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more and more tweaks... now lunch

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Hugh Burton 2020-11-19 13:03:17 +00:00
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@ -519,10 +519,11 @@ correctly modelling the physics of the system with the two electrons on opposing
\end{subfigure}
\caption{%
\hugh{(\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 quasi-EP $\lambda_{\text{c}}$ (black dot).
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 quasi-exceptional point.}
point at the \textit{quasi}-EP.}
\label{fig:HF_cplx}}
\end{figure*}
%%%%%%%%%%%%%%%%%
@ -577,7 +578,7 @@ the holomorphic HF approach.\cite{Hiscock_2014,Burton_2016,Burton_2018}
Remarkably, the coalescence point in this analytic continuation emerges as a
\textit{quasi}-EP on the real $\lambda$ axis (Fig.~\ref{fig:HF_cplx}), where
the different HF solutions become equivalent but not self-orthogonal.\cite{Burton_2019}
By analogy with perturbation theory, the regime where this quasi-EP occurs
By analogy with perturbation theory, the regime where this \textit{quasi}-EP occurs
within $\lambda_{\text{c}} \le 1$ can be interpreted as an indication that
the symmetry-pure reference orbitals no longer provide a qualitatively
accurate representation for the true HF ground state at $\lambda = 1$.
@ -596,7 +597,7 @@ 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 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}