PTEROSOR/EPAWTFT
10
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

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

Browse Source
This commit is contained in:
Hugh Burton 2020-11-19 15:01:27 +00:00
commit 222fbd15d8
2 changed files with 35 additions and 62 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

72
Manuscript/EPAWTFT.bbl
View File

@ -6,7 +6,7 @@
%Control: page (0) single
%Control: year (1) truncated
%Control: production of eprint (0) enabled
\begin{thebibliography}{94}%
\begin{thebibliography}{91}%
\makeatletter
\providecommand \@ifxundefined [1]{%
\@ifx{#1\undefined}
@ -529,6 +529,15 @@
{Thom}},\ }\href {\doibase 10.1021/ct5007696} {\bibfield {journal} {\bibinfo
{journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {10}},\
\bibinfo {pages} {4795} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Burton}\ and\ \citenamefont
{Thom}(2016)}]{Burton_2016}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~G.~A.}\
\bibnamefont {Burton}}\ and\ \bibinfo {author} {\bibfnamefont {A.~J.~W.}\
\bibnamefont {Thom}},\ }\href {\doibase 10.1021/acs.jctc.5b01005} {\bibfield
{journal} {\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo
{volume} {12}},\ \bibinfo {pages} {167} (\bibinfo {year} {2016})}\BibitemShut
{NoStop}%
\bibitem [{\citenamefont {Burton}\ \emph {et~al.}(2018)\citenamefont {Burton},
\citenamefont {Gross},\ and\ \citenamefont {Thom}}]{Burton_2018}%
\BibitemOpen
@ -539,15 +548,6 @@
{\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume}
{14}},\ \bibinfo {pages} {607} (\bibinfo {year} {2018})}\BibitemShut
{NoStop}%
\bibitem [{\citenamefont {Burton}\ and\ \citenamefont
{Thom}(2016)}]{Burton_2016}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~G.~A.}\
\bibnamefont {Burton}}\ and\ \bibinfo {author} {\bibfnamefont {A.~J.~W.}\
\bibnamefont {Thom}},\ }\href {\doibase 10.1021/acs.jctc.5b01005} {\bibfield
{journal} {\bibinfo {journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo
{volume} {12}},\ \bibinfo {pages} {167} (\bibinfo {year} {2016})}\BibitemShut
{NoStop}%
\bibitem [{\citenamefont {Langreth}\ and\ \citenamefont
{Perdew}(1979)}]{Langreth_1975}%
\BibitemOpen
@ -574,46 +574,6 @@
}\href {\doibase 10.1103/PhysRevA.69.052510} {\bibfield {journal} {\bibinfo
{journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {69}},\ \bibinfo
{pages} {052510} (\bibinfo {year} {2004})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Seidl}\ \emph {et~al.}(2018)\citenamefont {Seidl},
\citenamefont {Giarrusso}, \citenamefont {Vuckovic}, \citenamefont
{Fabiano},\ and\ \citenamefont {Gori-Giorgi}}]{Seidl_2018}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Seidl}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Giarrusso}},
\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Vuckovic}}, \bibinfo
{author} {\bibfnamefont {E.}~\bibnamefont {Fabiano}}, \ and\ \bibinfo
{author} {\bibfnamefont {P.}~\bibnamefont {Gori-Giorgi}},\ }\href {\doibase
10.1063/1.5078565} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
Phys.}\ }\textbf {\bibinfo {volume} {149}},\ \bibinfo {pages} {241101}
(\bibinfo {year} {2018})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Gilbert}\ \emph {et~al.}(2008)\citenamefont
{Gilbert}, \citenamefont {Besley},\ and\ \citenamefont
{Gill}}]{Gilbert_2008}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~T.~B.}\
\bibnamefont {Gilbert}}, \bibinfo {author} {\bibfnamefont {N.~A.}\
\bibnamefont {Besley}}, \ and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\
\bibnamefont {Gill}},\ }\href {\doibase 10.1021/jp801738f} {\bibfield
{journal} {\bibinfo {journal} {J. Phys. Chem. A}\ }\textbf {\bibinfo
{volume} {112}},\ \bibinfo {pages} {13164} (\bibinfo {year}
{2008})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Thom}\ and\ \citenamefont
{{Head-Gordon}}(2008)}]{Thom_2008}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~J.~W.}\
\bibnamefont {Thom}}\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{{Head-Gordon}}},\ }\href {\doibase 10.1103/PhysRevLett.101.193001}
{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf
{\bibinfo {volume} {101}},\ \bibinfo {pages} {193001} (\bibinfo {year}
{2008})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Shea}\ and\ \citenamefont
{Neuscamman}(2018)}]{Shea_2018}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.~R.}\
\bibnamefont {Shea}}\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Neuscamman}},\ }\href {\doibase 10.1063/1.5045056} {\bibfield {journal}
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {149}},\
\bibinfo {pages} {081101} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {M{\o}ller}\ and\ \citenamefont
{Plesset}(1934)}]{Moller_1934}%
\BibitemOpen
@ -702,6 +662,18 @@
{Epstein}},\ }\href {\doibase 10.1103/PhysRev.28.695} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {28}},\
\bibinfo {pages} {695} (\bibinfo {year} {1926})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Seidl}\ \emph {et~al.}(2018)\citenamefont {Seidl},
\citenamefont {Giarrusso}, \citenamefont {Vuckovic}, \citenamefont
{Fabiano},\ and\ \citenamefont {Gori-Giorgi}}]{Seidl_2018}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Seidl}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Giarrusso}},
\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Vuckovic}}, \bibinfo
{author} {\bibfnamefont {E.}~\bibnamefont {Fabiano}}, \ and\ \bibinfo
{author} {\bibfnamefont {P.}~\bibnamefont {Gori-Giorgi}},\ }\href {\doibase
10.1063/1.5078565} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
Phys.}\ }\textbf {\bibinfo {volume} {149}},\ \bibinfo {pages} {241101}
(\bibinfo {year} {2018})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Cremer}\ and\ \citenamefont
{He}(1996)}]{Cremer_1996}%
\BibitemOpen

25
Manuscript/EPAWTFT.tex
View File

@ -468,8 +468,8 @@ the total spin $\hat{\mathcal{S}}^2$ operator, leading to ``spin-contamination''
\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).}}
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.}
\end{figure}
%%%%%%%%%%%%%%%%%
@ -515,12 +515,12 @@ correctly modelling the physics of the system with the two electrons on opposing
\subcaption{\label{subfig:UHF_cplx_energy}}
\end{subfigure}
\caption{%
\hugh{(\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$.
(\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 $\pi/2$.
(\subref{subfig:UHF_cplx_energy}) The corresponding HF energy surfaces show a non-analytic
point at the \textit{quasi}-EP.}
point at the \textit{quasi}-EP.
\label{fig:HF_cplx}}
\end{figure*}
%%%%%%%%%%%%%%%%%
@ -550,7 +550,7 @@ of the HF energy rather than a minimum.
%============================================================%
% INTRODUCE PARAMETRISED FOCK HAMILTONIAN
\hugh{The inherent non-linearity in the Fock eigenvalue problem arises from self-consistency
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{SzaboBook}
Alternatively, the non-linear terms arising from the Coulomb and exchange operators can
be considered as a perturbation from the core Hamiltonian by introducing the
@ -559,10 +559,10 @@ transformation $U \rightarrow \lambda\, U$, giving the parametrised Fock operato
\Hat{f}_{\lambda}(\vb{x}) = \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.}
Hamiltonian, while self-consistent solutions at $\lambda = 1$ represent the orbitals of the true HF solution.
% INTRODUCE COMPLEX ANALYTIC-CONTINUATION
\hugh{For real $\lambda$, the self-consistent HF energies at given (real) $U$ and $t$ values
For real $\lambda$, the self-consistent HF energies at given (real) $U$ and $t$ values
in the Hubbard dimer directly mirror the energies shown in Fig.~\ref{fig:HF_real},
with coalesence points at
\begin{equation}
@ -583,10 +583,9 @@ For example, in the Hubbard dimer with $U > 2t$, one finds $\lambda_{\text{c}} <
do not provide a good representation of the HF ground state.
In contrast, $U < 2t$ yields $\lambda_{\text{c}} > 1$ and corresponds to
the regime where the HF ground state is correctly represented by symmetry-pure orbitals.
}
% COMPLEX ADIABATIC CONNECTION
\hugh{We have recently shown that the complex scaled Fock operator Eq.~\eqref{eq:scaled_fock}
We have recently shown that the complex scaled Fock operator Eq.~\eqref{eq:scaled_fock}
also allows states of different symmetries to be interconverted by following a well-defined
contour in the complex $\lambda$-plane.\cite{Burton_2019}
In particular, by slowly varying $\lambda$ in a similar (yet different) manner
@ -594,12 +593,12 @@ to an adiabatic connection in density-functional theory,\cite{Langreth_1975,Gunn
a ground-state wave function can be ``morphed'' into an excited-state wave function
via a stationary path of HF solutions.
This novel approach to identifying excited-state wave functions demonstrates the fundamental
role of \textit{quasi}-EPs in determining the behaviour of the HF approximation.}
role of \textit{quasi}-EPs in determining the behaviour of the HF approximation.
%\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}
%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}
%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.
%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.
%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.
%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.}
@ -659,6 +658,7 @@ iii) the strong coupling partitioning where the two operators are inverted compa
Let us illustrate the behaviour of the RMP and UMP series on the Hubbard dimer (see Fig.~\ref{fig:RMP}).
Within the RMP partition technique, we have
\begin{widetext}
\begin{equation}
\label{eq:H_RMP}
\bH_\text{RMP} =
@ -669,12 +669,13 @@ Within the RMP partition technique, we have
\lambda U/2 & 0 & 0 & 2t + U - \lambda U/2 \\
\end{pmatrix},
\end{equation}
\end{widetext}
which yields the ground-state energy
\begin{equation}
\label{eq:E0MP}
E_{-}(\lambda) = U - \frac{\lambda U}{2} - \frac{1}{2} \sqrt{(4t)^2 + \lambda ^2 U^2}.
\end{equation}
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}).
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}).
Equation \eqref{eq:E0MP} can be Taylor expanded in terms of $\lambda$ to obtain the $n$th-order MP correction
\begin{equation}
E_\text{MP}^{(k)} = U \delta_{0,k} - \frac{1}{2} \frac{U^k}{(4t)^{k-1}} \mqty( 1/2 \\ k/2),