Added some references, reorganised some of Section II
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Hugh Burton
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@ -472,34 +472,6 @@
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{Ernzerhof}},\ }\href {\doibase 10.1063/1.2348880} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {125}},\
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\bibinfo {pages} {124104} (\bibinfo {year} {2006})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Carrascal}\ \emph {et~al.}(2015)\citenamefont
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{Carrascal}, \citenamefont {Ferrer}, \citenamefont {Smith},\ and\
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\citenamefont {Burke}}]{Carrascal_2015}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont
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{Carrascal}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Ferrer}},
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\bibinfo {author} {\bibfnamefont {J.~C.}\ \bibnamefont {Smith}}, \ and\
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\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Burke}},\ }\href
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{\doibase 10.1088/0953-8984/27/39/393001} {\bibfield {journal} {\bibinfo
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{journal} {J. Phys. Condens. Matter}\ }\textbf {\bibinfo {volume} {27}},\
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\bibinfo {pages} {393001} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Carrascal}\ \emph {et~al.}(2018)\citenamefont
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{Carrascal}, \citenamefont {Ferrer}, \citenamefont {Maitra},\ and\
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\citenamefont {Burke}}]{Carrascal_2018}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont
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{Carrascal}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Ferrer}},
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\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Maitra}}, \ and\ \bibinfo
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{author} {\bibfnamefont {K.}~\bibnamefont {Burke}},\ }\href {\doibase
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10.1140/epjb/e2018-90114-9} {\bibfield {journal} {\bibinfo {journal} {Eur.
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Phys. J. B}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo {pages} {142}
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(\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Wigner}(1934)}]{Wigner_1934}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
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{Wigner}},\ }\href {\doibase 10.1103/PhysRev.46.1002} {\bibfield {journal}
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{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {46}},\
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\bibinfo {pages} {1002} (\bibinfo {year} {1934})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Taut}(1993)}]{Taut_1993}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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@ -530,6 +502,34 @@
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{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
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{volume} {108}},\ \bibinfo {pages} {083002} (\bibinfo {year}
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{2012})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Carrascal}\ \emph {et~al.}(2015)\citenamefont
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{Carrascal}, \citenamefont {Ferrer}, \citenamefont {Smith},\ and\
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\citenamefont {Burke}}]{Carrascal_2015}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont
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{Carrascal}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Ferrer}},
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\bibinfo {author} {\bibfnamefont {J.~C.}\ \bibnamefont {Smith}}, \ and\
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\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Burke}},\ }\href
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{\doibase 10.1088/0953-8984/27/39/393001} {\bibfield {journal} {\bibinfo
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{journal} {J. Phys. Condens. Matter}\ }\textbf {\bibinfo {volume} {27}},\
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\bibinfo {pages} {393001} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Carrascal}\ \emph {et~al.}(2018)\citenamefont
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{Carrascal}, \citenamefont {Ferrer}, \citenamefont {Maitra},\ and\
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\citenamefont {Burke}}]{Carrascal_2018}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont
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{Carrascal}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Ferrer}},
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\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Maitra}}, \ and\ \bibinfo
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{author} {\bibfnamefont {K.}~\bibnamefont {Burke}},\ }\href {\doibase
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10.1140/epjb/e2018-90114-9} {\bibfield {journal} {\bibinfo {journal} {Eur.
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Phys. J. B}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo {pages} {142}
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(\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Wigner}(1934)}]{Wigner_1934}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
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{Wigner}},\ }\href {\doibase 10.1103/PhysRev.46.1002} {\bibfield {journal}
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{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {46}},\
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\bibinfo {pages} {1002} (\bibinfo {year} {1934})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Goodson}(2011)}]{Goodson_2011}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~Z.}\ \bibnamefont
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@ -573,6 +573,20 @@
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{journal} {\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo
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{volume} {7}},\ \bibinfo {pages} {2667} (\bibinfo {year} {2011})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Stuber}\ and\ \citenamefont
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{Paldus}(2003)}]{StuberPaldus}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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{Stuber}}\ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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{Paldus}},\ }\enquote {\bibinfo {title} {{Symmetry Breaking in the
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Independent Particle Model}},}\ in\ \href@noop {} {\emph {\bibinfo
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{booktitle} {Fundamental World of Quantum Chemistry: A Tribute to the Memory
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of Per-Olov L\"{o}wdin}}},\ Vol.~\bibinfo {volume} {1},\ \bibinfo {editor}
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{edited by\ \bibinfo {editor} {\bibfnamefont {E.~J.}\ \bibnamefont
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{Br\"{a}ndas}}\ and\ \bibinfo {editor} {\bibfnamefont {E.~S.}\ \bibnamefont
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{Kryachko}}}\ (\bibinfo {publisher} {Kluwer Academic},\ \bibinfo {address}
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{Dordrecht},\ \bibinfo {year} {2003})\ p.~\bibinfo {pages} {67}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Coulson}\ and\ \citenamefont
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{Fischer}(1949)}]{Coulson_1949}%
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\BibitemOpen
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@ -591,20 +605,6 @@
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{title} {Quantum {Theory} of the {Electron} {Liquid}}}}\ (\bibinfo
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{publisher} {Cambridge University Press},\ \bibinfo {year}
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{2005})\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Stuber}\ and\ \citenamefont
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{Paldus}(2003)}]{StuberPaldus}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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{Stuber}}\ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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{Paldus}},\ }\enquote {\bibinfo {title} {{Symmetry Breaking in the
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Independent Particle Model}},}\ in\ \href@noop {} {\emph {\bibinfo
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{booktitle} {Fundamental World of Quantum Chemistry: A Tribute to the Memory
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of Per-Olov L\"{o}wdin}}},\ Vol.~\bibinfo {volume} {1},\ \bibinfo {editor}
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{edited by\ \bibinfo {editor} {\bibfnamefont {E.~J.}\ \bibnamefont
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{Br\"{a}ndas}}\ and\ \bibinfo {editor} {\bibfnamefont {E.~S.}\ \bibnamefont
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{Kryachko}}}\ (\bibinfo {publisher} {Kluwer Academic},\ \bibinfo {address}
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{Dordrecht},\ \bibinfo {year} {2003})\ p.~\bibinfo {pages} {67}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Fukutome}()}]{Fukutome_1981}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
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@ -148,7 +148,7 @@ In particular, we highlight the seminal work of several research groups on the c
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\tableofcontents
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%%%%%%%%%%%%%%%%%%%%%%%
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\section{Background}
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\section{Introduction}
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%
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@ -186,12 +186,48 @@ Through a complex-scaling of the electronic or atomic coordinates,\cite{Moiseyev
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We refer the interested reader to the excellent book of Moiseyev for a general overview. \cite{MoiseyevBook}}
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%%%%%%%%%%%%%%%%%%%%%%%
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\section{Exceptional points in electronic structure}
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\section{Exceptional Points in Electronic Structure}
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\label{sec:EPs}
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%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Time-Independent Schr\"odinger Equation}
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\label{sec:TDSE}
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%%%%%%%%%%%%%%%%%%%%%%%
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Within the Born-Oppenheimer approximation, the exact molecular Hamiltonian with $\Ne$ electrons and
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$\Nn$ (clamped) nuclei is defined \hugh{for a given nuclear framework} as
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\begin{equation}\label{eq:ExactHamiltonian}
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\hugh{\hH(\vb{R})} =
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- \frac{1}{2} \sum_{i}^{\Ne} \grad_i^2
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- \sum_{i}^{\Ne} \sum_{A}^{\Nn} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}}
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+ \sum_{i<j}^{\Ne}\frac{1}{\abs{\vb{r}_i-\vb{r}_j}},
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\end{equation}
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where $\vb{r}_i$ defines the position of the $i$-th electron, $\vb{R}_{A}$ and $Z_{A}$ are the position
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and charge of the $A$-th nucleus respectively, \hugh{and $\vb{R} = (\vb{R}_{1}, \dots, \vb{R}_{\Nn})$ is a
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collective vector for the nuclear positions.}
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The first term represents the kinetic energy of the electrons, while
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the two following terms account for the electron-nucleus attraction and the electron-electron repulsion.
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% EXACT SCHRODINGER EQUATION
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The exact many-electron wave function \hugh{at a given nuclear geometry} $\Psi(\vb{R})$ corresponds
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to the solution of the (time-independent) Schr\"{o}dinger equation
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\begin{equation}
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\hugh{\hH(\vb{R})\, \Psi(\vb{R}) = E(\vb{R})\, \Psi(\vb{R}),}
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\label{eq:SchrEq}
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\end{equation}
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with the eigenvalues $E(\vb{R})$ providing the exact energies.
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\hugh{The energy $E(\vb{R})$ can be considered as a ``one-to-many'' function since each input nuclear geometry
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yields several eigenvalues corresponding to the ground and excited states of the exact spectrum.}
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However, exact solutions to Eq.~\eqref{eq:SchrEq} are only possible in the simplest of systems, such as
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the one-electron hydrogen atom and some specific two-electron systems with well-defined mathematical
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properties.\cite{Taut_1993,Loos_2009b,Loos_2010e,Loos_2012}
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\hugh{In practice, approximations to the exact Schr\"{o}dinger equation must be introduced, including
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the perturbation theories and Hartree--Fock approximation considered in this review
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In what follows, we will drop the parametric dependence on the nuclear geometry and,
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unless otherwise stated, atomic units will be used throughout.}
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%===================================%
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\subsection{Exceptional points in the Hubbard dimer}
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\subsection{Exceptional Points in the Hubbard Dimer}
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\label{sec:example}
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%===================================%
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@ -213,9 +249,10 @@ We refer the interested reader to the excellent book of Moiseyev for a general o
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\end{figure*}
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To illustrate the concepts discussed throughout this article, we consider the symmetric Hubbard dimer at half filling, \ie\ with two opposite-spin fermions.
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Analytically solvable model systems are essential in theoretical chemistry and physics as the simplicity of the
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mathematics compared to realistic systems (e.g., atoms and molecules) readily allows concepts to be illustrated and new methods to be tested wile retaining much
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of the key physics.
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Analytic\trashHB{ally solvable} model systems are essential in theoretical chemistry and physics as their \hugh{mathematical} simplicity \trashHB{of the
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mathematics} compared to realistic systems (e.g., atoms and molecules) allows new concepts and methods to be
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easily \trashHB{illustrated and} tested while retaining the key physical phenomena.
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\hugh{(HGAB: This sentence felt too long to me. Feel free to re-instate words if you think they are neccessary)}
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Using the (localised) site basis, the Hilbert space of the Hubbard dimer comprises the four configurations
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\begin{align}
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@ -237,10 +274,12 @@ where $t$ is the hopping parameter and $U$ is the on-site Coulomb repulsion.
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We refer the interested reader to Refs.~\onlinecite{Carrascal_2015,Carrascal_2018} for more details about this system.
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The parameter $U$ controls the strength of the electron correlation.
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In the weak correlation regime (small $U$), the kinetic energy dominates and the electrons are delocalised over both sites.
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In the large-$U$ (or strong correlation) regime, the electron repulsion term drives the physics and the electrons localise on opposite sites to minimise their Coulomb repulsion.
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In the large-$U$ (or strong correlation) regime, the electron repulsion term \hugh{becomes dominant} \trashHB{drives the physics}
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and the electrons localise on opposite sites to minimise their Coulomb repulsion.
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This phenomenon is often referred to as Wigner crystallisation. \cite{Wigner_1934}
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To illustrate the formation of an EP, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t\rightarrow \lambda t$.
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To illustrate the formation of an EP, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t\rightarrow \lambda t$
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\hugh{to give the parameterised Hamiltonian $\hH(\lambda)$.}
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When $\lambda$ is real, the Hamiltonian~\eqref{eq:H_FCI} is Hermitian with the distinct (real-valued) (eigen)energies
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\begin{subequations}
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\begin{align}
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@ -281,36 +320,31 @@ such that $E_{\pm}(2\pi) = E_{\mp}(0)$ and $E_{\pm}(4\pi) = E_{\pm}(0)$.
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As a result, completely encircling an EP leads to the interconversion of the two interacting states, while a second complete rotation returns the two states to their original energies.
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Additionally, the wave functions pick up a geometric phase in the process, and four complete loops are required to recover their starting forms.\cite{MoiseyevBook}
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% LOCATING EPS
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\hugh{To locate EPs in practice, one must simultaneously solve
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\begin{subequations}
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\begin{align}
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\label{eq:PolChar}
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\det[E\hI-\hH(\lambda)] & = 0,
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\\
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\label{eq:DPolChar}
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\pdv{E}\det[E\hI-\hH(\lambda)] & = 0,
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\end{align}
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\end{subequations}
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where $\hI$ is the identity operator.\cite{Cejnar_2007}
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Equation \eqref{eq:PolChar} is the well-known secular equation providing the (eigen)energies of the system.
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If the energy is also solution of Eq.~\eqref{eq:DPolChar}, then this energy value is at least two-fold degenerate.
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These degeneracies can be conical intersections between two states with different symmetries
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for real values of $\lambda$,\cite{Yarkony_1996} or EPs between two states with the
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same symmetry for complex values of $\lambda$.}
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%============================================================%
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\subsection{Rayleigh-Schr\"odinger perturbation theory}
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\subsection{Rayleigh-Schr\"odinger Perturbation Theory}
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%============================================================%
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Within the Born-Oppenheimer approximation, the exact molecular Hamiltonian with $\Ne$ electrons and
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$\Nn$ (clamped) nuclei is defined as
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\begin{equation}\label{eq:ExactHamiltonian}
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\hH =
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- \frac{1}{2} \sum_{i}^{\Ne} \grad_i^2
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- \sum_{i}^{\Ne} \sum_{A}^{\Nn} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}}
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+ \sum_{i<j}^{\Ne}\frac{1}{\abs{\vb{r}_i-\vb{r}_j}},
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\end{equation}
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where $\vb{r}_i$ defines the position of the $i$-th electron, and $\vb{R}_{A}$ and $Z_{A}$ are the position
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and charge of the $A$-th nucleus respectively.
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The first term represents the kinetic energy of the electrons, while
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the two following terms account for the electron-nucleus attraction and the electron-electron repulsion.
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Atomic units are used unless otherwise stated.
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% EXACT SCHRODINGER EQUATION
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The exact many-electron wave function $\Psi$ corresponds to the solution of the (time-independent)
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Schr\"{o}dinger equation
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\begin{equation}
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\hH\Psi = E \Psi,
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\label{eq:SchrEq}
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\end{equation}
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with the eigenvalue $E$ providing the exact energy.
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Exact solutions to Eq.~\eqref{eq:SchrEq} are only possible in the simplest of systems, such as
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the one-electron hydrogen atom and some specific two-electron systems with well-defined mathematical properties. \cite{Taut_1993,Loos_2009b,Loos_2010e,Loos_2012}
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In practice, one of the most common approximations involves
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a perturbative expansion of the energy.
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\hugh{One of the most common routes to approximately solving the Schr\"odinger equation
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is to introduce a perturbative expansion of the exact energy.}
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% SUMMARY OF RS-PT
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Within Rayleigh-Schr\"odinger perturbation theory, the time-independent Schr\"odinger equation
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is recast as
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@ -363,13 +397,13 @@ For example, the simple function
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\end{equation}
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is smooth and infinitely differentiable for $x \in \mathbb{R}$, and one might expect that its Taylor series expansion would
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converge in this domain.
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However, this series diverges $x \ge 1$.
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However, this series diverges for $x \ge 1$.
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This divergence occurs because $f(x)$ has four singularities in the complex
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($\e^{\i\pi/4}$, $\e^{-\i\pi/4}$, $\e^{\i3\pi/4}$, and $\e^{-\i3\pi/4}$) with a modulus equal to $1$, demonstrating
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that complex singularities are essential to fully understand the series convergence on the real axis.\cite{BenderBook}
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\titou{Include Antoine's example $\sum_{n=1}^\infty \lambda^n/n$ which is divergent at $\lambda = 1$ but convergent at $\lambda = -1$.}
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The radius of convergence for the perturbation series is therefore dictated by the magnitude \titou{$\abs{\lambda_0}$} of the
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The radius of convergence for the perturbation series Eq.~\eqref{eq:E_expansion} is therefore dictated by the magnitude \titou{$\abs{\lambda_0}$} of the
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singularity in $E(\lambda)$ that is closest to the origin.
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Like the exact system in Sec.~\ref{sec:example}, the perturbation energy $E(\lambda)$ represents
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a ``one-to-many'' function with the output elements representing an approximation to both the ground and excited states.
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@ -378,26 +412,8 @@ $\lambda$ plane where two states become degenerate.
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Later we will demonstrate how the choice of reference Hamiltonian controls the position of these EPs, and
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ultimately determines the convergence properties of the perturbation series.
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Practically, to locate EPs, one must simultaneously solve
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\begin{subequations}
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\begin{align}
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\label{eq:PolChar}
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\det[E\hI-\hH(\lambda)] & = 0,
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\\
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\label{eq:DPolChar}
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\pdv{E}\det[E\hI-\hH(\lambda)] & = 0,
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\end{align}
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\end{subequations}
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where $\hI$ is the identity operator.\cite{Cejnar_2007}
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Equation \eqref{eq:PolChar} is the well-known secular equation providing the (eigen)energies of the system.
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If the energy is also solution of Eq.~\eqref{eq:DPolChar}, then this energy value is at least two-fold degenerate.
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These degeneracies can be conical intersections between two states with different symmetries
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for real values of $\lambda$,\cite{Yarkony_1996} or EPs between two states with the
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same symmetry for complex values of $\lambda$.
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%===========================================%
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\subsection{Hartree-Fock theory}
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\subsection{Hartree-Fock Theory}
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\label{sec:HF}
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%===========================================%
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@ -407,7 +423,7 @@ This Slater determinant is defined as an antisymmetric combination of $\Ne$ (rea
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\begin{equation}\label{eq:FockOp}
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\Hat{f}(\vb{x}) \phi_p(\vb{x}) = \qty[ \Hat{h}(\vb{x}) + \Hat{v}_\text{HF}(\vb{x}) ] \phi_p(\vb{x}) = \epsilon_p \phi_p(\vb{x}).
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\end{equation}
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Here the core Hamiltonian is
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Here the \hugh{(one-electron)} core Hamiltonian is
|
||||
\begin{equation}
|
||||
\label{eq:Hcore}
|
||||
\Hat{h}(\vb{x}) = -\frac{\grad^2}{2} + \sum_{A}^{M} \frac{Z_A}{\abs{\vb{r}-\vb{R}_A}}
|
||||
|
|
@ -454,14 +470,15 @@ From hereon, $i$ and $j$ denote occupied orbitals, $a$ and $b$ denote unoccupied
|
|||
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.\cite{Mayer_1993,Jimenez-Hoyos_2011}
|
||||
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 orbitals for different spins leads to the so-called unrestricted HF (UHF) approach.
|
||||
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 orbitals for different spins leads to the so-called unrestricted HF (UHF) approach.\cite{StuberPaldus}
|
||||
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}
|
||||
such as antiferromagnetic phases\cite{Slater_1951} or 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 ``spin-contamination'' in the wave function.
|
||||
|
||||
%================================================================%
|
||||
\subsection{The Hartree-Fock approximation in the Hubbard dimer}
|
||||
\subsection{Hartree-Fock in the Hubbard Dimer}
|
||||
\label{sec:HF_hubbard}
|
||||
%================================================================%
|
||||
|
||||
|
|
@ -472,13 +489,13 @@ the total spin $\hat{\mathcal{S}}^2$ operator, leading to ``spin-contamination''
|
|||
\includegraphics[width=\linewidth]{HF_real.pdf}
|
||||
\caption{\label{fig:HF_real}
|
||||
RHF and UHF energies as a function of the correlation strength $U/t$.
|
||||
The symmetry-broken UHF solution emerges at the coalescence point $U=2t$ (black dot) known as the Coulson-Fischer point.}
|
||||
The symmetry-broken UHF solution emerges at the coalescence point $U=2t$ (black dot), often known as the Coulson-Fischer point.}
|
||||
\end{figure}
|
||||
%%%%%%%%%%%%%%%%%
|
||||
|
||||
Returning to the Hubbard dimer, the UHF energy can be parametrised in terms of two rotation angles $\ta$ and $\tb$ as
|
||||
In the Hubbard dimer, the UHF energy can be parametrised using two rotation angles $\ta$ and $\tb$ as
|
||||
\begin{equation}
|
||||
E_\text{HF}(\ta, \tb) = -t \qty( \sin \ta + \sin \tb ) + \frac{U}{2} \qty( 1 + \cos \ta \cos \tb ),
|
||||
E_\text{HF}(\ta, \tb) = -t\, \qty( \sin \ta + \sin \tb ) + \frac{U}{2} \qty( 1 + \cos \ta \cos \tb ),
|
||||
\end{equation}
|
||||
where we have introduced bonding $\mathcal{B}^{\sigma}$ and anti-bonding $\mathcal{A}^{\sigma}$ molecular orbitals for
|
||||
the spin-$\sigma$ electrons as
|
||||
|
|
@ -502,8 +519,8 @@ and the ground-state RHF energy (Fig.~\ref{fig:HF_real})
|
|||
\begin{equation}
|
||||
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 $U>2t$, the closed-shell restriction on the orbitals prevents RHF from
|
||||
correctly modelling the physics of the system with the two electrons on opposing sites.
|
||||
However, in the strongly correlated regime $U>2t$, the closed-shell orbital restriction prevents RHF from
|
||||
modelling the correct physics with the two electrons on opposing sites.
|
||||
|
||||
%%% FIG 3 (?) %%%
|
||||
% Analytic Continuation of HF
|
||||
|
|
@ -521,7 +538,7 @@ correctly modelling the physics of the system with the two electrons on opposing
|
|||
(\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$.
|
||||
Symmetry-broken solutions correspond to individual sheets and become equivalent at
|
||||
the \textit{quasi}-EP $\lambda_{\text{c}}$ (black dot).
|
||||
The RHF solution is independent of $\lambda$, giving constant plane at $\pi/2$.
|
||||
The RHF solution is independent of $\lambda$, giving the constant plane at $\pi/2$.
|
||||
(\subref{subfig:UHF_cplx_energy}) The corresponding HF energy surfaces show a non-analytic
|
||||
point at the \textit{quasi}-EP.
|
||||
\label{fig:HF_cplx}}
|
||||
|
|
@ -547,21 +564,23 @@ with the corresponding UHF ground-state energy (Fig.~\ref{fig:HF_real})
|
|||
\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}.
|
||||
This type of symmetry breaking is also called a spin-density wave in the physics community as the system ``oscillates'' between the two symmetry-broken configurations. \cite{GiulianiBook}
|
||||
There also exist symmetry-breaking processes at the RHF level where a charge-density wave is created via the oscillation between the situations where the two electrons are one side or the other. \cite{StuberPaldus,Fukutome_1981}
|
||||
This type of symmetry breaking is also called a spin-density wave in the physics community as the system
|
||||
``oscillates'' between the two symmetry-broken configurations. \cite{GiulianiBook}
|
||||
\hugh{Symmetry breaking can also occur in RHF theory when a charge-density wave is formed from an oscillation
|
||||
between the two closed-shell configurations with both electrons localised on one site or the other.\cite{StuberPaldus,Fukutome_1981}}
|
||||
|
||||
%============================================================%
|
||||
\subsection{Self-consistency as a perturbation} %OR {Complex adiabatic connection}
|
||||
\subsection{Self-Consistency as a Perturbation} %OR {Complex adiabatic connection}
|
||||
%============================================================%
|
||||
|
||||
% INTRODUCE PARAMETRISED FOCK HAMILTONIAN
|
||||
The inherent non-linearity in the Fock eigenvalue problem \eqref{eq:FockOp} arises from self-consistency
|
||||
in the HF approximation, and is usually solved through an iterative approach.\cite{SzaboBook}
|
||||
The inherent non-linearity in the Fock eigenvalue problem arises from self-consistency
|
||||
in the HF approximation, and is usually solved through an iterative approach.\cite{Roothaan1951,Hall1951}
|
||||
Alternatively, the non-linear terms arising from the Coulomb and exchange operators can
|
||||
be considered as a perturbation from the core Hamiltonian \eqref{eq:Hcore} by introducing the
|
||||
transformation $U \rightarrow \lambda\, U$, giving the parametrised Fock operator
|
||||
\begin{equation}
|
||||
\Hat{f}_{\lambda}(\vb{x}) = \Hat{h}(\vb{x}) + \lambda\, \Hat{v}_\text{HF}(\vb{x}).
|
||||
\Hat{f}(\vb{x} \hugh{; \lambda}) = \Hat{h}(\vb{x}) + \lambda\, \Hat{v}_\text{HF}(\vb{x}).
|
||||
\end{equation}
|
||||
The orbitals in the reference problem $\lambda=0$ correspond to the symmetry-pure eigenfunctions of the one-electron core
|
||||
Hamiltonian, while self-consistent solutions at $\lambda = 1$ represent the orbitals of the true HF solution.
|
||||
|
|
|
|||
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Reference in New Issue