bibfnamefont {P.}~\bibnamefont {Gori-Giorgi}},\ }\href {\doibase
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10.1063/1.5078565} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
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Phys.}\ }\textbf {\bibinfo {volume} {149}},\ \bibinfo {pages} {241101}
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(\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Cremer}\ and\ \citenamefont
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{He}(1996)}]{Cremer_1996}%
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\BibitemOpen
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@ -471,8 +471,8 @@ the total spin $\hat{\mathcal{S}}^2$ operator, leading to so-called ``spin-conta
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\begin{figure}
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\includegraphics[width=\linewidth]{HF_real.pdf}
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\caption{\label{fig:HF_real}
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\hugh{RHF and UHF energies as a function of the correlation strength $U/t$.
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The symmetry-broken UHF solution emerges at the coalescence point $U=2t$ (black dot).}}
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RHF and UHF energies as a function of the correlation strength $U/t$.
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The symmetry-broken UHF solution emerges at the coalescence point $U=2t$ (black dot) known as the Coulson-Fischer point.}
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\end{figure}
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%%%%%%%%%%%%%%%%%
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@ -518,12 +518,12 @@ correctly modelling the physics of the system with the two electrons on opposing
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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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\hugh{(\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$.
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(\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$.
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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 constant plane at $\pi/2$.
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(\subref{subfig:UHF_cplx_energy}) The corresponding HF energy surfaces show a non-analytic
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point at the \textit{quasi}-EP.}
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point at the \textit{quasi}-EP.
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\label{fig:HF_cplx}}
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\end{figure*}
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%%%%%%%%%%%%%%%%%
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@ -553,7 +553,7 @@ of the HF energy rather than a minimum.
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%============================================================%
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% INTRODUCE PARAMETRISED FOCK HAMILTONIAN
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\hugh{The inherent non-linearity in the Fock eigenvalue problem arises from self-consistency
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The inherent non-linearity in the Fock eigenvalue problem arises from self-consistency
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in the HF approximation, and is usually solved through an iterative approach.\cite{SzaboBook}
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Alternatively, the non-linear terms arising from the Coulomb and exchange operators can
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be considered as a perturbation from the core Hamiltonian by introducing the
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@ -562,10 +562,10 @@ transformation $U \rightarrow \lambda\, U$, giving the parametrised Fock operato
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\Hat{f}_{\lambda}(\vb{x}) = \Hat{h}(\vb{x}) + \lambda\, \Hat{v}_\text{HF}(\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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Hamiltonian, while self-consistent solutions at $\lambda = 1$ represent the orbitals of the true HF solution.
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% INTRODUCE COMPLEX ANALYTIC-CONTINUATION
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\hugh{For real $\lambda$, the self-consistent HF energies at given (real) $U$ and $t$ values
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For real $\lambda$, the self-consistent HF energies at given (real) $U$ and $t$ values
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in the Hubbard dimer directly mirror the energies shown in Fig.~\ref{fig:HF_real},
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with coalesence points at
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\begin{equation}
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@ -586,10 +586,9 @@ For example, in the Hubbard dimer with $U > 2t$, one finds $\lambda_{\text{c}} <
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do not provide a good representation of the HF ground state.
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In contrast, $U < 2t$ yields $\lambda_{\text{c}} > 1$ and corresponds to
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the regime where the HF ground state is correctly represented by symmetry-pure orbitals.
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}
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% COMPLEX ADIABATIC CONNECTION
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\hugh{We have recently shown that the complex scaled Fock operator Eq.~\eqref{eq:scaled_fock}
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We have recently shown that the complex scaled Fock operator Eq.~\eqref{eq:scaled_fock}
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also allows states of different symmetries to be interconverted by following a well-defined
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contour in the complex $\lambda$-plane.\cite{Burton_2019}
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In particular, by slowly varying $\lambda$ in a similar (yet different) manner
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@ -597,12 +596,12 @@ to an adiabatic connection in density-functional theory,\cite{Langreth_1975,Gunn
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a ground-state wave function can be ``morphed'' into an excited-state wave function
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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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role of \textit{quasi}-EPs in determining the behaviour of the HF approximation.
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%\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}).
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%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}
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%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}
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%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.
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%One of the fundamental discovery we made was that Coulson-Fischer 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.
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%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.
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%It perfectly illustrates the deeper topology of electronic states revealed using a complex-scaled electron-electron interaction.
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%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.}
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@ -662,6 +661,7 @@ iii) the strong coupling partitioning where the two operators are inverted compa
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Let us illustrate the behaviour of the RMP and UMP series on the Hubbard dimer (see Fig.~\ref{fig:RMP}).
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Within the RMP partition technique, we have
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\begin{widetext}
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\begin{equation}
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\label{eq:H_RMP}
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\bH_\text{RMP} =
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@ -672,12 +672,13 @@ Within the RMP partition technique, we have
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\lambda U/2 & 0 & 0 & 2t + U - \lambda U/2 \\
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\end{pmatrix},
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\end{equation}
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\end{widetext}
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which yields the ground-state energy
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\begin{equation}
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\label{eq:E0MP}
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E_{-}(\lambda) = U - \frac{\lambda U}{2} - \frac{1}{2} \sqrt{(4t)^2 + \lambda ^2 U^2}.
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\end{equation}
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From this expression, it is easy to identify that the radius of convergence is defined by the EPs at $\lambda = \pm i 4t / U$ (which are actually located at the same position than the exact EPs discussed in Sec.~\ref{sec:example}).
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From this expression, it is easy to identify that the radius of convergence is defined by the EPs at $\lambda_c = \pm i 4t / U$ (which are actually located at the same position than the exact EPs discussed in Sec.~\ref{sec:example}).
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Equation \eqref{eq:E0MP} can be Taylor expanded in terms of $\lambda$ to obtain the $n$th-order MP correction
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\begin{equation}
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E_\text{MP}^{(k)} = U \delta_{0,k} - \frac{1}{2} \frac{U^k}{(4t)^{k-1}} \mqty( 1/2 \\ k/2),
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