Merge branch 'master' of github.com:pfloos/EPAWTFT

Manuscript/EPAWTFT.bbl

@@ -6,7 +6,7 @@
 %Control: page (0) single
 %Control: year (1) truncated
 %Control: production of eprint (0) enabled
-\begin{thebibliography}{93}%
+\begin{thebibliography}{94}%
 \makeatletter
 \providecommand \@ifxundefined [1]{%
  \@ifx{#1\undefined}
@@ -504,6 +504,15 @@
   {Ostlund}},\ }\href@noop {} {\emph {\bibinfo {title} {Modern quantum
   chemistry: {Introduction} to advanced electronic structure}}}\ (\bibinfo
   {publisher} {McGraw-Hill},\ \bibinfo {year} {1989})\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Mayer}\ and\ \citenamefont
+  {L{\"o}wdin}(1993)}]{Mayer_1993}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {I.}~\bibnamefont
+  {Mayer}}\ and\ \bibinfo {author} {\bibfnamefont {P.-O.}\ \bibnamefont
+  {L{\"o}wdin}},\ }\href {\doibase
+  https://doi.org/10.1016/0009-2614(93)85341-K} {\bibfield  {journal} {\bibinfo
+   {journal} {Chemical Physics Letters}\ }\textbf {\bibinfo {volume} {202}},\
+  \bibinfo {pages} {1 } (\bibinfo {year} {1993})}\BibitemShut {NoStop}%
 \bibitem [{\citenamefont {Coulson}\ and\ \citenamefont
   {Fischer}()}]{Coulson_1949}%
   \BibitemOpen

Manuscript/EPAWTFT.bib

@@ -1,13 +1,30 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-11-18 21:23:03 +0100 
+%% Created for Pierre-Francois Loos at 2020-11-19 09:09:27 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Mayer_1993,
+	abstract = {A study is made of the general Hartree---Fock (GHF) method, in which the basic spin-orbitals may be mixtures of functions having α and β spins. The existence of the solutions to the GHF equations has been proven by Lieb and Simon, and the nature of the various types of solutions has been group theoretically classified by Fukutome. Some numerical applications using Gaussian bases are carried out for some simple systems: the beryllium and carbon atoms and the BH molecule. Some GHF solutions of the general Fukutome-type ``torsional spin density waves'' (TSDW) were found.},
+	author = {Istv{\'a}n Mayer and Per-Olov L{\"o}wdin},
+	date-added = {2020-11-19 09:09:18 +0100},
+	date-modified = {2020-11-19 09:09:26 +0100},
+	doi = {https://doi.org/10.1016/0009-2614(93)85341-K},
+	issn = {0009-2614},
+	journal = {Chemical Physics Letters},
+	number = {1},
+	pages = {1 - 6},
+	title = {Some comments on the general Hartree---Fock method},
+	url = {http://www.sciencedirect.com/science/article/pii/000926149385341K},
+	volume = {202},
+	year = {1993},
+	Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/000926149385341K},
+	Bdsk-Url-2 = {https://doi.org/10.1016/0009-2614(93)85341-K}}
+
 @article{Zhang_2004,
 	author = {Zhang, Fan and Burke, Kieron},
 	date-added = {2020-11-18 21:23:02 +0100},

Manuscript/EPAWTFT.out

@@ -6,13 +6,13 @@
 \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}{Complex adiabatic connection}{section*.7}% 9
-\BOOKMARK [2][-]{section*.12}{M\370ller-Plesset perturbation theory}{section*.7}% 10
-\BOOKMARK [1][-]{section*.15}{Historical overview}{section*.2}% 11
-\BOOKMARK [2][-]{section*.16}{Behavior of the M\370ller-Plesset series}{section*.15}% 12
-\BOOKMARK [2][-]{section*.17}{Insights from a two-state model}{section*.15}% 13
-\BOOKMARK [2][-]{section*.18}{The singularity structure}{section*.15}% 14
-\BOOKMARK [2][-]{section*.19}{The physics of quantum phase transitions}{section*.15}% 15
-\BOOKMARK [1][-]{section*.20}{Conclusion}{section*.2}% 16
-\BOOKMARK [1][-]{section*.21}{Acknowledgments}{section*.2}% 17
-\BOOKMARK [1][-]{section*.22}{References}{section*.2}% 18
+\BOOKMARK [2][-]{section*.11}{Complex adiabatic connection}{section*.7}% 9
+\BOOKMARK [2][-]{section*.13}{M\370ller-Plesset perturbation theory}{section*.7}% 10
+\BOOKMARK [1][-]{section*.16}{Historical overview}{section*.2}% 11
+\BOOKMARK [2][-]{section*.17}{Behavior of the M\370ller-Plesset series}{section*.16}% 12
+\BOOKMARK [2][-]{section*.18}{Insights from a two-state model}{section*.16}% 13
+\BOOKMARK [2][-]{section*.19}{The singularity structure}{section*.16}% 14
+\BOOKMARK [2][-]{section*.20}{The physics of quantum phase transitions}{section*.16}% 15
+\BOOKMARK [1][-]{section*.21}{Conclusion}{section*.2}% 16
+\BOOKMARK [1][-]{section*.22}{Acknowledgments}{section*.2}% 17
+\BOOKMARK [1][-]{section*.23}{References}{section*.2}% 18

Manuscript/EPAWTFT.tex

@@ -445,7 +445,7 @@ 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
-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 \cite{Mayer_1993}) the one-electron orbitals can be complex-valued
 and contain a mixture of spin-up and spin-down components.\cite{Mayer_1993}
 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.
@@ -463,6 +463,13 @@ the total spin $\hat{\mathcal{S}}^2$ operator, leading to so-called ``spin-conta
 %Because $Y_0(\theta) = 1/\sqrt{4\pi}$, it is clear that the RHF wave function yields a uniform one-electron density.
 %
 
+\begin{figure}
+\includegraphics[width=\linewidth]{HF_real.pdf}
+\caption{\label{fig:HF_real}
+\hugh{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).}}
+\end{figure}
+
 Returning to the Hubbard dimer, the UHF energy can be parametrised in terms of 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 ),
@@ -485,7 +492,7 @@ giving the symmetry-pure molecular orbitals
 	&
 	\mathcal{A}_\text{RHF}^{\sigma} & = \frac{\Lsi - \Rsi}{\sqrt{2}},
 \end{align}
-and the ground-state RHF energy
+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}
@@ -503,7 +510,7 @@ For $U \ge 2t$, the optimal orbital rotation angles for the UHF orbitals become
     \tb^\text{UHF} & = \arctan (+\frac{2t}{\sqrt{U^2 - 4t^2}}),
     \label{eq:tb_uhf}
 \end{align}
-with the corresponding UHF ground-state energy
+with the corresponding UHF ground-state energy (Fig.~\ref{fig:HF_real})
 \begin{equation}
 	E_\text{UHF} \equiv E_\text{HF}(\ta^\text{UHF}, \tb^\text{UHF}) = - \frac{2t^2}{U}.
 \end{equation}
@@ -515,7 +522,6 @@ of the HF energy rather than a minimum.
 %============================================================%
 \subsection{Complex adiabatic connection}
 %============================================================%
-
 Self-consistency in HF approximations leads to the inherently non-linear Fock eigenvalue
 problem that is normally solved using an iterative approach.
 Alternatively, the non-linear terms arising from the Coulomb and exchange can be considered
@@ -526,7 +532,20 @@ as a perturbation from the reference core Hamiltonian problem by introducing the
 The orbitals in the reference problem correspond to the symmetry-pure eigenfunctions of the one-electron core
 Hamiltonian, while self-consistent solutions at $\lambda = 1$ represent the orbitals of the exact HF solution.
 
-
+%%% FIG 1 %%%
+\begin{figure*}[t]
+	\begin{subfigure}{0.49\textwidth}
+    \includegraphics[height=0.65\textwidth,trim={0pt 0pt 0pt -30pt},clip]{HF_cplx_angle}
+	\subcaption{\label{subfig:UHF_cplx_angle}}
+    \end{subfigure}
+	\begin{subfigure}{0.49\textwidth}
+	\includegraphics[height=0.65\textwidth]{HF_cplx_energy}
+	\subcaption{\label{subfig:UHF_cplx_energy}}
+    \end{subfigure}
+	\caption{%
+    Analytic continuation of HF into the complex $\lambda$ plane.
+	\label{fig:HF_cplx}}
+\end{figure*}
 
 \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}
@@ -536,13 +555,6 @@ The findings reported in Ref.~\onlinecite{Burton_2019} represent the very first
 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}
 %=====================================================%
@@ -646,7 +658,7 @@ Interestingly, one can show that the convergent and divergent series start to di
     \end{subfigure}
 	\caption{
 	Convergence of the RMP series as a function of the perturbation order $n$ for the Hubbard dimer at $U/t = 3.5$ (where $r_c > 1$) and $4.5$ (where $r_c < 1$).
-	The Riemann surfaces associated with the exact energy of the RMP Hamiltonian \eqref{eq:H_RMP} are also represented for these two values of $U/t$. 
+	The Riemann surfaces associated with the exact energies of the RMP Hamiltonian \eqref{eq:H_RMP} are also represented for these two values of $U/t$ as functions of $\lambda$. 
 	\label{fig:RMP}}
 \end{figure*}
 
@@ -664,30 +676,27 @@ The UMP partitioning yield the following $\lambda$-dependent Hamiltonian:
 	\end{pmatrix},
 \end{equation}
 \end{widetext}
-A closed-form expression for the ground-state energy can be obtained but it is cumbersome, so we eschew reporting its expression.
-The radius of convergence of the UMP series can obtained numerically as a function of $U/t$ and is depicted in Fig.~\ref{eq:UMP_rc}.
+A closed-form expression for the ground-state energy can be obtained but it is cumbersome, so we eschew reporting it.
+The radius of convergence of the UMP series can obtained numerically as a function of $U/t$ and is depicted in \titou{Fig.~\ref{eq:UMP_rc}}.
+ 
+The convergence of the UMP as a function of the ratio $U/t$ is shown in Fig.~\ref{subfig:UMP_cvg} for two specific values: the first ($U = 3t$) is well within the RMP convergence region, while the second ($U = 7t$) falls outside.
 Note that in the case of UMP, there are now two pairs of EPs as the open-shell singlet now couples strongly with both the ground and doubly-excited states.
 This has the clear tendency to move away from the origin the EP dictating the convergence of the ground-state energy, while deteriorating the convergence properties of the excited-state energy.
- 
-The convergence of the UMP as a function of the ratio $U/t$ is shown in Fig.~\ref{fig:UMP} for two specific values: the first ($U = 3t$) is well within the RMP convergence region, while the second ($U = 7t$) falls outside.
-
-For $U = 3t$ (see Fig.~\ref{subfig:UMP_3}), the ground-state energy is remarkably flat since the UHF energy is a pretty good estimate. 
+For $U = 3t$ (see Fig.~\ref{subfig:UMP_3}), the ground-state energy is remarkably flat since the UHF energy is already a pretty good estimate of the exact energy thanks to the symmetry-breaking process.
 Most of the UMP expansion is actually correcting the spin-contamination in the wave function.
-
 For $U = 7t$ (see Fig.~\ref{subfig:UMP_7}), we are well towards the strong correlation regime, where we see that the UMP series is slowly convergent while RMP diverges.
 We see a single EP on the ground-state surface which falls just outside (maybe on?) the radius of convergence. 
-An EP this close to the radius of convergence gives an increasingly slow convergence of the UMP series, but it will converge eventually!
-
-On the other hand, there is an exceptional point on the excited energy surface that is well within the radius of convergence.
-We can therefore say that the use of a symmetry-broken UHF wave function can retain a convergent ground-state perturbation series
-at the expense of a divergent excited-state perturbation series. (Note: the orbitals are not optimised for excited-state here).
-In contrast, the RMP expansion was always convergent for the open-shell excited state (which was a single CSF) while
-the radius of convergence for the doubly-excited state was identical to the ground-state as this was the only exceptional point.
+An EP this close to the radius of convergence gives an increasingly slow convergence of the UMP series, but it will converge eventually as observed in Fig.~\ref{subfig:UMP_cvg}.
+%On the other hand, there is an EP on the excited energy surface that is well within the radius of convergence.
+%We can therefore say that the use of a symmetry-broken UHF wave function can retain a convergent ground-state perturbation series
+%at the expense of a divergent excited-state perturbation series. (Note: the orbitals are not optimised for excited-state here).
+%In contrast, the RMP expansion was always convergent for the open-shell excited state (which was a single CSF) while
+%the radius of convergence for the doubly-excited state was identical to the ground-state as this was the only exceptional point.
 
 %%% FIG 3 %%%
 \begin{figure*}
 	\begin{subfigure}{0.32\textwidth}
-	\includegraphics[height=0.75\textwidth]{fig3c}	
+	\includegraphics[height=0.75\textwidth]{fig3a}	
 		\subcaption{\label{subfig:UMP_3} $U/t = 3$}
     \end{subfigure}
     %
@@ -697,11 +706,11 @@ the radius of convergence for the doubly-excited state was identical to the grou
     \end{subfigure}
     %
     \begin{subfigure}{0.32\textwidth}
-	\includegraphics[height=0.75\textwidth]{fig3a}	
+	\includegraphics[height=0.75\textwidth]{fig3c}	
 		\subcaption{\label{subfig:UMP_7} $U/t = 7$}
     \end{subfigure}	\caption{
 	Convergence of the UMP series as a function of the perturbation order $n$ for the Hubbard dimer at $U/t = 3$ and $7$.
-	The Riemann surfaces of the FCI energies are also represented for these two values of $U/t$. 
+	The Riemann surfaces associated with the exact energies of the UMP Hamiltonian \eqref{eq:H_UMP} are also represented for these two values of $U/t$ as functions of $\lambda$.
 	\label{fig:UMP}}
 \end{figure*}