From 1a9cd7d921775624711b5cdb91bd8fdafb7de7d0 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Fri, 20 Nov 2020 13:38:07 +0100 Subject: [PATCH] moving stuff around --- Manuscript/EPAWTFT.bbl | 60 ++++++------- Manuscript/EPAWTFT.tex | 191 ++++++++++++++++++++++------------------- 2 files changed, 132 insertions(+), 119 deletions(-) diff --git a/Manuscript/EPAWTFT.bbl b/Manuscript/EPAWTFT.bbl index dc130b2..ac54b4a 100644 --- a/Manuscript/EPAWTFT.bbl +++ b/Manuscript/EPAWTFT.bbl @@ -453,6 +453,25 @@ {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {150}},\ \bibinfo {pages} {031101} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Moiseyev}(1998)}]{Moiseyev_1998}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont + {Moiseyev}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rep.}\ }\textbf {\bibinfo {volume} {302}},\ \bibinfo {pages} {211} + (\bibinfo {year} {1998})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Riss}\ and\ \citenamefont {Meyer}(1993)}]{Riss_1993}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {U.~V.}\ \bibnamefont + {Riss}}\ and\ \bibinfo {author} {\bibfnamefont {H.-D.}\ \bibnamefont + {Meyer}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. + Phys. B}\ }\textbf {\bibinfo {volume} {26}},\ \bibinfo {pages} {4503} + (\bibinfo {year} {1993})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Ernzerhof}(2006)}]{Ernzerhof_2006}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont + {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}% @@ -481,25 +500,6 @@ {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 {Moiseyev}(1998)}]{Moiseyev_1998}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont - {Moiseyev}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} - {Phys. Rep.}\ }\textbf {\bibinfo {volume} {302}},\ \bibinfo {pages} {211} - (\bibinfo {year} {1998})}\BibitemShut {NoStop}% -\bibitem [{\citenamefont {Riss}\ and\ \citenamefont {Meyer}(1993)}]{Riss_1993}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {U.~V.}\ \bibnamefont - {Riss}}\ and\ \bibinfo {author} {\bibfnamefont {H.-D.}\ \bibnamefont - {Meyer}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. - Phys. B}\ }\textbf {\bibinfo {volume} {26}},\ \bibinfo {pages} {4503} - (\bibinfo {year} {1993})}\BibitemShut {NoStop}% -\bibitem [{\citenamefont {Ernzerhof}(2006)}]{Ernzerhof_2006}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont - {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 {Taut}(1993)}]{Taut_1993}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont @@ -754,17 +754,6 @@ }\href {\doibase 10.1021/jp952815d} {\bibfield {journal} {\bibinfo {journal} {J. Phys. Chem.}\ }\textbf {\bibinfo {volume} {100}},\ \bibinfo {pages} {6173} (\bibinfo {year} {1996})}\BibitemShut {NoStop}% -\bibitem [{\citenamefont {Surj{\'a}n}\ \emph {et~al.}(2018)\citenamefont - {Surj{\'a}n}, \citenamefont {Mih{\'a}lka},\ and\ \citenamefont - {Szabados}}]{Surjan_2018}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~R.}\ \bibnamefont - {Surj{\'a}n}}, \bibinfo {author} {\bibfnamefont {Z.~{\'E}.}\ \bibnamefont - {Mih{\'a}lka}}, \ and\ \bibinfo {author} {\bibfnamefont {{\'A}.}~\bibnamefont - {Szabados}},\ }\href {\doibase 10.1007/s00214-018-2372-3} {\bibfield - {journal} {\bibinfo {journal} {Theor. Chem. Acc.}\ }\textbf {\bibinfo - {volume} {137}},\ \bibinfo {pages} {149} (\bibinfo {year} - {2018})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Christiansen}\ \emph {et~al.}(1996)\citenamefont {Christiansen}, \citenamefont {Olsen}, \citenamefont {J{\o}rgensen}, \citenamefont {Koch},\ and\ \citenamefont {Malmqvist}}]{Christiansen_1996}% @@ -1053,4 +1042,15 @@ {journal} {\bibinfo {journal} {J. Phys. A: Math. Theor.}\ }\textbf {\bibinfo {volume} {40}},\ \bibinfo {pages} {581} (\bibinfo {year} {2007})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Surj{\'a}n}\ \emph {et~al.}(2018)\citenamefont + {Surj{\'a}n}, \citenamefont {Mih{\'a}lka},\ and\ \citenamefont + {Szabados}}]{Surjan_2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~R.}\ \bibnamefont + {Surj{\'a}n}}, \bibinfo {author} {\bibfnamefont {Z.~{\'E}.}\ \bibnamefont + {Mih{\'a}lka}}, \ and\ \bibinfo {author} {\bibfnamefont {{\'A}.}~\bibnamefont + {Szabados}},\ }\href {\doibase 10.1007/s00214-018-2372-3} {\bibfield + {journal} {\bibinfo {journal} {Theor. Chem. Acc.}\ }\textbf {\bibinfo + {volume} {137}},\ \bibinfo {pages} {149} (\bibinfo {year} + {2018})}\BibitemShut {NoStop}% \end{thebibliography}% diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 81ee796..31b4b17 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -146,15 +146,13 @@ In particular, we highlight the seminal work of several research groups on the c \maketitle +\tableofcontents + %%%%%%%%%%%%%%%%%%%%%%% -\section{Introduction} +\section{Background} \label{sec:intro} %%%%%%%%%%%%%%%%%%%%%%% -%======================% -\subsection{Background} -%=======================% - Due to the ubiquitous influence of processes involving electronic states in physics, chemistry, and biology, their faithful description from first principles has been one of the grand challenges faced by theoretical chemists since the dawn of computational chemistry. Accurately predicting ground- and excited-state energies (hence excitation energies) is particularly valuable in this context, and it has concentrated most of the efforts within the community. An armada of theoretical and computational methods have been developed to this end, each of them being plagued by its own flaws. @@ -184,8 +182,17 @@ In contrast, encircling Hermitian degeneracies at conical intersections only int More dramatically, whilst eigenvectors remain orthogonal at conical intersections, at non-Hermitian EPs the eigenvectors themselves become equivalent, resulting in a \textit{self-orthogonal} state. \cite{MoiseyevBook} More importantly here, although EPs usually lie off the real axis, these singular points are intimately related to the convergence properties of perturbative methods and avoided crossing on the real axis are indicative of singularities in the complex plane. \cite{BenderBook,Olsen_1996,Olsen_2000,Olsen_2019,Mihalka_2017a,Mihalka_2017b,Mihalka_2019} +\titou{The use of non-Hermitian Hamiltonians in quantum chemistry has a long history; these Hamiltonians have been used extensively as a method for describing metastable resonance phenomena. \cite{MoiseyevBook} +Through a complex-scaling of the electronic or atomic coordinates,\cite{Moiseyev_1998} or by introducing a complex absorbing potential,\cite{Riss_1993,Ernzerhof_2006,Benda_2018} outgoing resonance states are transformed into square-integrable wave functions that allow the energy and lifetime of the resonance to be computed. +We refer the interested reader to the excellent book of Moiseyev for a general overview. \cite{MoiseyevBook}} + +%%%%%%%%%%%%%%%%%%%%%%% +\section{Exceptional points in electronic structure} +\label{sec:EPs} +%%%%%%%%%%%%%%%%%%%%%%% + %===================================% -\subsection{Illustrative Example} +\subsection{Exceptional points in the Hubbard dimer} \label{sec:example} %===================================% @@ -275,14 +282,6 @@ such that $E_{\pm}(2\pi) = E_{\mp}(0)$ and $E_{\pm}(4\pi) = E_{\pm}(0)$. 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. 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} -\titou{The use of non-Hermitian Hamiltonians in quantum chemistry has a long history; these Hamiltonians have been used extensively as a method for describing metastable resonance phenomena. \cite{MoiseyevBook} -Through a complex-scaling of the electronic or atomic coordinates,\cite{Moiseyev_1998} or by introducing a complex absorbing potential,\cite{Riss_1993,Ernzerhof_2006,Benda_2018} outgoing resonance states are transformed into square-integrable wave functions that allow the energy and lifetime of the resonance to be computed. -We refer the interested reader to the excellent book of Moiseyev for a general overview. \cite{MoiseyevBook}} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Perturbation theory} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - %============================================================% \subsection{Rayleigh-Schr\"odinger perturbation theory} %============================================================% @@ -397,10 +396,9 @@ These degeneracies can be conical intersections between two states with differen 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} -%============================================================% +%===========================================% +\subsection{Hartree-Fock theory} +%===========================================% % SUMMARY OF HF In the Hartree-Fock (HF) approximation, the many-electron wave function is approximated as a single Slater determinant $\Psi^{\text{HF}}(\vb{x}_1,\ldots,\vb{x}_N)$, where $\vb{x} = (\sigma,\vb{r})$ is a composite vector gathering spin and spatial coordinates. @@ -460,14 +458,9 @@ 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 ``spin-contamination'' in the wave function. -% -%The spatial part of the RHF wave function is then -%\begin{equation}\label{eq:RHF_WF} -% \Psi_{\text{RHF}}(\theta_1,\theta_2) = Y_0(\theta_1) Y_0(\theta_2), -%\end{equation} -%where $\theta_i$ is the polar angle of the $i$th electron and $Y_{\ell}(\theta)$ is a zonal spherical harmonic. -%Because $Y_0(\theta) = 1/\sqrt{4\pi}$, it is clear that the RHF wave function yields a uniform one-electron density. -% +%================================================================% +\subsection{The Hartree-Fock approximation in the Hubbard dimer} +%================================================================% %%% FIG 2 (?) %%% % HF energies as a function of U/t @@ -612,8 +605,14 @@ role of \textit{quasi}-EPs in determining the behaviour of the HF approximation. %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.} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{M{\o}ller-Plesset perturbation theory} +\label{sec:MP} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + %=====================================================% -\subsection{M{\o}ller-Plesset perturbation theory} +\subsection{Basics} %=====================================================% In electronic structure, the HF Hamiltonian \eqref{eq:HFHamiltonian} is often used as the zeroth-order Hamiltonian @@ -622,14 +621,11 @@ This approach can recover a large proportion of the electron correlation energy, and provides the foundation for numerous post-HF approximations. With the MP partitioning, the parametrised perturbation Hamiltonian becomes \begin{multline}\label{eq:MPHamiltonian} - \hH(\lambda) = - \sum_{i}^{N} \Bigg( - -\frac{\grad_i^2}{2} - - \sum_{A}^{M} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}} + \hH(\lambda) = + \sum_{i}^{N} \qty[ - \frac{\grad_i^2}{2} - \sum_{A}^{M} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}} ] \\ - + (1-\lambda) v^{\text{HF}}(\vb{x}_i) - + \lambda\sum_{i