diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 12416bd..ff5ecf6 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -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}