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Antoine Marie 2020-07-28 18:11:18 +02:00
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RapportStage/Rapport.tex
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@ -553,7 +553,7 @@ Then the mono-electronic wave function are expand in the spatial basis set of th
\phi_\alpha(\theta_1)=\sum_{l=0}^{\infty}C_{\alpha,l}\frac{Y_{l0}(\Omega_1)}{R}
\end{equation}
It is possible to obtain the formula for the ground state UHF energy in this basis set (see Sec.~\ref{sec:UHF_NRJ} for the development):
It is possible to obtain the formula for the ground state UHF energy in this basis set \cite{Loos_2009}:
\begin{equation}
E_{\text{UHF}} = E_{\text{c},\alpha} + E_{\text{c},\beta} + E_{\text{p}}
@ -778,15 +778,11 @@ For a non-Hermitian Hamiltonian the exceptional points can lie on the real axis.
\section{Conclusion}
We have seen that the $\beta$ singularities modeling the ionization phenomenon described by Sergeev and Goodson are actually part of a more general class of singularities. Indeed, those singularities close to the real axis are connected to quantum phase transition. In this work we have shown that $\beta$ singularities are also involved in the spin symmetry breaking of the unrestricted Hartree-Fock wave function. It would be interesting to investigate the difference between the different type of symmetry breaking and how it affects the singularity structure. Moreover the singularity structure in the non-Hermitian case still need to be investigated. In the holomorphic domain some singularities lie on the real axis and it would also be interesting to look at the differences between the different symmetry breaking and their respective holomorphic domain. To conclude this work shows that our understanding of the singularity structure of the energy is still incomplete but we hope that it opens new perspectives for the understanding of the physics of exceptional points in electronic structure theory.
\newpage
\printbibliography
\newpage
\appendix
\section{The unrestricted energy}\label{sec:UHF_NRJ}
\end{document}