From d1cf2785c195f89580c50966a3b94093eedcf4c9 Mon Sep 17 00:00:00 2001 From: Hugh Burton Date: Mon, 23 Nov 2020 13:29:42 +0000 Subject: [PATCH] Added some references, reorganised some of Section II --- Manuscript/EPAWTFT.bbl | 84 +++++++++--------- Manuscript/EPAWTFT.tex | 163 +++++++++++++++++++---------------- References/Hall_1951.pdf | Bin 0 -> 2399897 bytes References/Roothaan_1951.pdf | Bin 0 -> 1645025 bytes References/Slater_1951.pdf | Bin 0 -> 469121 bytes 5 files changed, 133 insertions(+), 114 deletions(-) create mode 100644 References/Hall_1951.pdf create mode 100644 References/Roothaan_1951.pdf create mode 100644 References/Slater_1951.pdf diff --git a/Manuscript/EPAWTFT.bbl b/Manuscript/EPAWTFT.bbl index 7e5ab74..18ecaee 100644 --- a/Manuscript/EPAWTFT.bbl +++ b/Manuscript/EPAWTFT.bbl @@ -472,34 +472,6 @@ {Ernzerhof}},\ }\href {\doibase 10.1063/1.2348880} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {125}},\ \bibinfo {pages} {124104} (\bibinfo {year} {2006})}\BibitemShut {NoStop}% -\bibitem [{\citenamefont {Carrascal}\ \emph {et~al.}(2015)\citenamefont - {Carrascal}, \citenamefont {Ferrer}, \citenamefont {Smith},\ and\ - \citenamefont {Burke}}]{Carrascal_2015}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont - {Carrascal}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Ferrer}}, - \bibinfo {author} {\bibfnamefont {J.~C.}\ \bibnamefont {Smith}}, \ and\ - \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Burke}},\ }\href - {\doibase 10.1088/0953-8984/27/39/393001} {\bibfield {journal} {\bibinfo - {journal} {J. Phys. Condens. Matter}\ }\textbf {\bibinfo {volume} {27}},\ - \bibinfo {pages} {393001} (\bibinfo {year} {2015})}\BibitemShut {NoStop}% -\bibitem [{\citenamefont {Carrascal}\ \emph {et~al.}(2018)\citenamefont - {Carrascal}, \citenamefont {Ferrer}, \citenamefont {Maitra},\ and\ - \citenamefont {Burke}}]{Carrascal_2018}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont - {Carrascal}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Ferrer}}, - \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Maitra}}, \ and\ \bibinfo - {author} {\bibfnamefont {K.}~\bibnamefont {Burke}},\ }\href {\doibase - 10.1140/epjb/e2018-90114-9} {\bibfield {journal} {\bibinfo {journal} {Eur. - Phys. J. B}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo {pages} {142} - (\bibinfo {year} {2018})}\BibitemShut {NoStop}% -\bibitem [{\citenamefont {Wigner}(1934)}]{Wigner_1934}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont - {Wigner}},\ }\href {\doibase 10.1103/PhysRev.46.1002} {\bibfield {journal} - {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {46}},\ - \bibinfo {pages} {1002} (\bibinfo {year} {1934})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Taut}(1993)}]{Taut_1993}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont @@ -530,6 +502,34 @@ {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {108}},\ \bibinfo {pages} {083002} (\bibinfo {year} {2012})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Carrascal}\ \emph {et~al.}(2015)\citenamefont + {Carrascal}, \citenamefont {Ferrer}, \citenamefont {Smith},\ and\ + \citenamefont {Burke}}]{Carrascal_2015}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont + {Carrascal}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Ferrer}}, + \bibinfo {author} {\bibfnamefont {J.~C.}\ \bibnamefont {Smith}}, \ and\ + \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Burke}},\ }\href + {\doibase 10.1088/0953-8984/27/39/393001} {\bibfield {journal} {\bibinfo + {journal} {J. Phys. Condens. Matter}\ }\textbf {\bibinfo {volume} {27}},\ + \bibinfo {pages} {393001} (\bibinfo {year} {2015})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Carrascal}\ \emph {et~al.}(2018)\citenamefont + {Carrascal}, \citenamefont {Ferrer}, \citenamefont {Maitra},\ and\ + \citenamefont {Burke}}]{Carrascal_2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont + {Carrascal}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Ferrer}}, + \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Maitra}}, \ and\ \bibinfo + {author} {\bibfnamefont {K.}~\bibnamefont {Burke}},\ }\href {\doibase + 10.1140/epjb/e2018-90114-9} {\bibfield {journal} {\bibinfo {journal} {Eur. + Phys. J. B}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo {pages} {142} + (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Wigner}(1934)}]{Wigner_1934}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont + {Wigner}},\ }\href {\doibase 10.1103/PhysRev.46.1002} {\bibfield {journal} + {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {46}},\ + \bibinfo {pages} {1002} (\bibinfo {year} {1934})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Goodson}(2011)}]{Goodson_2011}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~Z.}\ \bibnamefont @@ -573,6 +573,20 @@ {journal} {\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {7}},\ \bibinfo {pages} {2667} (\bibinfo {year} {2011})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Stuber}\ and\ \citenamefont + {Paldus}(2003)}]{StuberPaldus}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont + {Stuber}}\ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont + {Paldus}},\ }\enquote {\bibinfo {title} {{Symmetry Breaking in the + Independent Particle Model}},}\ in\ \href@noop {} {\emph {\bibinfo + {booktitle} {Fundamental World of Quantum Chemistry: A Tribute to the Memory + of Per-Olov L\"{o}wdin}}},\ Vol.~\bibinfo {volume} {1},\ \bibinfo {editor} + {edited by\ \bibinfo {editor} {\bibfnamefont {E.~J.}\ \bibnamefont + {Br\"{a}ndas}}\ and\ \bibinfo {editor} {\bibfnamefont {E.~S.}\ \bibnamefont + {Kryachko}}}\ (\bibinfo {publisher} {Kluwer Academic},\ \bibinfo {address} + {Dordrecht},\ \bibinfo {year} {2003})\ p.~\bibinfo {pages} {67}\BibitemShut + {NoStop}% \bibitem [{\citenamefont {Coulson}\ and\ \citenamefont {Fischer}(1949)}]{Coulson_1949}% \BibitemOpen @@ -591,20 +605,6 @@ {title} {Quantum {Theory} of the {Electron} {Liquid}}}}\ (\bibinfo {publisher} {Cambridge University Press},\ \bibinfo {year} {2005})\BibitemShut {NoStop}% -\bibitem [{\citenamefont {Stuber}\ and\ \citenamefont - {Paldus}(2003)}]{StuberPaldus}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont - {Stuber}}\ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont - {Paldus}},\ }\enquote {\bibinfo {title} {{Symmetry Breaking in the - Independent Particle Model}},}\ in\ \href@noop {} {\emph {\bibinfo - {booktitle} {Fundamental World of Quantum Chemistry: A Tribute to the Memory - of Per-Olov L\"{o}wdin}}},\ Vol.~\bibinfo {volume} {1},\ \bibinfo {editor} - {edited by\ \bibinfo {editor} {\bibfnamefont {E.~J.}\ \bibnamefont - {Br\"{a}ndas}}\ and\ \bibinfo {editor} {\bibfnamefont {E.~S.}\ \bibnamefont - {Kryachko}}}\ (\bibinfo {publisher} {Kluwer Academic},\ \bibinfo {address} - {Dordrecht},\ \bibinfo {year} {2003})\ p.~\bibinfo {pages} {67}\BibitemShut - {NoStop}% \bibitem [{\citenamefont {Fukutome}()}]{Fukutome_1981}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index d74f7ff..38f59e7 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -148,7 +148,7 @@ In particular, we highlight the seminal work of several research groups on the c \tableofcontents %%%%%%%%%%%%%%%%%%%%%%% -\section{Background} +\section{Introduction} \label{sec:intro} %%%%%%%%%%%%%%%%%%%%%%% @@ -186,12 +186,48 @@ Through a complex-scaling of the electronic or atomic coordinates,\cite{Moiseyev We refer the interested reader to the excellent book of Moiseyev for a general overview. \cite{MoiseyevBook}} %%%%%%%%%%%%%%%%%%%%%%% -\section{Exceptional points in electronic structure} +\section{Exceptional Points in Electronic Structure} \label{sec:EPs} %%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Time-Independent Schr\"odinger Equation} +\label{sec:TDSE} +%%%%%%%%%%%%%%%%%%%%%%% +Within the Born-Oppenheimer approximation, the exact molecular Hamiltonian with $\Ne$ electrons and +$\Nn$ (clamped) nuclei is defined \hugh{for a given nuclear framework} as +\begin{equation}\label{eq:ExactHamiltonian} + \hugh{\hH(\vb{R})} = + - \frac{1}{2} \sum_{i}^{\Ne} \grad_i^2 + - \sum_{i}^{\Ne} \sum_{A}^{\Nn} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}} + + \sum_{i2t$, 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. diff --git a/References/Hall_1951.pdf b/References/Hall_1951.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b31ab68331f606648f5b7f9f971169ac560fca5e GIT binary patch literal 2399897 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